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SECTION 1.1: OPERATIONS WITH REAL NUMBERS
Skill Review and Development
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Review of Operations With Signed Numbers
Adding Signed Numbers:
positive + positive - Add the absolute values. The sign of the answer is positive.
negative + negative
Add the absolute values. The sign of the answer is negative.
positive + negative or negative + positive
Subtract the absolute values (smallest from largest). The sign of the answer is the sign of the largest numeral.
Link to practice worksheet on adding integers:
Subtracting Signed Numbers: Some problems can be easily done directly. Others are easier to do if you do a little reformatting first. When you reformat, change the subtraction to the sum of the opposite, and then follow the rules for addition.
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Link to practice worksheet on subtracting integers:
Multiplying and Dividing Signed Numbers:
Multiply/Divide the absolute values. If the signs of the values are:
• Two Positives - The answer is positive.
• Two Negatives - The answer is positive.
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• One Positive and One Negative - The answer is negative.
Link to practice worksheet on multiplying integers:
Link to practice worksheet on dividing integers:
Review of Operations With Fractions
|Operation |Rule |Example |
|Addition/Subtraction | |[pic] |
|Same Denominators |Add/Subtract the numerators. Denominator stays | |
| |the same. Simplify if possible. | |
|Addition/Subtraction | |[pic] |
|Different Denominators |Convert the fractions to LCD, then Add/Subtract | |
| |the numerators. Denominator stays the same. | |
| |Simplify if possible. | |
|Multiplication | |[pic] |
| |Multiply the numerators. |Reduce before or after multiplying. |
| |Multiply the denominators. | |
| |Simplify if possible. | |
|aDivision | |[pic] |
| |Change to multiplication of the inverse. Multiply| |
| |the numerators. Multiply the denominators. | |
| |Simplify if possible. | |
|Fractions with Signs (+, -) | | |
| |Use the rules above in addition to the sign rules | |
| |on the previous page. | |
Link to practice worksheet on adding fractions with like denominators.
Link to practice worksheet on adding fractions with unlike denominators.
Link to practice worksheet on adding/subtracting mixed numbers.
Link to practice worksheet on multiplying and dividing fractions.
Using a factor tree to find the LCD – see power point
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Review of Operations with Mixed Numbers When adding and subtracting, it is usually a good idea to work the whole numbers and fractions separately.
Example 1: Find the space between each fence slat if the spaces are all equal.
Review of Operations with Decimals
|Operation |Rule |Example |
|Addition/Subtraction |Write the problem vertically lining up the decimals |14.68 – 3.901 |
| |points. Add/Subtract the values and bring down the | |
| |decimal. |14.68 |
| | |- 3.901 |
| | |10.779 |
|Multiplication |Right justify the numbers. Multiply the numbers, insert |14.68 x 3.901 |
| |the decimal the total number of places moving from right | |
| |to left. |14.68 2 places |
| | |x 3.901 3 places |
| | |1468 |
| | |132120 |
| | |4404___ |
| | |57.26668 5 places |
|Division |Move the decimal in the divisor to the far right. Move | |
| |the decimal in the dividend the same number of places. |___10 .1_ |
| |Bring the decimal up in the quotient. Divide the |2.43 ) 24.600 |
| |numbers. Directions for rounding would probably apply. |243 |
| | |300 |
| | |243 |
| | |57 |
Link to practice worksheets on decimals
Example 2:
Find the new dimensions for the part below if it is scaled up by a multiple of 6.
Find the new dimensions for the part below if it is scaled down by a factor of 4.
Example 3:
Calculate the overall width and height of plate steel necessary to lay out 3 rows and 5 columns of the part below if they are spaces ¾ in apart to allow for the cutting tool. Assume there is no allowance between the part and the edge of the plate (the part goes all the way to the edge). All dimensions are in inches. Round your answer to the nearest 16th of an inch. Hint: Drawing a picture will probably help.
Example 4:
Calculate dimensions X & Y so that all 8 gaps (between the circles and between the circle and edge of the plate) are the same size. Give your answer to the nearest 16th of an inch.
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From -
Instructions
Move the decimal to the right or the left three places at a time while keeping track of the total number of decimal places moved. If the number is very large, move it to the left; stop before the number would become less than one. If the number is very small, move it to the right until the number is greater than one and less than one thousand.
Rewrite the number with the decimal place in the new location. If the number is not greater than one and less than one thousand, Step 1 has been performed incorrectly.
Add to the end of the number "x 10" with the proper exponent. The exponent will be equivalent to the the number of places the decimal was moved. It will be negative for small numbers and positive for large numbers. The exponent should always be a multiple of 3 or -3.
Homework: You are assigned problems 1,2,3,5,6,7,8,9,11,12,13,14,17,19.
Show all of your work in an organized manner. Label your work so it can be easily followed. If you draw pictures, do so with care. Be careful with your units and label what you are doing. You will be graded on your presentation.
We will do problems 15 and 16 together in class; they are in your lecture notes. You are expected to turn these in (completed) separately from the
homework from section 1.1.
Here are some additional links you may find helpful:
Adding and Subtracting Positive and Negative Numbers -
Scientific Notation -
Decimals -
Fractions -
Section 1.1
1. 16 [pic]
2. A = 12.672 in B = 3.934 in
C = 7.134 in D = 2.67 in
3. S = 1 [pic]
H = [pic]
4. R = 20 [pic]
Width between centers = 58 [pic]
5. Width = 76 [pic]
Height = 31 [pic]
6. a) [pic]
b) Yes there is [pic] to spare, although this is not much when divided among 9 spaces.
7. A = 3 [pic]
B = 5 [pic]
C = 11 [pic]
8. Total Rise = 68 [pic]
Total Run = 104 [pic]
9. A = 109 [pic]
B = 122 [pic], this design will produce 84 parts.
10. A = 102.506 in
B = 75.39 in, this design will produce 96 parts.
11. Area = 89 [pic] square inches
Perimeter = 39 [pic]
12. Width = 2 [pic]
Height = 1 [pic]
13. A = 2 [pic]
B = 3 [pic],
14. 32 [pic]
15. a) 6,320,000,000 V
b) .0000000098 A
c) .00000047 F
d) 14,200,000 W
|4.8 x 105 |480 x 103 |480 kHz |
|4.3 x 10-4 |430 x 10-6 |430 µV |
|7.5 x 103 |7.5x 103 |7.5 kW |
|4.2 x 10-8 |42 x 10-9 |42 nF |
|6.2 x 107 |62 x 106 |62MH |
16.
|maximum |minimum |
|17.351 in |17.283 in |
|18.27 cm |17.97 cm |
|26[pic] in |26[pic] in |
|320.06 cm |319.94 cm |
17.
|Difference |
|- [pic] in |
|- .05 mm |
|+ .041 in |
18.
19. 27 Rows, 41 Columns, this design will produce 1107 parts.
20. 32 Rows, 35 Columns, this design will produce 1120 parts.
21.
|Design Length|Tolerance |Fraction |Max |Min |
|3.7 in |±[pic] in |3[pic] in |3[pic] in |3[pic] in |
|36.62 in |±[pic] in |36[pic] in |36[pic] in |36[pic] in |
|80.5 in |±[pic] in |80[pic] in |80[pic] in |80[pic] in |
|100.29 in |±[pic] in |100[pic] in |100[pic] in |100[pic] in |
|Design Dimension|Tolerance |Maximum |Minimum |
| | |Dimension |Dimension |
|[pic] in |±[pic] in |[pic] in |[pic] in |
|[pic] in |±[pic] in |[pic] in |[pic] in |
|[pic] in |±[pic] in |[pic] in |[pic] in |
|[pic] in |±[pic] in |[pic] in |[pic] in |
|[pic] in |±[pic] in |[pic] in |[pic] in |
|[pic] in |±[pic] in |[pic] in |[pic] in |
22.
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TAKE THE SHORTCUT When adding and subtracting integers (positive and negative numbers) there is a much easier approach.
1. Get rid of all the double signs.
+ + becomes + - - becomes +
+ - becomes – - + becomes –
2. Now look at each integer separately and combine them as if they were numbers tossed on a table. You will still need to follow the rules for addition, but this is a much easier approach.
Link to practice worksheet on adding integers:
TAKE THE SHORTCUT
1. If the problem is multiplication, just multiply.
If the problem is division, just divide.
2. If the signs are the same, the answer is positive.
If the signs are different, the answer is negative.
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