Gavilan College



SS 6.1 Percent Notation

A percent is a part of a hundred. We can think of it in 3 different manners:

As a ratio of n parts to 100: n% = n

100

As a fractional part of 100: n% = n x 1

100. (which is really the same as the

above ratio)

As a decimal written in 100ths: n% = n x 0.01 (which is reading the fraction

above and writing it using

decimals)

All these are called equivalent forms, because they indicate the same thing

Example: Express each of the following using the above 3 forms:

a) 25 %

b) 72.65%

c) 275%

We can use any 3 of the above forms to change a percentage to a decimal. I believe, however, that division by 100 is the easiest, because if you recall that when dividing by a factor of 10, we simply move the decimal to the left the same number of places as the number of zeros! All this requires is remembering what the definition of a percentage is and then knowing where the decimal is located in the percentage. By the way, the decimal is always located between the whole number and the percentage sign if it is not explicitly written (i.e. 36% is 36.0%) This whole concept leads to the shortcut:

Percentage to a Decimal ( Move the decimal two places to the left

Example: Express each of the following as a decimal:

d) 25 %

e) 72.65%

f) 275%

When converting a decimal to a percentage we must again think of the definition of a percentage. If we can write the decimal in 100ths, we are then able to write the decimal in percentage form. Let’s get some practice.

Example: Write each decimal as some hundredths

a) 0.38

b) 0.015

c) 0.285

Now, by using what we learned at the beginning, that a percentage is a part of 100, we should be able to write each of the above as percentages with no problem. Let’s try.

Example: Write each decimal as a percentage, relying on the above example

d) 0.38

e) 0.015

f) 0.285

Here is a shortcut for converting a decimal to a percentage:

Decimal to a Percentage ( Move the decimal two places to the right

HW

p. 307-308 #2-64 even

SS 6.2 Percent Notation & Fractional Notation

Converting to percentages from fractions is just a matter of converting to decimals, which is a skill that we acquired in chapter 5.

Steps to Change Fraction to Percentage

Step 1 Convert fraction to a decimal by dividing numerator by denominator

Step 2 Convert decimal to percentage by moving decimal 2 places to right

Note 1: If repeating decimal, you may use the fractional notation for the

repeating portion as well as the bar notation.

Note 2: For any decimal portion that can be represented by a fraction, the fraction

can be used instead of the decimal portion, just as if you were writing a

mixed number.

Step 3 Attach a percentage sign

Example: Convert each of the following to a percentage

a) ½

b) ¼

c) 1/3

d) 5/6

e) 9/10

f) 3/5

At this time I would like to remind you of the fraction to decimal conversions that you should have been attempting to memorize. It is not a stretch at all to also attach the percentage conversion to this memorization. You could expect to see such a conversion table on the final, and no a copy of the table is not permissible for the final’s cheat sheet!

|Fraction |Decimal |Percentage |

|½ |0.5 |50% |

|1/3 |0.333( |33.33 % or 331/3 % |

|2/3 |0.666( |66.66( % or 662/3% |

|¼ |0.25 |25% |

|¾ |0.75 |75% |

|1/5 |0.2 |20% |

|2/5 |0.4 |40% |

|3/5 |0.6 |60% |

|4/5 |0.8 |80% |

|1/6 |0.1666( |16.66(% or 16 2/3% |

|5/6 |0.8333( |83.33(% or 83 1/3% |

|1/8 |0.125 |12.5% or 12 ½% |

|3/8 |0.375 |37.5% or 37 ½% |

|5/8 |0.625 |62.5% or 62 ½% |

|7/8 |0.875 |87.5% or 87 ½% |

|1/9 |0.111( |11.11(% or 11 1/9% |

|2/9 |0.222( |22.22(% or 22 2/9% |

|4/9 |0.444( |44.44(% or 44 4/9% |

|5/9 |0.555( |55.55(% or 55 5/9% |

|7/9 |0.777( |77.77(% or 77 7/9% |

|8/9 |0.888( |88.88(% or 88 8/9% |

We will also need to convert percentages back to fractions. It is easiest to use the definition of a percentage and then reduce. Recall that a percentage is a part of 100.

Steps for Converting Percentages to Fractions

Step 1 Convert any fractions in the percentage to decimals (See note about repeating decimals below)

Step 2 Use the definition of a percentage to write it as a fraction

Step 3 Multiply by factors of 10 to remove decimals from numerator

Step 4 Reduce fraction to lowest terms

Note: For any decimal or percentage with repeating decimals there is no way to convert to a fraction unless you have the decimal to fraction conversion memorized!

Example: Write each of the following percentages as fractions in their lowest

terms

a) 75%

b) 82%

c) 40%

d) 83 1/3%

e) 66 2/3%

HW

p. 313-316 #2-64 even & #65-68 all & #70-82 even

SS 6.3 Solving Percent Problems Using Equations

We will be seeing percentage problems in word problems. The following will be the forms that we will be seeing and their corresponding algebraic equation.

|Forms |Algebra Equation |

|Percent of a total is what part | (%)(total) = x |

|What percent of a total is the part | (x%)(total) = part |

|Percent of what total is the part | (%)(x) = part |

In each case we will use some very basic rules to set the equation up so that it is the same each time, but so that the variable is all that changes. I will want you to always start by writing the problem in the below form and then using the key words of and is and percent to fill in the blanks. Since of translates to multiplication and is translates to an equal sign, this gives us the perfect set up every time!!

Percentages

|Rewrite all problems into this form |Make Algebraic Equation |

|------% of -----(total) is --------(part) |Percent Missing 72x = 36 |

| |Note: x will be a decimal, convert to % |

| |Total Missing 0.5x = 36 |

| |Note: Change percent to a decimal to solve |

| |Part Missing 0.5(72) = x |

| |Note: Change percent to a decimal to solve |

Note: If the part is greater than or equal to the total, then the percentage will be greater than or equal to 100%

Example: For each of the following write the form, fill in the blanks, write

the corresponding equation and then solve the problem as we

learned in the introduction to chapter 5.

a) 35% of 18 is what?

b) 16% of 10 is what?

_____% of 10 is 6?

c) 17% of _______ is 12?

d) 81% of _______ is 62?

HW

p. 321-322 #2-42 even

Note: We will skip section 6.4 as I do not teach percentages this way!

SS 6.5 Applications of Percent & 6.6 Consumer Applications

The key to doing word problems with percentages is to remember the form and to put each problem in that form! % * whole = part

Example: I am 20% shorter than my dad. My dad is 75 inches tall.

How tall am I?

Example: Tax is 7 ¼ % in some areas of California. What amount of

tax will be paid on an item that costs $12.97?

Example: The price on the item was $7.25, but I had to pay $7.85 for

the item. What was the tax?

Example: Bonnie and Clyde rob banks for a living. In each robbery,

Bonnie spends 7 hours planning and 2 hours in the actual robbery. Clyde on the other hand spends 10 hours planning and 2 hours in the robbery. What percent of the total time does Bonnie help? If they rob a band and get $17,761, how much should Bonnie get? Clyde?

Example: Tom and Huck spent 18% of the time that they told Tom's

aunt it took them to white wash the fence fishing. If they

spent 7 hours fishing, how long, to the nearest hour, did

they tell Aunty that it took them to paint the fence?

Example: Becky earns a commission by selling merchandise at her job. If

she sells $2000 worth of merchandise, and receives $250 in commission, what is her rate of commission?

Example: John makes 5% commission for each selling video tapes promoting

learning at his school. If John sells 35 tapes and each tape is $25, how much commission will he receive?

Example: A dress is discounted 15%. The original price of the dress is $35.

What is the discount ? What will the dress cost before tax? If tax

is 8 ¼ % what will the dress cost after tax?

Precent increase or decrease problems always involve the original price. They are usually 2 step problems, where you must first find the amount of increase or decrease, and then find the percentage of the original price.

% of original = increase/decrease

Example: Hallahan’s Construction Company increased their estimate for building a new house from $95,500 to $110,000. Find the percent increase.

Example: A pair of shoes originally costs $56 and they are now marked $42,

what is the discount being offered? (Although this problem does

not appear to be a decrease problem, it really is, as a discount is a decrease!)

HW

p. 335-338 #4-40 mult. of 4 & #41 (This should show you why discounts on discounts don’t add up to the total of the perctages!)

p. 345-348 #4-52 mult. of 4 & #57 (Again percentages don’t add!)

SS 6.7 Consumer Applications: Interest

Simple interest (interest=I) is calculated by multiplying the amount of money invested (principle=P) by the rate of investment (rate=R) by the amount of time in years (time=T).

I = P ( R ( T

It is important to remember that simple interest is computed in terms of years for time and therefore all times must be put in terms of a year. Recall that a year has 365 days or 12 months or 52 weeks.

Example: I have $2000 dollars to invest. I wish to invest it for one year in an account that pays 8% interest. How much will I earn in the year?

Example: I have $1500 to invest for 6 months. If it is invested in an account that pays 10% annually, how much will I make in 6 months?

Example: If I invest $25,000 in an account that pays 15% interest annually, but I only invest it for 25 days, how much interest will I earn?

Example: I invest $12,000 in an account that pays 12 ½ % interest annually.

After 24 weeks I withdraw my money. Assuming that there is no penalty, how much money will I have?

There is another kind of interest called compound interest, but at this time I do not wish to cover this topic. If time allows at the end of the class I will return to this topic and give it the time that it deserves. This type of interest is important, as it is the type used by most lending institutions and banks to compute the interest that you earn or that you are charged.

HW

p. 353-356 #2-12 even & #26-36 even

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