Constructing Math
Constructing Math
By German Moreno
2. Introduction to signed numbers
Integers are the set of whole numbers and their opposites. The number line can be used to represent the set of integers. Look carefully at the number line below and the definitions that follow.
[pic]
Definitions
• The number line goes on forever in both directions. This is indicated by the arrows.
• Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line.
• Whole numbers less than zero are called negative integers. These numbers are to the left of zero on the number line.
• The integer zero is neutral. It is neither positive nor negative.
• The sign of an integer is either positive (+) or negative (-), except zero, which has no sign.
Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above, +3 and -3 are labeled as opposites.
Example 1
The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the distance from the top of Mt. McKinley to the bottom of Death Valley?
Understanding the problem
[pic]
The distance from the top of Mt. McKinley to sea level is 20,320 feet and the distance from sea level to the bottom of Death Valley is 282 feet.
Devising a Plan
The total distance is the sum of 20,320 and 282,
Carrying out the Plan
20,320+ 282 = 20,602 feet.
Looking Back
The distance from the top of Mt. McKinley to the bottom of Death Valley is the same as the distance from +20,320 to -282 on the number line. We add the distance from +20,320 to 0, and the distance from 0 to -282, for a total of 20,602 feet.
|Elevation |Integer |
|20,320 feet above sea level |+20,320 |
|sea level |0 |
|282 feet below sea level |-282 |
The problem above uses the notion of opposites: Above sea level is the opposite of below sea level. Here are some more examples of opposites:
top, bottom increase, decrease forward, backward positive, negative
Practice 1
Write an integer to represent each situation then graph each number on the number line:
|10 degrees above zero |[p| |
| |ic| |
| |] | |
|a loss of 16 dollars |[p| |
| |ic| |
| |] | |
|a gain of 5 points |[p| |
| |ic| |
| |] | |
|8 steps backward |[p| |
| |ic| |
| |] | |
[pic]
Example 2
Name the opposite of each integer then graph each integer.
-5 -4 2 3 -10
[pic]
Practice 2
Name 4 real life situations in which integers can be used.
Comparing and Ordering Integers
|The set of integers is composed of the counting (natural ) numbers, their | |
|opposites, and zero. | |
|Begining with zero, numbers increase in value to the right (0, 1, 2, 3, …) and| |
|decrease in value to the left (…-3, -2, -1, 0) | |
|When comparing numbers the order in which they are placed on the number line | |
|will determine if it is greater than or less than another number. | |
|If a number is to the left of a number on the number line, it is less than the| |
|other number. If it is to the right then it is greater than that number. | |
Example 3: If the lowest score wins, order the following golf scores from best to worst: Tigre Madera –4, Jack Nickles +1, Nick Cost –2, Freddy Pairs –5, John Weekly +3
Practice 3
The Picksburg running back, nicknamed "the trolley," had five carries with the following results: +18, -3, +4, 0, -1. List these yardage figures from best to worst.
Using the < and > symbols
If there are two numbers we can compare them. One number is either greater than, less than or equal to the other number.
If the first number has a higher count than the second number, it is greater than the second number. The symbol ">" is used to mean greater than. In this example, we could say either "15 is greater than 9" or "15 > 9". The greater than symbol can be remembered because the larger open end is near the larger number and the smaller pointed end is near the smaller number.
If one number is larger than another, then the second number is smaller than the first. In this example, 9 is less than 15. We would have to count up from 9 to reach 15. We could either write "9 is less than 15" or "9 < 15". Once again the smaller end goes toward the smaller number and the larger end toward the larger number.
If both numbers are the same size we say they are equal to each other. We would not need to count up or down from one number to arrive at the second number. We could write "15 is equal to 15" or use the equal symbol "=" and write " 15 = 15".
Example 4
6 -6
Practice 4
-11 1
9 -10
Absolute Value
Absolute value is the distance a number is from zero on a number line. We use |n | to indicate absolute value.
|-n| = |n| =n; where n is an integer.
Example 5
|-7|
Practice 5
|-1000|
|-4|
|21|
Example 6
[pic]
Practice 6
[pic]
Example 7
Jack Nickles is –4 after 36 holes of golf. How many strokes away from even par is he? (Par for a course is a score of zero strokes above or below.)
1.3 Adding Integers
Adding Integers That Have The Same Sign
When adding integers of the same sign, we add their absolute values, and give the result the same sign.
Example 1:
2 + 5 =
(-7) + (-2) =
(-80) + (-34) =
Adding Integers That Have Different Signs
When adding integers of the opposite signs, we take their absolute values, subtract the smaller from the larger, and give the result the sign of the integer with the larger absolute value.
Example 2:
8 + (-3) =
Example 3:
8 + (-17) =
Example 4:
-22 + 11 =
Example 5:
53 + (-53) =
[pic]
Properties of Addition
Addition Property of 0
A + 0 = A
Commutative Property of Addition
A + B = B + A
Associative Property of Addition
(A + B) + C = A + (B + C)
Example 6
Application Problems
Example 7
Cynthia had $ in her checking account. She wrote a check for $83 and was charged $17 for overdrawing her account last month. What was her account balance?
Example 8
Which sum is farther from zero, the sum of 101 and 85, or the sum of -98 and -104?
Example 9
Ms. Wilburson's candy store is selling lots of Super Chompers (a kind of candy bar). The numbers of Super Chompers she sold per hour for the first 5 hours of the day are 100, 70, 77, 59 and 34. How many did she sell in those first five hours?
Example 10
If a dense plastic block is dropped into a tank of water, it experiences a change in velocity of -3 m/s. If the original velocity was 20 m/s, what was the velocity immediately after it hit the water?
1.4 Subtracting Integers
Subtracting Integers
Subtracting an integer is the same as adding its opposite.
Examples:
In the following examples, we convert the subtracted integer to its opposite, and add the two integers.
7 - 4 = 7 + (-4) = 3
12 - (-5) = 12 + (5) = 17
-8 - 7 = -8 + (-7) = -15
-22 - (-40) = -22 + (40) = 18
Note that the result of subtracting two integers could be positive or negative.
Practice
1. 5 – 7 =
2. 8 – 6 =
3. 4 – (-7)=
4. -5 – 8=
5. -8 – (-8)=
6. -3 – (-5)=
Application Problem
Example 2
In Buffalo, New York, the temperature was -14 ° F in the morning. If the temperature dropped 7° F, what is the temperature now?
Example 3
Roman Civilization began in 509 B.C. and ended in 476 A.D. How long did Roman Civilization last?
Practice 1
In the Sahara Desert one day it was 136° F. In the Gobi Desert a temperature of -50° F was recorded. What is the difference between these two temperatures?
Practice 2
The Punic Wars began in 264 B.C. and ended in 146 B.C. How long did the Punic Wars last?
Practice 3
Metal mercury at room termperature is a liquid. Its melting point is -39°C. The freezing point of alcohol is -144°C. How much warmer is the melting point of mercury than the freezing point of alcohol?
Practice 4
The water potential in one plant cell was calculated to be -3 bar. The water potential in another cell was found to be -11 bar. What was the difference in water potentials? Note: the bar is a pressure unit equal to about 14.5 PSI.
1.6 Multiplying Integers
To multiply a pair of integers if both numbers have the same sign, their product is the product of their absolute values (their product is positive). If the numbers have opposite signs, their product is the opposite of the product of their absolute values (their product is negative). If one or both of the integers is 0, the product is 0.
Example 1:
In the product below, both numbers are positive, so we just take their product.
4 × 3 = 12
In the product below, both numbers are negative, so we take the product of their absolute values.
(-4) × (-5) = |-4| × |-5| = 4 × 5 = 20
In the product of (-7) × 6, the first number is negative and the second is positive, so we take the product of their absolute values, which is |-7| × |6| = 7 × 6 = 42, and give this result a negative sign: -42, so (-7) × 6 = -42.
In the product of 12 × (-2), the first number is positive and the second is negative, so we take the product of their absolute values, which is |12| × |-2| = 12 × 2 = 24, and give this result a negative sign: -24, so 12 × (-2) = -24.
Practice 1
1. [pic]
2. [pic]
3. [pic]
4. [pic]
To multiply any number of integers:
1. Count the number of negative numbers in the product.
2. Take the product of their absolute values.
3. If the number of negative integers counted in step 1 is even, the product is just the product from step 2, if the number of negative integers is odd, the product is the opposite of the product in step 2 (give the product in step 2 a negative sign). If any of the integers in the product is 0, the product is 0.
Example 2
4 × (-2) × 3 × (-11) × (-5) = ?
Counting the number of negative integers in the product, we see that there are 3 negative integers: -2, -11, and -5. Next, we take the product of the absolute values of each number:
4 × |-2| × 3 × |-11| × |-5| = 1320.
Since there were an odd number of integers, the product is the opposite of 1320, which is -1320, so
4 × (-2) × 3 × (-11) × (-5) = -1320.
Practice 2
1. [pic]
2. [pic]
Application Problems
Example 3
Which product is closer to zero, the product of 85 and -102, or the product of -63 and -126?
Example 4
Stephanie is saving for a trip to her cousin’s house in another state. She figures she needs $216 to have a comprtable trip. To earn money she mows lawns. Each mowing earns her $16. She already has mowed seven lawns. How many more lawns must she mow to get at least $216?
Practice 3
Chantele has three children. Her older daughter had a throat culture taken at the clinic today. Her baby received three immunization shots and her son received two shots. THE co-pay amounts we e $8 for each shot, an $18 office charge for each child an a $12 charge for the throat culture. How much did Chantele pay?
Practice 4
There is a 3 degree drop in temperature for every thousand feet that an airplane clibs into the sky. If the temperature on the ground is 50 degrees, what will be the termperature when th plane reaches an altitude of 24,000 feet?
1.7 Dividing Integers
Dividing Integers
To divide a pair of integers if both integers have the same sign, divide the absolute value of the first integer by the absolute value of the second integer.
To divide a pair of integers if both integers have different signs, divide the absolute value of the first integer by the absolute value of the second integer, and give this result a negative sign.
Example 1
In the division below, both numbers are positive, so we just divide as usual.
4 ÷ 2 = 2.
In the division below, both numbers are negative, so we divide the absolute value of the first by the absolute value of the second.
(-24) ÷ (-3) = |-24| ÷ |-3| = 24 ÷ 3 = 8.
In the division (-100) ÷ 25, both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |-100| ÷ |25| = 100 ÷ 25 = 4, and give this result a negative sign: -4, so (-100) ÷ 25 = -4.
In the division 98 ÷ (-7), both number have different signs, so we divide the absolute value of the first number by the absolute value of the second, which is |98| ÷ |-7| = 98 ÷ 7 = 14, and give this result a negative sign: -14, so 98 ÷ (-7) = -14.
Practice 1
1. [pic]
2. [pic]
3. [pic]
4. [pic]
Application Problems
Example 2
Because of sea floor spreading, the Atlantic Ocean is getting wider at a rate of about one cm per year. At that rate of expansion, how much wider will the Atlantic be at the end of one-fifth of a century?
Example 3
If there were 1,500,220,586 hydrogen atoms in a collection of H2O molecules, how many oxygen atoms were there?
Example 4
The melting point of trifluoroacetophenone is -40[pic]C. The melting point of 4–nitroacetophenone is 81[pic]C. Another similar compound melts at -46[pic]C. What is the average melting point of the compounds?
Practice 2
Three cars came to a sudden stop on the highway. The acceleration of one was -28 m/s2. The acceleration of another was -32 m/s2. The acceleration of the other car was exactly halfway between the other two accelerations. What was the acceleration of the third car?
Practice 3
It was a close race for the pennant (division championship) in the baseball league this year. The Rattlers won. Each team is awarded a point for every game won, and the team with the most points is the winner of the pennant. The Rattlers won 66 games. The Tigers were -2 (two games behind), the lizards were -6, and the rollers were -12. What was the average number of games won by the teams that did not win the pennant?
Practice 4
Mr. Bloop likes to hit fly balls to his nephew on Saturday afternoons. His nephew catches about two out of every five fly balls hit to him. If Mr. Bloop hits 70 fly balls, how many will he probably catch?
5. Rounding and Estimating
Example 1
1. 96,547
Tens
2. 771,978
hundred thousands
3. -842
tens
4. -63,217
ten thousands
5. -6,024,047
hundreds
Practice
1. 705,072
thousands
2. 6,665
tens
3. -9,897
hundreds
4. -4,838,348
millions
5. -33,892
thousands
Application Problems
Example 2
Sebastian recorded his observations at the bird feeder for a Year. He counted: 226 cardinals, 532 house sparrows, 213 goldfinches, 119 blue jays, 86 mourning doves, 64 downy woodpeckers, and 416 grackles. Estimate the total number of birds he counted by rounding to the nearest tens place?
Practice
Amanda performs an endothermic reaction in a flask on the bench. At the beginning of the reaction the temperature in the flask is 129[pic]C. The temperature decreases by 213[pic]C. Estimate the new temperature?
1.8 Order of Operations
The Order of Operations is very important when simplifying expressions and equations. The Order of Operations is a standard that defines the order in which you should simplify different operations such as addition, subtraction, multiplication and division.
This standard is critical to simplifying and solving different algebra problems. Without it, two different people may interpret an equation or expression in different ways and come up with different answers. The Order of Operations is shown below.
Example 1
64 ÷ 8 × 5
(9 - 1) + 2 × 2
82 × 12 - 4
46 ÷ 2 + 1
84 ÷ 4 × 44
(9 - 1) - (35 ÷ 7)
Practice 1
5 x 8 + 6 ÷ 6 - 12 x 2
150 ÷ (6 + 3 x 8) – 5
5 - 2×2 = ?
(8 - 3)×4 = ?
(8 - 7)×6 - 10/5 + 4 = ?
7 - (11 - 8) + 14 = ?
(2 + 8)/(6 - 1) + 7×2 = ?
(1 + 2×3) - 7/(4 - 3) + 2 = ?
(12/(3×2) + 4)/(13 - (8 + 2)) = ?
2.1 Introduction to Variable Expressions
Variable
A variable is a letter that represents a number.
Don't let the fact that it is a letter throw you. Since it represents a number, you treat it just like you do a number when you do various mathematical operations involving variables.
x is a very common variable that is used in algebra, but you can use any letter (a, b, c, d, ....) to be a variable.
Algebraic Expressions
An algebraic expression is a number, variable or combination of the two connected by some mathematical operation like addition, subtraction, multiplication, division, exponents, and/or roots.
2x + y, a/5, and 10 - r are all examples of algebraic expressions.
Evaluating an Expression
You evaluate an expression by replacing the variable with the given number and performing the indicated operation.
Value of an Expression
When you are asked to find the value of an expression, that means you are looking for the result that you get when you evaluate the expression.
Example 1
Write a variable expression that represents the perimeter of the following rectangle and fill in the table below.
[pic]
|Side |Perimeter |
|8 | |
|10 | |
|12 | |
|14 | |
|16 | |
Example 2
[pic]
|r |C |
| | |
| | |
| | |
| | |
| | |
| | |
Example 3
Write a variable expression for a cell phone plan and fill in the table below?
|Minutes (m) |Cost |
| |(C) |
|500 | |
|1000 | |
|1500 | |
|2000 | |
|2500 | |
|3000 | |
Example 4
Write an expression for the volume of object below and fill in the table.
[pic]
|s |t |V |
| | | |
| | | |
| | | |
| | | |
| | | |
2.2 Combining Like Terms
A term is a constant, a variable or the product of a constant and variable(s)in an expression. In the equation 12+3x+2x2=5x-1, the terms on the left are 12, 3x and 2x2, while the terms on the right are 5x, and -1.
Combining Like Terms is a process used to simplify an expression or an equation using addition and subtraction of the coefficients of terms. Consider the expression below
5 + 7
By adding 5 and 7, you can easily find that the expression is equivalent to 12
What Does Combining Like Terms Do?
Algebraic expressions can be simplified like the example above by Combining Like Terms. Consider the algebraic expression below:
12x + 7 + 5x
As you will soon learn, 12x and 5x are like terms. Therefore the coefficients, 12 and 5, can be added. This is a simple example of Combining Like Terms. You get 17x + 7.
What are Like Terms?
The key to using and understanding the method of Combining Like Terms is to understand like terms and be able to identify when a pair of terms is a pair of like terms. Some examples of like terms are presented below.
Example 1
The following are like terms because each term consists of a single variable, x, and a numeric coefficient.
2x,
5x,
x,
0x,
-2x,
-x
Each of the following are like terms because they are all constants.
5,
-2,
27,
9043,
0.6
Each of the following are like terms because they are all y2 with a coefficient.
3y2,
y2,
-y2,
-6y2
For comparison, below are a few examples of unlike terms.
The following two terms both have a single variable with an exponent of 1, but the terms are not alike since different variables are used.
17x,
17z
Each y variable in the terms below has a different exponent, therefore these are unlike terms.
15y,
19y2,
31y5
Although both terms below have an x variable, only one term has the y variable, thus these are not like terms either.
19x,
14xy
Combining Like Terms
In an Expression
Consider the expression below:
5x2 + 7x + 2 - 2x2 + 7 + x2
We will demonstrate how to simplify this expression by combining like terms. First, we identify sets of like terms. Both 2 and 7 are like terms because they are both constants. The terms 5x2, -2x2, and x2 are like terms because they each consist of a constant times x squared.
Now the coefficients of each set of like terms are added. The coefficients of the first set are the constants themselves, 2 and 7. When added the result is 9. The coefficients of the second set of like terms are 5, -2, and 1. Therefore, when added the result is 4.
With the like terms combined, the expression becomes
9 + 7x + 4x2
The Combining Like Terms process is also used to make equations easier to solve.
Example
Combine like terms.
6x + 2x
6x − 2x
5x + x
Practice
Combine like terms.
5x − x
−4x + 5x
4x − 5x
Example
Combine like terms.
−5x − 3x
−x − x
−3x − 4 + 2x + 6
Practice
Combine like terms.
x − 2 − 4x − 5
4x + y − 2x + y
3x − y − 8x + 2y
The Distributive Property is an algebra property which is used to multiply a single term and two or more terms inside a set of parentheses. Take a look at the problem below.
2(3 + 6)
Because the binomial "3 + 6" is in a set of parentheses, when following the Order of Operations, you must first find the answer of 3 + 6, then multiply it by 2. This gives an answer of 18.
2(3 + 6)
2(9)
18
Example
Simplify.
6(2 + 4x)
The Distributive Property tells us that we can remove the parentheses if the term that the polynomial is being multiplied by is distributed to, or multiplied with each term inside the parentheses.
This definition is tough to understand without a good example, so observe the example below carefully.
6(2 + 4x)
now by applying the Distributive Propery
6 * 2 + 6 * 4x
The parentheses are removed and each term from inside is multiplied by the six.
Now we can simplify the multiplication of the individual terms:
12 + 24x
Practice
[pic]
Distributing a Negative Sign
Example
The next problem does not have a number outside the parentheses, only a negative sign.
-(3 + x2)
There are two easy ways to simplify this problem. The first and simplest way is to change each positive or negative sign of the terms that were inside the parentheses. Negative or minus signs become positive or plus signs. Similarly, positive or plus signs become negative or minus signs. Recall that in the case of 3, no positive or negative sign is shown, so a positive sign is assumed.
-3 - x2
We will now work through this problem again, but using a different method.
-(3 + x2)
Recall that any term that does not have a coefficient has an implied coefficient of 1. Because of the negative sign on the parentheses, we instead assume a coefficient of negative one. Thus, we can rewrite the problem as
-1(3 + x2)
Now the -1 can be distributed to each term inside the parentheses as in the first example in this lesson.
-1 * 3 + -1 * x2
-3 + -x2
Practice
[pic]
2.3 Solving Equations Using Addition
Equation
An equation is a mathematical sentence built from expressions using one or more equal signs (=).
Solution and Solution Set
Any and all value(s) of the variable(s) that satisfies an equation.
Example 1
Find a solution to the following equations by using trial and error.
[pic]
Example 2
Find solutions to the following equation by using trial and error.
[pic]
Practice 1
[pic]
Practice 2
[pic]
Addition Property of Equality
If a = b, then a + c = b + c.
Example 3
[pic]
[pic]
[pic]
Practice 3
[pic]
[pic]
[pic]
Application Problems
Example 1
A mixture contains 8 more gallons of water than of alcohol. If the mixture contains 47 gallons of water, how many gallons of alcohol does it contain?
Example 2
Sabrina and Hector took turns driving to their destination. Sabrina drove 81 more miles than Hector drove. If Sabrina drove 350 miles, how far did they drive in all?
Practice 1
The difference between the height of a column and the peak of the roof is 6 feet. If the peak of the roof is at 15 feet what is the height of the column.
Practice 2
A carpenter is charging $1100 for labor in order to build a shed for Juan’s 1965 Mustang. He will be charging Juan $3500 including materials and labor. Determine the cost of the materials per square foot if the shed is 250 square feet.
2.4 Solving Equations Using Multiplication and Division
Example 1
Solve the following equations using the trial and error method.
[pic]
[pic]
[pic]
Practice 1
[pic]
[pic]
[pic]
Division Property of Equality
If [pic], then [pic] as long as [pic].
Example 2
[pic]
[pic]
[pic]
Practice 2
[pic]
[pic]
[pic]
Explain
How do you solve equations that have a coefficient in front of the variable?
Application Problems
Example 3
John makes eight dollars an hour working with a moving company. He made 720 dollars last month. How many hours did John work last month?
Example 4
Nick has six children. He works 3 Jobs and makes 5 dollars an hour at each job. He made 240 dollars this month. How many hours does Nick work at each job if he works the same amount of hours at each job?
Example 5
The residents of a housing development use 360 feet of fencing to enclose a rectangular playground. The length of the playground is twice the width. What are the dimensions of the playground?
Practice 3
In a right triangle, one of the acute angles is 2 times as large as the other acute angle. Find the measure of the two acute angles.
Practice 4
Two sides of a triangle are the same length and the third side is 3 times the length of the two equal sides. The perimeter is 125 feet. What are the lengths of the s ides of th triangles?
Practice 5
Rita is very good at making three-pointers. In the last 6 basketball games Rita scored 234 points. Assuming she shot and madee the same number of three-pointers at each game and only shot three pointers, how many three pointers did she shoot and make per game?
5. Solving Equations Using Multiplication and Division
Two-Step Equations
When solving equations of this form, we must carefully simplify the equation using a special two-step order. The term that is either adding to or subtracting from the variable must be canceled first by doing the opposite, otherwise known as the inverse operation. For instance given the equation 3x - 5 = -26, the subtraction by five must be cancelled by doing the opposite. We must add five to both sides. Doing so we get, 3x - 5 + 5 = -26 + 5. This simplifies to 3x = -21.
The next step involves cancelling the number next to the variable, and that number is called the coefficient. If the variable is being multiplied by a number, then we divide both sides of the equation by that number. If the variable is being divided by a number, then we multiply both sides of the equation by that number. Performing this step allows us to cancel the coefficient. In order to finish up our example, let's perform the last step to 3x = -21. We must divide both sides by 3 to get, 3x/3 = -21/3, which reduces to x = -7.
A nice fact about solving equations is that the solutions can be checked. The numeric solution is substituted into the original problem. Then the order of operations is used to simplify the remaining solution. Once simplified, both sides of the remaining equation should be equal to each other if the original answer is correct.
Using our example above, we can see that our solution, x = -7 is correct because it checks. Substituting it into the original equation yields, 3(-7) - 5 = -26. This simplifies to -21 - 5 = -26. Finally we see that the left side further simplifies to -26 which is the value of the right side. Since both sides are now equal to each other, x = -7 is without doubt the correct solution. In fact, any other number that is substituted into the equation will not work.
Try solving -4x + 1 = 21. First we must cancel the addition by 1 by subtracting 1 from both sides. Doing so, we get -4x = 20. Then we must divide both sides by -4 to cancel the multiplication by -4. This allows us to get the solution, x = -5. Upon checking we can see that -4 times -5 plus 1 is in fact 21. Therefore, x = -5 is the correct solution.
Three-Step (Type 1) Equations ax + b + cx = d
This three-step problem must be simplified in order for it to be solved. In fact, we can actually simplify equations of this type into becoming two-step equations like the type of equations mentioned above. Doing this will make solving these equations relatively painless.
To solve these quations we must combine like terms. The terms that have the exact same variables are called like and can be combined. All we need to do is combine the numbers in front of the variables. For instance, the equation -5x + 4 + 2x = 16 has like terms that need to be combined. The x-terms can be combined to make -3x + 4 = 16.
Once this combination occurs, the problem can be solved using the procedure outlined under solving two-step equations above. Our example lands up having a solution, x = -4.
The final solution can be checked by substututing it within the original equation wherever the variable is placed. Each variable must be replaced with the solution and the order of operations must be used to simplify the expression, similar to the steps used for solving two-step equations. Substituting the solution, x = -4 into the original equation yields, -5(-4) + 4 + 2(-4) = 16. Simplifying further, we get 20 + 4 + (-8) = 16. Finally we see that 16 = 16, which tells us that our solution, x = -4, is a correct one.
Let's use the same procedure to solve 7x - 5 - 4x = 13. Combining the variables gives us 3x - 5 = 13. Solving the remaining two-step equation, we get x = 6. Substituting this solution into the problem gives us 7(6) - 5 - 4(6) = 13. Simplifying that expression gives us 42 - 5 - 24 = 13. Further work gives us 13 = 13, which indicates that our solution, x = 6, is correct.
Three-Step ax + b = cx + d
This type of equation can be simplified in such a way that it can be transformed into a two-step equation. The procedure is similar to that mentioned immediately above, under solving three-step type 1 equations mentioned above, but with a slight twist.
Equations of this type have like terms too but these terms are on opposite sides of the equal sign. In order to combine them, we must target one of the variable terms and add its opposite to both sides of the equation. This will cancel a term from one side and make the equation into a two-step equation.
For instance, let's solve 4x + 3 = -5x + 21. We must cancel a variable term, so let's cancel the -5x by adding 5x to both sides of the equation. This gives us 4x + 5x + 3 = -5x + 5x + 21. Simplifying we get, 9x + 3 = 21. Solving the rest of the problem can be done following the procedure mentioned under solving two-step equations above. The result is x = 2.
Checking the solution can be done by substituting the solution into the original problem for all the x-values. This gives us 4(2) + 3 = -5(2) + 21. Doing further work yields 8 + 3 = -10 + 21 and finally 11 = 11. Therefore the solution x = 2 is the correct solution.
Let's try the same steps to solve -6x - 13 = 4x + 27. Cancelling the 4x by adding -4x to both sides of the equation gives us -6x - 4x - 13 = 4x - 4x + 27. Furthermore we can combine like terms to get -10x - 13 = 27. We can use the procedure for solving two-step equations to get the final solution, x = -4. This solution can be tested by replacing the variable with -4, which gives us -6(-4) - 13 = 4(-4) + 27. Simplifying both sides of the equation using the order of operations gives us 24 - 13 = -16 + 27 and finally 11 = 11. This is proof that x = -4 is the correct solution.
Example 2
[pic]
[pic]
[pic]
[pic]
[pic]
Practice 2
[pic]
[pic]
[pic]
[pic]
[pic]
Explain
Explain some of the mistakes that somebody could make in solving the problems above.
Application Problems
Example 2
The sum of three consecutive numbers is 18. What are these numbers?
Example 3
Find three consecutive even integers so that the largest is 2 times more than the smallest.
Example 4
Separate 100 into two parts so that one part is 3 times the other.
Practice 2
A man is nine times as old as his son. In nine years, he will be only three times as old as his son. How old are each now?In a right triangle, one of the acute angles is 2 times as large as the other acute angle. Find the measure of the two acute angles.
Practice 3
Bob's father is 3 times old as Bob. 4 years ago, he was 4 times older. How old is Bob?
Practice 4
Write a problem for the following equation: [pic]
4.1
Definition: A fraction is a number of the form [pic] where a and b are integers and [pic].
Definition: The denominator of a fraction shows the number of equal parts in the whole, and the numerator shows how may parts are being considered.
Definition: If the numerator of a fraction is smaller than the denominator, the fraction is a proper fraction. A proper fraction is less than 1.
Definition: If the numerator is greater than or equal to the denominator, the fraction is an improper fraction. An improper fraction is greater than or equal to 1.
Example 1
[pic]
Example 2
[pic]
Practice 1
Give an example of situation in which you use a proper fraction.
Practice 2
Give an example of situation in which you use an improper fraction.
Definition: Fractions that represent the same number are equivalent fractions.
Writing Equivalent Fractions
If a, b, and c are numbers with b and c not equal to zero, then [pic] and [pic].
Example 3
[pic]
[pic]
Practice 4
[pic]
[pic]
Graphing Positive and Negative Fractions
Example 5
[pic]
Practice 5
[pic]
Example 6
[pic]
Practice 6
[pic]
4.2 Writing Fractions in Lowest Terms
Definition: A prime number is a whole number that has exactly two different factors, itself and 1.
Definition: A number with a factor other than itself or 1 is called a composite number.
Example 1
Label as prime or composite.
0 5 9 13 14 18 31 46 47
Practice 1
Write down 5 prime numbers greater than 10 and 5 composite numbers greater than 10.
Definition: A prime factorization of a number is a factorization in which every factor is a prime number.
Example 2
Factor by dividing 102960
Factor 102960 by using a factor tree.
Practice 2
Factor each of the following using any method.
1356600
4210800
Using Prime Factorization to Write a Fraction in Lowest Terms
1. Write the prime factorization of both numerator and denominator
2. Use slashes to show where you are dividing the numerator and denominator by any common factors..
3. Multiply the remaining factors in the numerator and in the denominator.
Example 3
[pic]
Practice 3
[pic]
[pic]
Example 4
[pic]
Practice 4
[pic]
4. Adding and Subtracting Signed Fractions
Adding and Subtracting Like Fractions
You can add or subtract fractions only when they have a common denominator. If a,b, and c are numbers and b is not zero, then [pic] and [pic].
Example 1
[pic]
[pic]
[pic]
[pic]
Practice 1
[pic]
[pic]
[pic]
[pic]
Least Common Denominator (LCD)
The least common denominator (LCD) for two fractions is the smallest positive number divisible by both denominators of the original fractions.
Example 2
What is the LCD for [pic]?
Practice 2
What is the LCD for [pic]?
Adding and Subtracting Unlike Fractions
Step 1 Find the LCD.
Step 2 Rewrite each original fraction as an equivalent fraction whose denominator is the LCD.
Step 3 Add or subtract the numerators of the like fractions. Keep the common denominator.
Step 4 Write the sum or difference in lowest terms.
Example 3
[pic]
[pic]
[pic]
[pic]
Practice 3
[pic]
[pic]
[pic]
[pic]
Application Problems
Example 4
An advertising agent has placed newspaper ads of [pic] c.i. (column inches), [pic] c.i., [pic] c.i., [pic] c.i., and [pic] c.i. At the rate of $4 per column inch, find the total cost of the ads.
Example 5
A cabinet 30 inches high must have a 4-inch thick base and a [pic] inch thick top. Four equal sized drawers must fit in the remaining space, with [pic] inch between the drawers. What is the width of each drawer?
Practice 4
In a welding job three pieces of I-beam with lengths of [pic] inches, [pic] inches, and [pic] inches are needed. What is the total length of I-beam needed?
Practice 5
Write and solve a problem from your past experience that involves adding and/or subtracting fractions.
3. Multiplying and Dividing Signed Fractions
Multiplying Fractions
If a, b and c are numbers, then [pic].
Example 1
[pic]
[pic]
[pic]
[pic]
Practice 1
[pic]
[pic]
[pic]
[pic]
Example 2
[pic]
[pic]
[pic]
[pic]
Practice 2
[pic]
[pic]
[pic]
[pic]
Example 3
[pic]
[pic]
[pic]
[pic]
Practice 3
[pic]
[pic]
[pic]
[pic]
Application Problems
Example 4
A rectangular dog bed is [pic]yd. by [pic]yd. Find its area.
Example 5
The perimeter of a square table is [pic] yd. Determine the area of the table.
Example 6
A plane is consuming fuel at a rate of [pic] gallons per hour. The pilot has 18 gallons remaining. How long will the fuel last?
Practice 4
The total interest on a 3-year loan is $150. The borrower had the loan for 8 months of the first tax year. What fraction of the interest is deductible for the year, and how much does this amount to?
Practice 5
The doctor orders [pic] of a grain of medication. The nurse has a vial labeled [pic]grains per cc (cubic centimeter). How many cc does she give the patient?
Practice 6
How many [pic]in. beads fit in a necklace that measures [pic][pic] in.?
[pic]
5. Mixed Numbers
To convert a mixed number into an improper fraction you must:
1. Multiply the whole number by the denominator.
2. Add the result from step 1 to the numerator. This is the new numerator.
3. The denominator of the result is the same as that of the original fraction.
Example 1
[pic]
[pic]
[pic]
Practice 1
[pic]
[pic]
[pic]
Writing an Improper Fraction as a Mixed Number
Divide the numerator by the denominator. The quotient is the whole number. The divisor is the denominator. And the remainder is the numerator.
Example 2
[pic]
[pic]
[pic]
Practice 2
[pic]
[pic]
[pic]
Application Problems
Example 1
Angelina is making sopaipillas to share with her class when she tells them about the history of Mexican Independence Day. The sopaipillas are so good, especially with honey, and Angelina is sure everyone will want to eat more than one! It takes three cups of flour, two teaspoons of sugar, one tablespoon of baking powder, one-fifth of a cup of shortening, one and a third teaspoons salt, and one and three-fourths cups of water to make fourteen servings. If Angelina wants to make 70 servings, how much of each ingredient will she need?
Example 2
In an introductory biology lab, students are asked to determine the average length of the red earthworm. Working in teams, they reported the data in Table 1. At the end of the experiment, the teams combined their data to find the overall average. What was the overall average? Round your answer to the nearest tenth.
Table 1
|Team # |Average length (cm) |# of worms measured |
|1 |5 1/2 |4 |
|2 |4 1/3 |3 |
|3 |8 1/2 |3 |
Practice 1
If one-fifth of all type-Y organisms are bristly and six-sevenths of all type-Y organisms are wrinkly, what is the maximum number of type-Y organisms that are neither bristly nor wrinkly out of a sample population of 1,057?
Practice 2
Three labs measured the iron content of an ore sample sent to them by Terra-Chem, Inc. The first lab reported 24 mg of iron per cm3. The second lab reported a concentration of 9/10 that and the third lab reported a value of about 1 1/10 of the value reported by lab 2. What was the iron concentration reported by lab 3? Round your answer to the nearest hundredth of a milligram.
1. Writng Decimal Numbers
Place value
Look at the placement of the 9.
|millions |9,000,000.0 |
|hundred thousands | 900,000.0 |
|ten thousands | 90,000.0 |
|thousands | 9,000.0 |
|hundreds | 900.0 |
|tens | 90.0 |
|ones | 9.0 |
|tenths | 0.9 |
|hundredths | 0.09 |
|thousandths | 0.009 |
|ten thousandths | 0.0009 |
|hundred thousandths | 0.00009 |
|millionths | 0.000009 |
Writing a Decimal as a Fraction or Mixed Number
Step 1 The digits to the right of a decimal point are the numerator of the fraction.
Step 2 The denominator is 10 for tenths, 100 for hundredths, 1000 for thousandths, 10000 for ten-thousandths and so on.
Step 3 If the decimal has a whole number part, the fraction will be a mixed number with the same whole number part.
Step 4 Write fractions in lowest terms.
Example 1
.125 12.6
1.23 .0012025
.404 65.00128
Practice 1
.375 23.64
5.03 .0062026
.6762 95.00125
2. Rounding Decimal Numbers
Rounding Decimals
The first thing to know about rounding decimals is that it works
exactly like rounding whole numbers. To round to the nearest tenth, you do the same thing you do to round to the nearest ten, except that you are working with a different digit in the number. I can use the same picture to explain both:
| | X-------->|
+----+----+----+----+----+----+----+----+----+----+
140 141 142 143 144 145 146 147 148 149 150
This shows that 148 is between 140 and 150, and is closer to 150
because it's bigger than 145.
The next picture shows that 1.48 is between 1.40 and 1.50, and is closer
to 1.50 because it is bigger than 1.45:
| | X-------->|
+----+----+----+----+----+----+----+----+----+----+
1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50
To round to the nearest ten, you drop all digits to the right of the
tens place, replacing them with zero, so that you are left with a
multiple of 10. Then you check to see whether your original number is
actually nearer to the next multiple of 10, by checking whether the
next digit (the leftmost one you dropped) was 5 or more:
148 rounded to nearest 10:
|\
| \ replace 8 with 0
V \
140 8 is greater than 5, so add 10:
|
|
V
150
What all this means is that 148 is between 140 and 150, but is closer
to 150 so we round it to that.
Now, to round to the nearest tenth, you do the same thing, but using
the tenths digit rather than the tens:
1.48 rounded to nearest 0.1:
|\
| \ replace 8 with 0
V \
1.40 8 is greater than 5, so add 0.1:
|
|
V
1.50
Again, this means that 1.48 is between 1.4 and 1.5, but is closer to
1.5. If the number had been 1.45, exactly between the two, you would
still round up to 1.5. If it were 1.43, you would have rounded down,
leaving it at 1.40, because 3 is less than 5.
The only real difference is that when you put zeroes to the right of a decimal, you can ignore them completely because they don't affect the number. So you can write the answer as 1.5 rather than 1.50.
Example 1
a) round 72.3478 to the nearest tenth
b) round 15.4201 to the nearest tenth
c) round 5.74999 to the nearest thousandth
d) round 87.5555 to the nearest hundredth
Practice 1
a) round 12.3974 to the nearest tenth
b) round 65.201 to the nearest hundredth
c) round 5.7499469 to the nearest ten-thousandth
d) round 92.558855 to the nearest thousandth
Exercise 2
$12.345
$78.944
$51.6013
$99100.4749
Practice 2
$82.362
$898.2494
$5.013
$999100.981749
1. Airplane Mechanic:
The “mean chord” of the wings of an airplane is equal to its wing area divided by its wing span. Find the mean chord of a plane if its wing area is 275 square feet, and its span is 42.25 feet.
2. Architect:
If a beam weighs 32 ponds per foot, what is the weight of a bema 4.8 feet long?
3. Civil Engineer:
A civil engineer working for the County Building an Safety Department must determine whether or not building plans conform to code specifications. For, the two supporting side walls of a building are only allowed to bear a certain maximum amount of weight per linear foot. To compute this, one would find the total weight per square foot of all roofing materials, multiply by the number of square feet of roof surface, and divide by the total number of linear feet in the two side walls.
Suppose plans are submitted for a 20-foot square garage. The 400 square feet of built-up roofing weighs 5.5 pounds per square foot. The joists supporting the roofing weigh 1.54 pounds per square foot, and the ½ in. plywood is between the joist and the roof weights 1.5 pounds per square foot. How much weight will the 40 feet of side walls have to bear per linear foot?
4. Nurse:
The doctor orders .2 grams of nicotinic acid for a patient. If it is only available in .05 gram tablets, how many tablets do you give the patient?
5. Stock Broker:
If you sold 250 shares of Exxon at $48 7/8 per share, how many shares of Burroght’s could you buy at $71 3/4 per share?
6. Surveyor:
A condominium development has a flooding problem which requires the installation of a gutter. The surveyor finds the exact point of origin of the gutter by determining its elevation above sea level. The lower end of the gutter is at an elevation of 126.58 feet. Between there and the upper end there must be a grade of 0.8% meaning the gutter must rise 0.008 feet in elevation for every horizontal foot of gutter. If the total horizontal distance is 80 feet, find the elevation of the upper end of the gutter.
5.7
The MEAN is the arithmetic average, the average you are probably used
to finding for a set of numbers - add up the numbers and divide by how
many there are
Example 1
80, 90, 90, 100, 85, 90
(80 + 90 + 90 + 100 + 85 + 90) / 6 = 89 1/6.
Example 2:
Four tests results: 15, 18, 22, 20
The sum is: 75
Divide 75 by 4: 18.75
The 'Mean' (Average) is 18.75
(Often rounded to 19)
Practice 1
Practice 2
The MEDIAN is the number in the middle. In order to find the median,
you have to put the values in order from lowest to highest, then find
the number that is exactly in the middle
Example 3
80 85 90 90 90 100
^
since there is an even number of values, the MEDIAN is
between these two, or it is 90. Notice that there is
exactly the same number of values ABOVE the median as
BELOW it!
Example 4
Find the Median of: 9, 3, 44, 17, 15 (Odd amount of numbers)
Line up your numbers: 3, 9, 15, 17, 44 (smallest to largest)
The Median is: 15 (The number in the middle)
Example 5
Find the Median of: 8, 3, 44, 17, 12, 6 (Even amount of numbers)
Line up your numbers: 3, 6, 8, 12, 17, 44
Add the 2 middles numbers and divide by 2: 8 12 = 20 ÷ 2 = 10
The Median is 10.
Practice 3
Practice 4
The MODE is the value that occurs most often.
Example 6
In this case, since
there are 3 90's, the mode is 90. A set of data can have more than one
mode.
Examples 7
Find the mode of:
9, 3, 3, 44, 17 , 17, 44, 15, 15, 15, 27, 40, 8,
Put the numbers is order for ease:
3, 3, 8, 9, 15, 15, 15, 17, 17, 27, 40, 44, 44,
The Mode is 15 (15 occurs the most at 3 times)
*It is important to note that there can be more than one mode and if no number occurs more than once in the set, then there is no mode for that set of numbers.
Practice 5
Practice 6
The RANGE is the difference between the lowest and highest values.
Example 8
9, 3, 3, 44, 17 , 17, 44, 15, 15, 15, 27, 40, 8,
In this case 100 - 80 = 20, so the range is 20. The range tells you
something about how spread out the data are. Data with large ranges
tend to be more spread out.
if your set is 9, 3, 44, 15, 6 - The range would be 44-3=41. Your range is 41.
Practice 7
5.8
Pythagorean Theorem
Over 2,500 years ago, a Greek mathematician named Pythagoras developed a proof that the relationship between the hypotenuse and the legs is true for all right triangles.
[pic]
"In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs."
This relationship can be stated as:
[pic]
and is known as the
Pythagorean Theorem.
Example 1:
[pic]
Find x.
[pic]
[pic]
[pic]
a, b are legs.
c is the hypotenuse
(c is across from the right angle).
There are certain sets of numbers that have a very special property. Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem.
For example: the numbers 3, 4, and 5 satisfy the Pythagorean Theorem. If you multiply all three numbers by 2 (6, 8, and 10), these new numbers ALSO satisfy the Pythagorean theorem.
The special sets of numbers that possess this property are called
Pythagorean Triples.
The most common Pythagorean Triples are:
3, 4, 5
5, 12, 13
8, 15, 17
REMEMBER: The Pythagorean Theorem ONLY works in Right Triangles!
Example 2:
A triangle has sides 6, 7 and 10.
Is it a right triangle?
Let a = 6, b = 7 and c = 10. The longest side MUST be the hypotenuse, so c = 10. Now, check to see if the Pythagorean Theorem is true.
[pic]
Since the Pythagorean Theorem is NOT true, this triangle is NOT a right triangle.
Example 3:
[pic]
A ramp was constructed to load a truck. If the ramp is 9 feet long and the horizontal distance from the bottom of the ramp to the truck is 7 feet, what is the vertical height of the ramp?
Since the ramp is described as having horizontal and vertical measurements, a right angle is implied. Solve using the Pythagorean Theorem:
[pic]
The height of the ramp is 5.7 feet. The ramp will allow packages to be loaded into an area of the truck that is too high to be reached from the ground.
Practice 1
Practice 2
Practice 3
Practice 4
Practice 5
6.1 Ratios
A ratio compares two quantities that have the same type of units.
Writing a ratio as a fraction.
Order is important. The quantity mentioned first is the numerator and the quantity mentioned second is the denominator.
Example 1
If the ratio of your height to you arm span is 63 to 65 you write 63/65.
Practice 1
Write down the ratio of the cost of your vehicle to your income.
Writing Ratios in lowest terms.
As with any fraction you can write a ratio in lowest terms.
Example 2
The ratio of students to teachers at Onate high school is 50 students to 2 teachers. Write this ratio in lowest terms.
Practice 2
The the ratio of 8 days to 28 days in lowest terms.
Application Problem
Pg. 426 Historical Ratios
6. 2 Rates
A ratio compares two measurements with the same type of units, but many comparisons we make use measurements with different types of units. This type of comparison is called a rate. If a rate is not in lowest terms you should write the rate in lowest terms.
Example 1
If the number of miles you traveled this morning was 5 and the number of minutes it took you to travel was 10 then your rate of travel was 5 miles/10 minutes. We can write this rate as 1mile/2minutes.
Practice 1
Write down the rate at which you traveled this morning. Write the rate in lowest terms.
Write down 3 other rates which you have encountered this week.
Unit Rates
When the denominator of the rate is 1, it is called a unit rate. If the rate is in terms of the cost of an item we call the unit ratio the cost per unit.
Example 2
The average cost of square foot in a home is $110. We can write this as a rate as follows:
$110/1 square foot.
Example 3
You can divide in order to write a rate as a unit rate. For example $5.45 for 16 ounces of shampoo can be written as a unit rate as follows:
$5.45(16oz=$0.340625/ 1 oz.
Practice 3
Write down 3 unit rates that you recall from the last shopping trip you made.
6.3 Proportions
A proportion states that two ratios or rates are equivalent.
Example 1
10 feet is to 5 feet as 5 feet is to 2.5 feet
[pic]
Practice 1
$5.40 is to 18 ounces as $3.00 is to 10 ounces
Example 2
Determine whether the proportions are true.
[pic]
[pic]
[pic]
Practice 2
[pic]
[pic]
[pic]
Use Cross Products to Determine Whether a Proportion is True
To see whether a proportion is true, first multiply along one diagonal then multiply along the other diagonal as shown below:
[pic]
If the cross products are equal then the proportion is true. If the cross products are unequal then the proportion is false.
Example 3
[pic]
[pic]
Practice 3
[pic]
[pic]
Solving Proportions
Step 1 Find the cross products.
Step 2 Show that the cross products are equivalent
Step 3 Divide both sides of the equation by the coefficient of the variable term.
Step 4 Check by writing the solution in the original proportion and finding the cross
products.
Example 4
[pic]
[pic]
[pic]
[pic]
Practice 4
[pic]
[pic]
[pic]
[pic]
6.4 Problem Solving with Proportions
Example 1
Attorney: A sampling of recent cases showed the ratio of monthly child support to father’s yearly income to be 1:40. If a client makes $18,000 annually how much should he expect to pay in montly child support?
Example 2
Brandon analyzed some pollen grains found near a site that is thought to have been inhabited by humans many thousands of years ago. He found 75 examples of grass pollen, 16 examples of oak pollen, and 30 examples of unknown kinds of pollen. Let's assume his analysis was typical of the site and that another scientist could study the site and determine the exact same proportions of the three classes of pollen. Assuming this, if there were 19,500 pollen grains in the area of study, how many of them would be of an unknown type?
Example 3
One group (A) contains 172 people. One-fourth of the people in group A will be selected to win free tickets to a concert. There is another group (B) in a nearby town that will receive the same number of tickets, but there are 396 people in that group. What will be the ratio of non-winners in group A to non-winners in group B after the selections are made?
Practice 1
Pharmacist:
A certain crème mixture contains 12 grams of Crème A, 18.6 grams petrolatum, and 30 grams univase. The pharmacist needs to make 480-gram mixture containing the same ratio of ingredients. How much of each ingredient should he include?
Practice 2
If the ratio of saturated to unsaturated fatty acids in a cell membrane is 9 to 1, and there are a total of 85 billion fatty acid molecules, how many of them are saturated?
Practice 3
According to the instructions for a chemical procedure, Jordan mixed salt, baking soda and water in a 2:4:21 ratio by mass. How many grams of water would be required to make a mixture that contained 16 grams of salt?
More Application Problems
Practice 4
The human body contains 90 pounds of water for every 100 pounds of body weight. How many pounds of water are in child who weighs 80 pounds.
Practice 5
A 150 pound person burns 189 calories during 45 minutes of grocery shopping. How many calories would a 115 pound person burn , to the nearest whole number?
Practice 6
About 9 out of 10 adults think it is a good idea to exercise regularly. But of the ones who think it is a good idea, only 1 in 6 actually exercises at least three times a week. At this rate, how many of the 300 employees in our company exercise regularly?
6.6 Triangles
Similar Triangles:
If you have any triangle, and another one that looks identicle except that it is either larger or smaller (i.e. looks "blown-up" or "shrunk-down"), then those two triangles are similar. The mathematical definition for similar triangles is that they both have correponding angles that are equal, while the lengths of the corresponding sides are in proportion.
Recall that the sum of all 3 angles in a triangle total 180o. This means that if you know that 2 angles in 2 different triangles are the same, then the 3rd angle must also be the same in both triangles. Therefore, the triangles are similar.
Notation: We use [pic]ABC to denote the measure of the angle formed by the line segments AB and BC. We use |AB| to denote the distance from point A to point B.
Proposition: Let [pic]ABC and [pic]DEF be two triangles. If [pic]ABC = [pic]DEF and [pic]ACB = [pic]DFE, then [pic]ABC and DEF are similar. Moreover,
|AC| |AB| |BC|
---- = ---- = ---- (see diagram below)
|DF| |DE| |EF|
[pic]
1)
[pic]
[pic]
2) A clever outdoorsman whose eye-level is 2 meters above the ground, wishes to find the height of a tree. He places a mirror horizontally on the ground 20 meters from the tree, and finds that if he stands at a point C which is 4 meters from the mirror B, he can see the reflection of the top of the tree. How high is the tree?
Solution:
[pic]
We make the assumption that the man and the tree are both standing up
straight and that the ground is flat. So [pic]PBC = [pic]QBA also,
the triangles [pic]PCB and [pic]QAB are similar (by the proposition stated on the
tutorial page). Thus,
|QA| |AB|
---- = ----
|PC| |CB|
|QA| 20
= ---- = ----
2 4
|QA| = 10
Therefore, the height of the tree is 10 meters.
3) A child 1.2 meters tall is standing 11 meters away from a tall building. A spotlight on the ground is located 20 meters away from the building and shines on the wall. How tall is the child's shadow on the building?
Solution:
Let h be the height of the shadow on the building. Then draw a diagram
assuming the ground to be flat, as in the diagram below.
[pic]
There are two triangles: one formed by the spotlight and the child, and one
formed by the spotlight and the height of the shadow, h. These two triangles
share a common angle A at the spotlight. If we assume that the child and the
wall of the building are perpendicular to the ground, then the angle formed by
the child and the ground (angle C) are both right angles. So the triangles have
another pair of equal angles. Therefore, the trianlges are similar.
Now we must look at the lenghts of the corresponding sides. We know that the
child must be 9 meters from the spotlight (i.e. 20 m- 11 m). This length in the
smaller triangle corresponds to the distance from the spotlight to the building
in the larger triangle (i.e. 20 m). The height of the child in the smaller
triangle (1.2 m) corresponds to the height of the shadow in the larger triangle
(h). Since the triangles are similar, these lengths are in proportion.
Therefore: 9 1.2
--- = ----
20 h
9h = 20 (1.2)
h = 24/9 = 8/3 = 2.67 meters
The height of the shadow is 8/3 meters (approx. 2.67 meters).
7.1 The Basics of Percent
Problem What fraction of each grid is shaded?
|Grid 1 |Grid 2 |Grid 3 |
|[pic] |[pic] |[pic] |
|Answer |Answer |Answer |
Each grid above has 100 boxes. For each grid, the ratio of the number of shaded boxes to the total number of boxes can be represented as a fraction.
|Comparing Shaded Boxes to Total Boxes |
|Grid |Ratio |Fraction | |
|1 |96 to 100 |[pic] | |
|2 | 9 to 100 |[pic] | |
|3 |77 to 100 |[pic] | |
We can represent each of these fractions as a percent using the symbol %.
|[pic] |= |
Definition: A percent is a ratio whose second term is 100. Percent means parts per hundred. The word percent comes from the Latin phrase per centum, which means per hundred. In mathematics, we use the symbol % for percent.
|Let's look at our comparison table again. This time the table includes percents. |
|Comparing Shaded Boxes to Total Boxes |
|Grid |Ratio |Fraction |Percent |
|1 |96 to 100 |[pic] |96% |
|2 | 9 to 100 |[pic] | 9% |
|3 |77 to 100 |[pic] |77% |
Writing a Percent as an Equivalent Decimal
To write a percent as a decimal, drop the % symbol and divide by 100.
Writing a Decimal as a Percent
To write a decimal as a percent, multiply by 100 and attach a % symbol.
Writing a Percent as a Fraction
To write a percent as a fraction, drop the % symbol and write the number over 100. Then write the fraction In lowest terms.
Let's look at some examples in which we are asked to convert between ratios, fractions, decimals and percents.
|Example 1: |Write each ratio as a fraction, a decimal, and a percent: 4 to 100, 63 to 100, 17 to 100 |
|[pic] |
| |Solution |
| | |
| |Ratio |
| |Fraction |
| |Decimal |
| |Percent |
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|Example 2: |Write each percent as a ratio, a fraction in lowest terms, and a decimal: 24%, 5%, 12.5% |
|[pic] |
| |Solution |
| | |
| |Percent |
| |Ratio |
| |Fraction |
| |Decimal |
| | |
| | |
| | |
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|Example 3: |Write each percent as a decimal: 91.2%, 4.9%, 86.75% |
|[pic] |
| |Solution |
| | |
| |Percent |
| |Decimal |
| | |
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[pic]
|Summary: |A percent is a ratio whose second term is 100. Percent means parts per hundred. We use the symbol % |
| |for percent. In this lesson, we learned how to make basic conversions between ratios, fractions, |
| |decimals and percents. |
[pic]
Practice 1
Write each percent as a decimal.
68%
40.6%
350%
900%
0.9%
Practice 23
Write each number as a percent
.18
.617
.4
5.34
4
Practice 4
Write each percent as a fraction or mixed number in lowest terms.
50%
6%
300%
Practice 5
Write each percent as a fraction in lowest terms.
87.5%
6.5%
66 2/3 %
12 1/3%
Practice 6
Write each fraction as a percent.
½
¾
1/10
7/8
5/6
2/3
2. The Percent Proportion
Using the Proportion Method to Solve Percent Problems
There are a variety of ways to solve percent problems, many of which can be VERY confusing. Fortunately, the PROPORTION METHOD will work for all three types of questions:
What number is 75% of 4?
3 is what percent of 4?
75% of what number is 3?
Using the PROPORTION METHOD, the set-up is always the same:
[pic]
|PERCENT |--|the number with the percent sign (%). |
|PART |--|the number with the word is. |
|WHOLE |--|the number with the word of. |
EXAMPLE #1:
|What number is 75% of 4? (or Find 75% of 4.) |
| |The PERCENT always goes over 100. | |
| | (It's a part of the whole 100%.) | |
| |4 appears with the word of: | |
| | It's the WHOLE and goes on the bottom. | |
|We're trying to find the missing PART (on the top). |
| |In a proportion the cross-products are equal: So 4 times 75 is equal to 100 times the PART. |
|The missing PART equals 4 times 75 divided by 100. |
| |(Multiply the two opposite corners with numbers; then divide by the other number.) |
| | | |
| |4 times 75 | |
| |= | |
| |100 times the part | |
| | | |
| | | |
| |300 | |
| |= | |
| |100 times the part | |
| | | |
| | | |
| |300/100 | |
| |= | |
| |100/100 times the part | |
| | | |
| | | |
| |3 | |
| |= | |
| | the part | |
| | | |
EXAMPLE #2:
|3 is what percent of 4? |
| |3 appears with the word is: It's the PART and goes on top.| |
| | | |
| |4 appears with the word of: | |
| | It's the WHOLE and goes on the bottom. | |
|We're trying to find the missing PERCENT (out of the whole 100%). |
| |In a proportion the cross-products are equal: So 3 times 100 is equal to 4 times the PERCENT. |
|The missing PERCENT equals 100 times 3 divided by 4. |
| |(Multiply the two opposite corners with numbers; then divide by the other number.) |
| | | |
| |3 times 100 | |
| |= | |
| |4 times the percent | |
| | | |
| | | |
| |300 | |
| |= | |
| |4 times the percent | |
| | | |
| | | |
| |300/4 | |
| |= | |
| |4/4 times the percent | |
| | | |
| | | |
| |75 | |
| |= | |
| | the percent | |
| | | |
EXAMPLE #3:
|75% of what number is 3? (or 3 is 75% of what number?) |
| |The PERCENT always goes over 100. | |
| | (It's a part of the whole 100%.) | |
| |3 appears with the word is: | |
| | It's the PART and goes on the top. | |
|We're trying to find the missing WHOLE (on the bottom). |
| |In a proportion the cross-products are equal: So 3 times 100 is equal to 75 times the WHOLE. |
|The missing WHOLE equals 3 times 100 divided by 75. |
| |(Multiply the two opposite corners with numbers; then divide by the other number.) |
| | | |
| |3 times 100 | |
| |= | |
| |75 times the whole | |
| | | |
| | | |
| |300 | |
| |= | |
| |75 times the whole | |
| | | |
| | | |
| |300/75 | |
| |= | |
| |75/75 times the whole | |
| | | |
| | | |
| |4 | |
| |= | |
| | the whole | |
| | | |
Even unfriendly-looking problems like this can be solved using the PROPORTION METHOD:
EXAMPLE #4:
|Find 83 2/3 % of 12.6 (or What number is 83 2/3 % of 12.6?) |
| |The PERCENT always goes over 100. | |
| | (It's a part of the whole 100%.) | |
| |12.6 appears with the word of: | |
| | It's the WHOLE and goes on the bottom. | |
|We're trying to find the missing PART (on the top). |
| |In a proportion the cross-products are equal: So 12.6 times 83 2/3 is equal to 100 times the PART. |
|The missing PART equals 12.6 times 83 2/3 divided by 100. |
| |(Multiply the two opposite corners with numbers; then divide by the other number.) |
| |12.6 times 83 2/3 | |
| |= | |
| |100 times the part | |
| | | |
| |(126/10)(251/3) | |
| |= | |
| |100 times the part | |
| | | |
| |31626/30 | |
| |= | |
| |100 times the part | |
| | | |
| |(31626/30)(1/100) | |
| |= | |
| |(100/1)(1/100)( the part) | |
| | | |
| |31626/3000 | |
| |= | |
| | the part | |
| | | |
| |10542/1000 | |
| |= | |
| | the part | |
| | | |
| |10.542 | |
| |= | |
| | the part | |
| | | |
|PLEASE NOTE: There are MANY other ways to do the arithmetic in this problem - hopefully this shows the steps in an |
|understandable manner; it is neither the easiest nor the best approach. |
| |
| |
| |
Practice Problems:
|Top of Form | |Top of Form |
|What is 20% of 60? | |60 is 3/8 of what number? |
|12 is 75% of what number? | |12 is what fractional part of 20? |
|6 is what percent of 8? | |12 is what percent of 20? |
|8 is 40% of what? | |9 is 37 1/2% of what? |
|33 1/3% of what number is 24? | |What is 0.25% of 30? |
|Find 12.5% of 400. | |15 is what percent of 40? |
|What is 5/8 of 32? 12 is 2/5 of what? | |What percent of 60 is 75? |
|What is 87 1/2% of 150? | |45 is 60% of what? |
|25 is what percent of 30? | |Find 150% of 30 |
|Bottom of Form | |Find 7/8 of 20.6 |
| | |Bottom of Form |
3. The Percent Equation
Percent Equation
Percent of whole = Part
Percent * whole = Part
Example 1
15% of $195 is how much money?
Practice 1
12% of 915 people is how many people?
Example 2
10 pounds is what percent of 213 pounds?
Practice 2
103 miles is what percent of 513 miles?
Example 3
What percent of $40 is $112?
Practice 3
What percent of 110 inches is 120 inches?
Example 4
121 credits are 95% of how many credits?
Practice 4
120 acres is 95% of how many acres?
Example 5
125% of what amount is $98?
Practice 5
250% of what amount is $68?
7.4 Problem Solving with Percent
Example 1
When Maribel received her first $180 paycheck as a math tutor, 12 ½ % was withheld for federal income tax. How much was withheld?
Example 2
On a 15 point quiz, Esteban, earned 13 points. What percent correct is this, to the nearest whole percent?
Example 3
A newspaper article stated that 648 pints of blood were donated at the blood bank last month, which was only 72% of the number of pints needed. How many pints of blood were needed?
Example 4
Roberto’s hourly wage as assistant manger of a fast-food restaurant was raised from $9.40 to $9.87. What was the percent increase?
Practice 1
Pg. 540#2
Practice 2
Pg. 541 #4
Practice 3
Pg. 542 5a
Practice 4
Pg. 5447b
Practice 5
Electrical Inspector
The electrical code specifies how to calculate the net load in watts for different types of appliances and lights. This information is then used to determine such things as current requirements.
For example, to compute the net load for general lighting, small appliances, and laundry in a home, add10% of the first 3000 watts to 35% of the remainder. The reason for this is that these items are not demanding electricity on a constant basis.
Determine the net load if the total load for general lighting, small appliances, an laundry is 8000 watts.
Practice 6
Social Worker:
When a mother on welfare hires an “in-home” child care worker, the county reimburses her for expenses. Since the worker must receive minimum wage, the mother must deduct social security tax of 6.05% and send it to the government. The social worker will usually compute these taxes for the mother.
A child care worker receives $3.10 per hour for 138 hours. Determine the social security tax on the total wages and the net amount paid to the worker after deducting the tax.
7.5 Consumer Applications: Sales Tax, Tips, Discounts and Simple Interest
Example 1
Ms. Ortiz bought a $21,950 pickup truck. She paid an additional $1646.25 in sales tax. What was the sales tax rate?
Example 2
Suppose that you buy a DVD player for $289 from A-1 Elect5ronics. The sales tax rate in your state is 7.25%. How much is the tax? What is the total cost of the DVD player?
Example 3
The Oak Mill Furniture Store has an oak entertainment center with an original price of $840 on sale at 15% off. Find the sale price of the entertainment center.
Example 4
Find the simple interest on a $2000 loan at 6% for 1 year. Use the formula I=p*r*t.
Practice 1
Joseph bought himself a watch normally priced at $200 while it was on sale at 10% off. The sales tax was 7.25%. What did he end up paying for the watch?
Practice 2
Adam likes to give tips at restaurants between 10% and 20% depending on the service. While on a date he wanted to impress his companion by giving the waitress an 18% tip. If the cost of their meals was $55.00 what was the tip that Adam left the waitress?
Practice 3
Find the simple interest of a $300 loan at 23% for 2 years.
8.1 English Customary Weights and Measures
Distance
In all traditional measuring systems, short distance units are based on the dimensions of the human body. The inch represents the width of a thumb; in fact, in many languages, the word for "inch" is also the word for "thumb." The foot (12 inches) was originally the length of a human foot, although it has evolved to be longer than most people's feet. The yard (3 feet) seems to have gotten its start in England as the name of a 3-foot measuring stick, but it is also understood to be the distance from the tip of the nose to the end of the middle finger of the outstretched hand.
In Anglo-Saxon England (before the Norman conquest of 1066), short distances seem to have been measured in several ways. The inch (ynce) was defined to be the length of 3 barleycorns, which is very close to its modern length. Several foot units were in use, including a foot equal to 12 inches
When the Normans arrived, they brought back to England the Roman tradition of a 12-inch foot. Although no single document on the subject can be found, it appears that during the reign of Henry I (1100-1135) the 12-inch foot became official, and the royal government took steps to make this foot length known. A 12-inch foot was inscribed on the base of a column of St. Paul's Church in London, and measurements in this unit were said to be "by the foot of St. Paul's" (de pedibus Sancti Pauli). Henry I also appears to have ordered construction of 3-foot standards, which were called "yards," thus establishing that unit for the first time in England.
Longer distances in England are traditionally measured in miles. The mile is a Roman unit, originally defined to be the length of 1000 paces of a Roman legion. A "pace" here means two steps, right and left, or about 5 feet, so the mile is a unit of roughly 5000 feet. In 1592, Parliament settled this question by setting the length of the mile 5280 feet. This decision completed the English distance system. Since this was just before the settling of the American colonies, British and American distance units have always been the same.
Area
In all the English-speaking countries, land is traditionally measured by the acre, a very old Saxon unit that is either historic or archaic, depending on your point of view. There are references to the acre at least as early as the year 732. The word "acre" also meant "field", and as a unit an acre was originally a field of a size that a farmer could plow in a single day. In practice, this meant a field that could be plowed in a morning, since the oxen had to be rested in the afternoon.
Most area units were eventually defined to be the area of a square having sides equal to some simple multiple of a distance unit, like the square yard.
Weight
The basic traditional unit of weight, the pound, originated as a Roman unit and was used throughout the Roman Empire. The Roman pound was divided into 12 ounces, but many European merchants preferred to use a larger pound of 16 ounces, perhaps because a 16-ounce pound is conveniently divided into halves, quarters, or eighths. During the Middle Ages there were many different pound standards in use, some of 12 ounces and some of 16.
Around 1300 the mercantile pound was replaced in English commerce by the 16-ounce avoirdupois pound. This is the pound unit still in common use in the U.S. and Britain. Modeled on a common Italian pound unit of the late thirteenth century, the avoirdupois pound weighs exactly 7000 grains.
Since at least 1400 a standard weight unit in Britain has been the hundredweight, which is equal to 112 avoirdupois pounds rather than 100. There were very good reasons for the odd size of this "hundred": 112 pounds made the hundredweight equivalent for most purposes with competing units of other countries, especially the German zentner and the French quintal. Furthermore, 112 is a multiple of 16, so the British hundredweight can be divided conveniently into 4 quarters of 28 pounds, 8 stone of 14 pounds, or 16 cloves of 7 pounds each. The ton, originally a unit of wine measure, was defined to equal 20 hundredweight or 2240 pounds.
During the nineteenth century, an unfortunate disagreement arose between British and Americans concerning the larger weight units. Americans, not very impressed with the history of the British units, redefined the hundredweight to equal exactly 100 pounds. The definition of the ton as 20 hundredweight made the disagreement carry over to the size of the ton: the British "long" ton remained at 2240 pounds while the American "short" ton became exactly 2000 pounds. (The American hundredweight became so popular in commerce that British merchants decided they needed a name for it; they called it the cental.) Today, most international shipments are reckoned in metric tons, which, coincidentally, are rather close in weight to the British long ton.
Volume
The names of the traditional volume units are the names of standard containers. Until the eighteenth century, it was very difficult to measure the capacity of a container accurately in cubic units, so the standard containers were defined by specifying the weight of a particular substance, such as wheat or beer, that they could carry. Thus the gallon, the basic English unit of volume, was originally the volume of eight pounds of wheat. This custom led to a multiplicity of units, as different commodities were carried in containers of slightly different sizes. Gallons are always divided into 4 quarts, which are further divided into 2 pints each.
The situation was still confused during the American colonial period, so the Americans were actually simplifying things by selecting just two of the many possible gallons. These two were the gallons that had become most common in British commerce by around 1700. For dry commodities, the Americans were familiar with the "Winchester bushel," defined by Parliament in 1696 to be the volume of a cylindrical container 18.5 inches in diameter and 8 inches deep. The corresponding gallon, 1/8 of this bushel, is usually called the "corn gallon" in England. This corn gallon holds 268.8 cubic inches.
For liquids Americans preferred to use the traditional British wine gallon, which Parliament defined to equal exactly 231 cubic inches in 1707. As a result, the U.S. volume system includes both "dry" and "liquid" units, with the dry units being about 1/6 larger than the corresponding liquid units.
On both sides of the Atlantic, smaller volumes of liquid are traditionally measured in fluid ounces, which are at least roughly equal to the volume of one ounce of water. To accomplish this in the different systems, the smaller U.S. pint is divided into 16 fluid ounces, and the larger British pint is divided into 20 fluid ounces.
One of These Things is Not Like the Other
A Discussion of Units Copyright (c) 1996 by Kenny Felder
This sheet will explain briefly the concept of units, and the use of a simple technique with a fancy name—"dimensional analysis"—to convert from one unit system to another.
Whatever You Measure, You Have to Use Units
When we measure something, we always have to specify what units we are measuring in. For instance, if I tell you I am 16 tall, you haven't learned much about my height: your proper response would be "16 what?" If I respond 16 inches, you know that I am a very short person; on the other hand, if I tell you I am 16 feet, you can sign me up for the basketball team. And there are many other units of length I could use, such as meters and kilometers.
The above examples are for units of length, but there are many other things that we measure, and all of them require units. For instance, time can be measured in seconds or minutes (or hours, or days, etc); angles can be measured in degrees or in radians; and so on. Some units are made up of other units: for instance, you might measure the speed of a car in miles/hour. So "50 mi/hr" means that every hour, the car travels 50 miles; very different from a car travelling 50 mi/s, for instance! No matter what you are measuring, you must specify the units in order to give a measurement.
Sometimes You Have to Convert Between Different Units
Suppose that a 60-inch man stands on the head of a 6-foot man. How tall are they together?
Obviously, you cannot find the answer to this question by adding 60 to 6. The reason is that the two figures are given in different units. Before you can add the two numbers, you have to convert one of them to the units of the other. Then, when you have two numbers in the same units, you can add them.
In order to perform this conversion, you need a conversion factor. That is, in this case, you have to know how many inches make up a foot. You probably already know the answer: 12 inches is 1 foot. So knowing that, you can perform the calculation in two steps as follows:
a) 60 inches is really 60/12 = 5 feet
b) 5 feet + 6 feet = 11 feet
The second half of the calculation is easy; once all your units are the same, calculation is a snap. The tricky part is the conversion: because sometimes you have to divide, and sometimes you have to multiply. In the above example, I converted from inches to feet by dividing by 12. If I were going the other way, converting from feet to inches, I would have multiplied by 12. How do I figure out which to do?
One way is by using common sense. I know that 60 inches can't possibly be 720 feet; there have to be less feet than inches, obviously. However, sometimes common sense falls down on you. For instance, if you want to convert miles/hour into meters/second, what you do you multiply or divide by what?
Fortunately, there is another way: relatively easy once you get used to it, and guaranteed to work in every situation. This technique goes by the somewhat intimidating name of dimensional analysis, and I apologize. I didn't name it.
Dimensional Analysis Tells You What to Multiply or Divide
Let's attack that problem I solved above, but use dimensional analysis, just so we can introduce the method. As you'll recall, the interesting part of the problem was step a), where I had to convert 60 inches into feet. I knew that there are 12 inches in a foot, so common sense told me to divide by 12. Here's the dimensional analysis way.
First of all, we write our conversion factor as a fraction. 12 inches = 1 foot, so the conversion factor is
[pic]or [pic]
Because 12 inches is the same thing as 1 foot, that fraction is just like 5/5; it's equal to 1. And we can multiply by it without changing anything.
So, let's try that on our 60 inches.
[pic]
The critical thing to note about the above equation is that the units behave like numbers do when you multiply fractions. That is, the inches on top and the inches on the bottom cancel out, leaving feet. Then all you have to worry about is the numbers.
Suppose that I had written in the wrong way, like the following.
[pic](Wrong)
I would have immediately realized that multiplying 60 * 12 is wrong, because the units of the answer are inches*inches/feet instead of just feet. So the dimensional analysis shows me immediately that dividing is correct, and multiplying is not.
Such conversions are even easier in the metric system, of course, since all the conversion factors are multiples of ten. For intstance, to convert 10 km to m, you would simply write
10 km * [pic]= 10,000 m
At this point, you may well be asking yourself "so what?" Dimensional analysis doesn't take much time, and it isn't very difficult, but it is certainly more trouble than the common-sense method.
To answer that, let's look at a more difficult unit conversion problem. Suppose that I want to convert 55 mi/hr into m/s. I know that 1 mile is 5280 feet, that 1 meter is 3.3 feet, that 1 hour is 60 minutes, and that 1 minute is 60 seconds. Now, what do I multiply, or divide, by what?
This is a very tricky question, but using dimensional analysis, it becomes a snap. First of all, I write all the above conversions as fractions, remembering that I might have to turn some of them upside-down.
[pic]* [pic]* [pic]* [pic]
Now I write out an equation to convert from miles/hour to m/s, watching the units to make sure they cancel.
[pic]* [pic]* [pic]* [pic]* [pic]= [pic](Wrong)
Did you follow that, and make sure that the units cancel to give us meters/second? If you did, you hopefully were not surprised by the annotation on the right: because this equation is most definitely incorrect. Do you see why? The "feet" on the top of the fractions don't cancel; so instead of being left with meters/seconds, we are left with the rather ungainly units of [pic]. That is most definitely not what we are looking for; which is a perfect example of how, in a complicated problem like this, dimensional analysis can prevent you from making errors. If we turn the feet/meters fraction around, we get:
[pic]* [pic]* [pic]* [pic]* [pic]= [pic]= 24.4 m/s.
If you think that was overly difficult, try to imagine what it would have been like without dimensional analysis! You would still have had to multiple the 55, 5280, 3.3, 60, and 60, but first you would have had to figure out which to multiply and divide. And if you had made a mistake, how would you have caught yourself?
In Conclusion
Units are a critical part of describing every measurement. Before you can work with units mathematically, you frequently must convert from one unit to another. Dimensional analysis does not do your math for you, but it makes sure you get your multiplications and divisions straight. After that, all you have to do is find the conversion factors (unless you are the sort who memorizes numbers like 3.3, which I don't) and plug into a calculator (unless you are the sort who enjoys long division).
Although the formalism of dimensional analysis (not to mention the name) can be intimidating at first, you will find that once you have worked a few problems, it's a snap. Try it!
Example 1
Convert
4. ½ ft to inches
64 oz to pounds
2 T to pounds
3 wk to days
36 in. to feet
39 ft to yards
2 mi to feet
16 qt to gallons
4 c to pints
3 ½ T to pounds
4 oz to pounds
Practice 1
Convert
54 oz to pounds
9 T to pounds
6 in. to feet
64 in. to feet
99 ft to yards
4 mi to feet
6 qt to gallons
8 c to pints
7 ½ T to pounds
12 oz to pounds
27 seconds to days
8 yd. to miles
10 quarts to cups
9 tons to ounces
8.2/8.3 The Metric System
The Metric System
|LENGTH |
|Unit |Abbreviation |Number of |
| | |Meters |
|kilometer |km |1,000 |
|hectometer |hm |100 |
|dekameter |dam |10 |
|meter |m |1 |
|decimeter |dm |0.1 |
|centimeter |cm |0.01 |
|millimeter |mm |0.001 |
|micrometer |µm |0.000001 |
| |
|CAPACITY/VOLUME |
|Unit |Abbreviation |Number of |
| | |Liters |
| | | |
|kiloliter |kl |1,000 |
|hectoliter |hl |100 |
|dekaliter |dal |10 |
|liter |l |1 |
|cubic decimeter |dm3 |1 |
|deciliter |dl |0.10 |
|centiliter |cl |0.01 |
|milliliter |ml |0.001 |
|microliter |µl |0.000001 |
|MASS AND WEIGHT |
|Unit |Abbreviation |Number of |
| | |Grams |
|metric ton |t |1,000,000 |
|kilogram |kg |1,000 |
|hectogram |hg |100 |
|dekagram |dag |10 |
|gram |g |1 |
|decigram |dg |0.10 |
|centigram |cg |0.01 |
|milligram |mg |0.001 |
|microgram |µg |0.000001 |
K H D M D C M
Dimensional Analysis
Example 1
.9 m to mm
561.4 m to km
20.7 cm to mm
85.6 mg to cg
3 dm to m
4 mm to hm
.8 hm to cm
3275 mL to L
6.3 kg to dag
5630 g to kg
Practice 1
13.62 m to cm
42.17 cm to m
.92 m to mm
121.4 cm to mm
7 cm to m
90 dg to kg
100 mL to L
275 cL to kL
16.1 kg to g
6301 dg to dag
4. Problem Solving with Metric Measurement
Example 1
Each piece of lead for a mechanical pencil has a thickness of .5 mm and is 60 mm long. Find the total length in centimeters of the lead in a package with 30 pieces. If the price of the package is $3.29 find the cost per centimeter for the lead.
Example 2
A cup of coffee contains 90 mg of caffeine per 8 oz cup. If Henry drinks a pint of coffee each week, how many grams of caffeine will he have consumed?
Example 3
Ethan lives at one end of Park Avenue. Brian lives at the other end of the avenue. It is 5.8 kilometers from one end of Park Avenue to the other. If Ethan walks 2.79 kilometers toward Brian's house, how many meters does he have to walk to get there?
Example 4
Aaron and Fermin wanted to have a contest to see which of their paper airplanes could fly the longest distance. Aaron's plane flew .4 dam. Noah's plane only flew 79 centimeters. How much further did Aaron's plane fly?
Practice 1
The world’s longest insect is the giant stick insect of Indonesia, measuring 33 cm. The fairy fly, the smallest insect is just .2 mm long. How much longer is the giant stick insect, in millimeters?
Practice 2
What is the length of the base of a triangle with height 1700 cm and area 10200 cm2? Give the length in km.
Practice 3
Medical Lab Technician
How many grams of NaCl would you add to a liter of water to make up a normal saline solution (.85 %, or .85 grams per 100 milliliters)?
Practice 4
Devin had the measles and had to stay inside even though he didn't feel very bad at all. He decided to make a cake to surprise his mother. The recipe said he needed four deciliters of milk. How many hectoliters of milk did he need?
8.5 Converting from English to Metric and from Metric to English
Example 1
|Kim's Cherry Chocolate Chip Bars | |
|Submitted by: LUCKY LEAF® Pie Filling |Yields: 16 servings |
"Cherry pie filling provides a sticky layer inside these chocolate and oat bars."
INGREDIENTS:
|1 (16 ounce) package of |1/3 cup sugar |
|refrigerated chocolate chip |1 egg, slightly beaten |
|cookie dough |1 teaspoon vanilla |
|3/4 cup quick-cooking oats |1 (21 ounce) can LUCKY |
|1 (8 ounce) package of cream |LEAF® Cherry Pie Filling |
|cheese, softened | |
DIRECTIONS:
|1. |Preheat oven to 350 degrees F. Lightly grease an 8x8x2-inch baking pan and set aside. In a large bowl, break the cookie |
| |dough into chunks. Knead in the oats. Press two-thirds of the mixture into the pan. Bake for 12 to 15 minutes or until crust|
| |is set. In another bowl, beat cream cheese and sugar until smooth. Stir in egg and vanilla. Carefully spread cream cheese |
| |mixture evenly over the crust. Top with LUCKY LEAF Cherry Pie Filling. Dot with the remaining cookie dough. Bake for 20 to |
| |25 minutes or until set and lightly browned. Cool completely, cover, and chill for 2 hours before serving. |
| | |
| |Practice 1 |
| | |
| | |
| |Banana Crumb Muffins |
| | |
| | |
| | |
| |Submitted by: Lisa Kreft |
| |Rated: 5 out of 5 by 3788 members |
| |Prep Time: 15 Minutes |
| |Cook Time: 20 Minutes |
| |Ready In: 35 Minutes |
| |Yields: 10 servings |
| | |
| |"A basic banana muffin is made extraordinary with a cinnamon-and-brown-sugar streusel topping." |
| |INGREDIENTS: |
| |190 g all-purpose flour |
| |5 g baking soda |
| |5 g baking powder |
| |3 g salt |
| |3 bananas, mashed |
| |150 g white sugar |
| |1 egg, lightly beaten |
| |75 g butter, melted |
| |75 g packed brown sugar |
| |15 g all-purpose flour |
| |0.3 g ground cinnamon |
| |15 g butter |
| | |
| |DIRECTIONS: |
| |1. |
| |Preheat oven to 375 degrees F (190 degrees C). Lightly grease 10 muffin cups, or line with muffin papers. |
| | |
| |2. |
| |In a large bowl, mix together 1 1/2 cups flour, baking soda, baking powder and salt. In another bowl, beat together bananas,|
| |sugar, egg and melted butter. Stir the banana mixture into the flour mixture just until moistened. Spoon batter into |
| |prepared muffin cups. |
| | |
| |3. |
| |In a small bowl, mix together brown sugar, 2 tablespoons flour and cinnamon. Cut in 1 tablespoon butter until mixture |
| |resembles coarse cornmeal. Sprinkle topping over muffins. |
| | |
| |4. |
| |Bake in preheated oven for 18 to 20 minutes, until a toothpick inserted into center of a muffin comes out clean. |
| | |
9.2 Pie charts
Constructing a pie chart
A pie chart is a way of summarizing a set of categorical data or displaying the different values of a given variable (e.g., percentage distribution). This type of chart is a circle divided into a series of segments. Each segment represents a particular category. The area of each segment is the same proportion of a circle as the category is of the total data set.
Pie charts usually show the component parts of a whole.
Example 1. Student and faculty response to the poll 'Should Avenue High School adopt student uniforms?'
[pic]
The pie chart above clearly shows that 90% of all students and faculty members at Avenue High School do not want to have a uniform dress code and that only 10% of the school population would like to adopt school uniforms. This point is clearly emphasized by its visual separation from the rest of the pie.
Pie Charts and Percentages
A pie chart uses percentages to compare information. Percentages are used because they are the easiest way to represent a whole. The whole is equal to 100%. The equation you need is as follows.
percent ÷ 100 x 360 degrees = the number of degrees
Example 2
If you spend 7 hours at school and 55 minutes of that time is spent eating lunch, then 13.1% of your school day was spent eating lunch. To present this in a pie chart, you would need to find out how many degrees represent 13.1%. This ratio works because the total percent of the pie chart represents 100% and there are 360 degrees in a circle. Therefore 47.1 degrees of the circle (13.1%) represents the time spent eating lunch.
Constructing a pie chart
A pie chart is constructed by converting the share of each component into a percentage of 360 degrees. In Figure 2, music preferences in 14- to 19-year-olds are clearly shown.
Example 3. Music preferences in young adults 14 to 19
[pic]
The pie chart quickly tells you that
• the majority of students like rap best (50%), and
• the remaining students prefer alternative (25%), rock and roll (13%), country (10%) and classical (2%).
Tip! When drawing a pie chart, ensure that the segments are ordered by size (largest to smallest) and in a clockwise direction.
In order to reproduce this pie chart, follow this step-by-step approach:
If 50% of the students liked rap, then 50% of the whole pie chart (360 degrees) would equal 180 degrees.
1. Draw a circle with your protractor.
2. Starting from the 12 o'clock position on the circle, measure an angle of 180 degrees with your protractor. The rap component should make up half of your circle. Mark this radius off with your ruler.
3. Repeat the process for each remaining music category, drawing in the radius according to its percentage of 360 degrees. The final category need not be measured as its radius is already in position.
Labeling the segments with percentage values often makes it easier to tell quickly which segment is bigger.
Example 3. Pets owned by students in Mr. Moreno’s Class
Figure 5. Favourite movie genres in Mr. Moreno’s class
9.3
Bar graphs
Vertical bar graphs
Horizontal bar graphs
Comparing several places or items
A bar graph may be either horizontal or vertical. The important point to note about bar graphs is their bar length or height—the greater their length or height, the greater their value.
Bar graphs usually present categorical and numeric variables grouped in class intervals. They consist of an axis and a series or labeled horizontal or vertical bars. The bars depict frequencies of different values of a variable or simply the different values themselves. The numbers on the x-axis of a bar graph or the y-axis of a column graph are called the scale.
When developing bar graphs, draw a vertical or horizontal bar for each category or value. The height or length of the bar will represent the number of units or observations in that category (frequency) or simply the value of the variable. Select an arbitrary but consistent width for each bar as well.
There are three types of graphs used to display time series data:
• horizontal bar graphs,
• vertical bar graphs and
• line graphs.
All three of these types of graphs work well when you need to compare values. However, in general, data comparisons are best represented vertically.
[pic]
Example 1 – Vertical bar graphs
Bar graphs should be used when you are showing segments of information. From the information given in the section on graph types, you will know that vertical bar graphs are particularly useful for time series data. The space for labels on the x-axis is small, but ideal for years, minutes, hours or months. At a glance you can see from the graph that the scales for both the x- and y-axis increase as they get farther away from the origin. Figure 1 below shows the number of police officers in Crimeville from 1993 to 2001.
Number of police officers in Crimeville, 1993 to 2001
[pic]
In Figure 1 you can see that the number of police officers decreased from 1993 to 1996, but started increasing again in 1996. The graph also makes it easy to compare or contrast the number of police officers for any combination of years. For example, in 2001 there were nine more police officers than in 1998.
The double (or group) vertical bar graph is another effective means of comparing sets of data about the same places or items. This type of vertical bar graph gives two or more pieces of information for each item on the x-axis instead of just one as in Figure 1. This allows you to make direct comparisons on the same graph by age group, sex, race, or anything else you wish to compare. However, if a group vertical bar graph has too many sets of data, the graph becomes cluttered and it can be confusing to read.
Figure 2, a double vertical bar graph, compares two sets of data: the numbers of boys and girls using the Internet at Redwood Secondary School from 1995 to 2002. One bar represents the number of boys who use the Internet and the other bar represents the girls.
Example 2. Internet use at Redwood Secondary School, by sex, 1995 to 2002
[pic]
One disadvantage of vertical bar graphs, however, is that they lack space for text labelling at the foot of each bar. When category labels in the graph are too long, you might find a horizontal bar graph better for displaying information.
[pic]
Practice 1
Practice 2
Example 2 – Horizontal bar graphs
The horizontal bar graph uses the y-axis (vertical line) for labelling. There is more room to fit text labels for categorical variables on the y-axis.
Figure 3 shows the number of students at Diversity College who are immigrants by their last country of permanent residence. The graph shows that 100 students immigrated from China, 380 from France, and 260 from Brazil.
A horizontal bar graph has been used to show a comparison of these data. This graph is the best method to present this type of information because the labels (in this case, the countries' names) are too long to appear clearly on the x-axis.
Example 3. Number of students at Diversity College who are immigrants, by last country of permanent residence
[pic]
Double or Group Horizontal Graphs
A double or group horizontal bar graph is similar to a double or group vertical bar graph, and it would be used when the labels are too long to fit on the x-axis.
In Figure 4, more than one piece of information is being delivered to the audience: drug use by 15-year-old boys is being compared with drug use by 15-year-old girls at Jamie's school. Having both pieces of information on the same graph makes it easier to compare. The graph indicates that 32% of boys and 29% of girls have used hashish or marijuana, and 3% of boys and 1% of girls have tried Lysergic acid diethylamide. The graph also shows that the same percent of boys and girls (4%) have used cocaine.
Example 4. Drug use by 15-year-old students in Jamie's school, by gender
[pic]
[pic]
Example 3 – Comparing several places or items
Create a double bar graph with the following categories.
• popcorn
• chips
• chocolate bars
• fruit
• candy
• vegetables.
Line graphs
Comparing two related variables
Multiple line graphs
A line graph is a visual comparison of how two variables—shown on the x- and y-axes—are related or vary with each other. It shows related information by drawing a continuous line between all the points on a grid.
Line graphs compare two variables: one is plotted along the x-axis (horizontal) and the other along the y-axis (vertical). The y-axis in a line graph usually indicates quantity (e.g., dollars, litres) or percentage, while the horizontal x-axis often measures units of time. As a result, the line graph is often viewed as a time series graph. For example, if you wanted to graph the height of a baseball pitch over time, you could measure the time variable along the x-axis, and the height along the y-axis.
In summary, line graphs
• show specific values of data well
• reveal trends and relationships between data
• compare trends in different groups of a variable
Example 4 – Plotting a trend over time
Figure 1 shows one obvious trend, the fluctuation in the labour force from January to July. The number of students at Andrew's high school who are members of the labour force is scaled using intervals on the y-axis, while the time variable is plotted on the x-axis.
The number of students participating in the labour force was 252 in January, 252 in February, 255 in March, 256 in April, 282 in May, 290 in June and 319 in July. When examined further, the graph indicates that the labour force participation of these students was at a plateau for the first four months covered by the graph (January to April), and for the next three months (May to July) the number increased steadily.
Labour force participation in Andrew's high school
[pic]
[pic]
Example 5 – Comparing two related variables
Figure 2 is a single line graph comparing two items; in this instance, time is not a factor. The graph compares the number of dollars donated by the age of the donors. According to the trend in the graph, the older the donor, the more money he or she donates. The 17-year-old donors donate, on average, $84. For the 19-year-olds, the average donation increased by $26 to make the average donation of that age group $110.
Figure 2. Average number of dollars donated at Evergreen High School, by age of donor
[pic]
Practice 4 –
Example 4 – Multiple line graphs
A multiple line graph can effectively compare similar items over the same period of time (Figure 5).
Figure 5. Cell phone use in Anytowne, 1996 to 2002
[pic]
Figure 5 is an example of a very good graph. The message is clearly stated in the title, and each of the line graphs is properly labelled. It is easy to see from this graph that the total cell phone use has been rising steadily since 1996, except for a two-year period (1999 and 2000) where the numbers drop slightly. The pattern of use for women and men seems to be quite similar with very small discrepancies between them.
Practice 5
9.4/9.5
The Rectangular Coordinate System
After completing this tutorial, you should be able to:
1. Plot points on a rectangular coordinate system.
2. Identify what quadrant or axis a point lies on.
3. Tell if an ordered pair is a solution of an equation in two variables or not.
Complete an ordered pair that has one missing value.
Learning Objectives
This section covers the basic ideas of graphing: rectangular coordinate system, ordered pairs and solutions to equations in two variables. Graphs are important in giving a visual representation of the correlation between two variables. Even though in this section we are going to look at it generically, using a general x and y variable, you can use two-dimensional graphs for any application where you have two variables. For example, you may have a cost function that is dependent on the quantity of items made. If you needed to show your boss visually the correlation of the quantity with the cost, you could do that on a two-dimensional graph.
Rectangular Coordinate System
The following is the rectangular coordinate system:
[pic]
It is made up of two number lines:
1. The horizontal number line is the x- axis.
2. The vertical number line is the y- axis.
The origin is where the two intersect. This is where both number lines are 0.
It is split into four quadrants which are marked on this graph with Roman numerals.
Each point on the graph is associated with an ordered pair. When dealing with an x, y graph, the x coordinate is always first and the y coordinate is always second in the ordered pair (x, y). It is a solution to an equation in two variables. Even though there are two values in the ordered pair, be careful that it associates to ONLY ONE point on the graph, the point lines up with both the x value of the ordered pair (x-axis) and the y value of the ordered pair (y-axis).
Example 1: Plot the ordered pairs and name the quadrant or axis in which the point lies. A(2, 3), B(-1, 2), C(-3, -4), D(2, 0), and E(0, 5). Remember that each ordered pair associates with only one point on the graph. Just line up the x value and then the y value to get your location.
[pic]
A(2, 3) lies in quadrant I.
B(-1, 2) lies in quadrant II.
C(-3, -4) lies in quadrant III.
D(2, 0) lies on the x-axis.
E(0, 5) lies on the y-axis.
Example 2: Find the x- and y- coordinates of the following labeled points
[pic]
Remember that each ordered pair associates with only one point on the graph. Just line up the x value and then the y value to get your ordered pair.
Solutions of Equations in Two Variables
The solutions to equations in two variables consist of two values that when substituted into their corresponding variables in the equation, make a true statement.
In other words, if your equation has two variables x and y, and you plug in a value for x and its corresponding value for y and the mathematical statement comes out to be true, then the x and y value that you plugged in would together be a solution to the equation.
Equations in two variables can have more than one solution.
We usually write the solutions to equations in two variables in ordered pairs.
Example 3: Determine whether each ordered pair is a solution of the given equation. y = 5x - 7; (2, 3), (1, 5), (-1, -12)
Let’s start with the ordered pair (2, 3).
Which number is the x value and which one is the y value? If you said x = 2 and y = 3, you are correct!
Let’s plug (2, 3) into the equation and see what we get:
Let’s plug (1, 5) into the equation and see what we get:
Let’s plug (-1, -12) into the equation and see what we get:
Note that you were only given three ordered pairs to check, however, there are an infinite number of solutions to this equation. It would very cumbersome to find them all.
Example 4: Determine whether each ordered pair is a solution of the given equation. x = 3; (3, 5), (2, 3), (3, 4) This equation looks a little different than the one on example 3. In this equation, we only have an x value to plug in. So as long as the x value is 3, then we have a solution to the equation. It doesn’t matter what y’s value is.
Let’s start with the ordered pair (3, 5).
Which number is the x value and which one is the y value? If you said x = 3 and y = 5, you are correct!
Let’s plug (3, 5) into the equation and see what we get:
Let’s plug (2, 3) into the equation and see what we get:
Let’s plug (3, 4) into the equation and see what we get:
Note that you were only given three ordered pairs to check, however, there are an infinite number of solutions to this equation. It would very cumbersome to find them all.
Finding the Corresponding Value in an Ordered Pair Given One Variable’s Value
Again, the solutions to equations in two variables consist of two values that when substituted into their corresponding variables in the equation, make a true statement.
Sometimes you are given a value of one of the variables and you need to find the corresponding value of the other variable. The steps involved in doing that are:
Step 1: Plug given value for variable into equation.
Step 2: Solve the equation for the remaining variable.
Example 5: Complete each ordered pair so that it is a solution of the equation [pic]. (1, ) and ( , -1).
In the ordered pair (1, ), is 1 that is given the x or the y value?
If you said x, you are correct.
Plugging in 1 for x into the given equation and solving for y we get:
Plugging in -1 for y into the given equation and solving for x we get:
Example 6: Complete the table of values for the equation [pic].
|x |y |
| |0 |
| |-1 |
| |1 |
The only difference between this one and example 5 above is that we are using a table to match up our values of our variables instead of writing it in an ordered pair. The concept is still the same, we need to find the corresponding values of our variables that are solutions to the given equation.
Plugging in 0 for y into the given equation and solving for x we get:
Plugging in -1 for y into the given equation and solving for x we get:
Filling in the table we get:
|x |y |
|-1/2 |0 |
|-1/2 |-1 |
|-1/2 |1 |
Practice Problems
Practice 1:
Plot each point and name the quadrant or axis in which the point lies.
1a. A(3, 1), B(-2, -1/2), C(2, -2), and D(0,1)
Practice 2:
Find the x- and y- coordinates of the following labeled points.
[pic]
Practice Problems 3a - 3b:
Determine if each ordered pair is a solution of the given equation.
3a. y = 4x - 10 ; (0, -10), (1, -14), (-1, -14)
3b. y = -5 ; (2, -5), (-5, 1), (0, -5)
Practice 4:
Complete each ordered pair so that it is a solution of the equation [pic].
(0, ) and ( , 1).
Practice 5:
Complete the table of values for the equation [pic].
|x |y |
|0 | |
|-1 | |
|1 | |
-----------------------
Rounding integers
Step 1 Locate the place to which the number is to be rounded. Draw a line under that place.
Step 2 Look only at the next digit to the right5 of the one you underlined. If the next digit is 5 or more, increase the underlined digit by 1. If the next digit is 4 or less, do not change the digit in the underlined place.
Step 3 Change all digits to the right of the underlined place to zeros.
1. Parentheses and Brackets -- Simplify the inside of parentheses and brackets before you deal with the exponent (if any) of the set of parentheses or remove the parentheses.
2. Exponents -- Simplify the exponent of a number or of a set of parentheses before you multiply, divide, add, or subtract it.
3. Multiplication and Division'8Mm§0 1 — š › § [
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*”•üñæüßƱ—±—±„ß|ß|ß|ß|ßtßißi^ißtßtßtß|j÷
hO;ØhkxU -- Simplify multiplication and division in the order that they appear from left to right.
4. Addition and Subtraction -- Simplify addition and subtraction in the order that they appear from left to right.
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