Middle Grades Math Pre-Algebra

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Middle Grades MathPre-Algebra 

• Chapter 6

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Chapter 6 Overview(Prerequisite Skills)

• write fractions in lowest terms
• solve equations using the multiplication property
  
• write fractions as decimals
• write decimals as fractions or a mixed number in
  simplest form
• write percents using the symbol

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Chapter 6 Overview(Chapter Objectives)

• learn how to find and use ratios and unit rates
• learn how to write and solve proportions
• learn how to find and use percents
• learn how to solve a problem by making a table

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Chapter 6 Key Terms
commission complement cross products discount
event indirect measurement markup odds
outcome
percent percent of change probability
proportion rate ratio scale drawing similar
figures unit rate
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Ratio
A ratio is a comparison of two quantities. The
ratio of solid squares to white squares is 2 to
5. The ratio of solid squares to total squares
is 2 to 7. Ratios may be written in three
different ways
2 to 7 27
All three written ways are read the same Two to
Seven.
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Rates and Unit Rates
A rate is a ratio that compares quantities in
different units.
The forward slash (/) is used to write rates
kilometers per hour km/h feet per second
ft/s miles per gallon mi/gal miles per hour
mi/hr
A unit rate compares a quantity with one quantity
. Example 1.20 for 24 oz. Find the cost per
ounce.
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Proportions
Two equal ratios form a proportion. In a
proportion, cross products are equal. To solve a
proportion, you can use cross products to find
the missing number.
A proportion can be used to find a unit rate.
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Converting between Measurement Systems
A ratio can be used to convert between
measurements systems.
Customary Units and Metric Units Conversion
Factor 1 in. 2.54 cm. 1 mi. 1.61
km. 1.06 qt. 1 L 1 oz. 28.4 g 2.20 lb.
1 kg
1 in 2.54 cm 2.54 cm 1 in 1 mi 1.61 km 1.61
km 1 mi 1.06 qt 1 L 1 L 1.06 qt 1 oz 28.4
g 28.4 g 1 oz 2.20 lb 1 kg 1 kg 2.20 lb
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Solving Proportions
n
800 2
Kurt gets an 800 paycheck every 2 weeks. At this
rate, how much does he earn in 1 year (52 weeks)?

52
2n 800 x 52 2n 41600 n 20800
n
3 100
During a test, 3 100 cars selected at random
were found to be defective. Use this ratio to
estimate the number of defective cars if 1,500
cars were tested.

1500
100n 3 x 1500 100n 4500 n 45
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Many problems are solved using proportions.
In 3 hours, Alex can walk 14 miles. How long will
it take him to walk 25 miles? Six oranges cost
1.00. What do 28 oranges cost? A girl 5.5 ft
tall casts a shadow 8.25 ft long. She is next to
a tree that has an 18 ft shadow. How tall is the
tree? The scale of a model airplane is 1in
3.5ft. The wing span of the plane is 40 ft. What
is the wing span of the model?
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Similar Figures
Similar figures are the same shape but not
necessarily the same size. Congruent figures are
the same size and the same shape. Proportions can
be used to determine if two shapes are similar.
Proportions can be used to find the missing
length of a side in two similar figures.
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Indirect Measurement
You can use similar figures to compute distances
that are difficult to measure directly. This
method is called indirect measurement.
n
6 10
A man who is 6 ft tall casts a shadow that is 10
feet long. How tall is a nearby tree that casts a
shadow that is 35 feet long? Answer 21 feet

35
3
7
6 x 35 10
21
2
1
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Scale Drawings
A scale drawing is an enlarged or reduced drawing
that is similar to an actual object or place. The
ratio of the distance in the drawing to the
corresponding actual distance is the scale of the
drawing.
6.5
The scale on a drawing is 1 in 25 ft. How tall
is a building that is 6.5 in. tall in the
drawing? Answer 162.5 feet
1 25

n
1n 6.5 x 25 162.5
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Probability
Probability and odds are two DIFFERENT ways of
measuring the chances of an event
occurring. PROBABILITY compares the number of
favorable outcomes to the total number of
possible outcomes. All probabilities range from 0
to 1. The complement of an event is the opposite
of that event. Probability is expressed as a
ratio or a percent.
Number of favorable outcomes Number of total
outcomes
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Odds
Odds describe the likelihood of an event.
Odds in favor of an event compare the number of
ways an event can occur to the number of ways it
cannot occur.
Number of favorable outcomes Number of
unfavorable outcomes
Odds against an event compare the number of ways
an event cannot occur to the number of ways it
can occur.
Number of unfavorable outcomes Number of
favorable outcomes
Odds are commonly expressed as a ratio.
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Fractions, Decimals, and Percents
To convert a fraction to a percent, write an
equivalent fraction with a denominator of 100.
If the fraction can not be written with a
denominator of 100, then convert the fraction to
a decimal by dividing the numerator by the
denominator. To convert a decimal to a percent,
multiply the decimal by 100 (move the decimal
point two places to the right)and write the
percent symbol. To multiply any number by 100,
move the decimal point 2 places to the right.
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Common Fraction/Percent Equivalents
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Percent and Proportion
Percent problems can be solved using proportions.
A percent is a ratio that makes a comparison to
100.
45 means 45 out of 100 or
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To find a percent of a number, you can use a
proportion. Ex. What is 38 of 45. To find
what percent one number is of another number, you
can use a proportion. Ex. 72 is what percent of
81
(Hint The whole usually follows the word of.)
20
To find the whole (base), you can use a
proportion. (Hint The whole usually follows the
word of.) Ex. 15 is 75 of what number.
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Percent and Equations
To find a percent of a number, you can use an
equation. Rewrite the problem as a number
sentence. Ex. What is 38 of 45. n .38 x 45
n 17.1 To find what percent one number is of
another number, you can also use an
equation. Rewrite the problem as a number
sentence. Ex. 72 is what percent of 81 72 n x
81 n 72/81 .889 88.9
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To find the whole (base), you can use an
equation. Rewrite the problem as a number
sentence. (Hint The whole usually follows the
word of.) Ex. 15 is 75 of what number. 15 .75n
n 15/.75 20
Commission
Some sales jobs pay an amount based on how
muchyou sell. This amount is called a
commission. Ex. A store clerk makes 5 commission
on her sales. How much does she make on 1235. of
sales? 5 of 1235 .05 x 1236 61.75
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Percent of Increase
To find the percent of increase, compare the
amount of increase to the original
amount. Example Last year the Smith family
paid 900 a month for rent. This year the rent
increased to 972 a month. By what percent did
their rent increase from last year to this year?
900n 7200 n 8
8
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Percent of Decrease
To find the percent of decrease, compare the
amount of decrease to the original
amount. Example The original price of a pair
of jeans was 30.00. The sale price is 24.00.
What is the percent of discount? (Discount is a
decrease in price)
20
30n 600 n 20
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Sales Tax
Example The tax on a jacket costing 35. is 6½.
What is the tax? What is the total cost of the
jacket?
100n 227.5 n 2.275
The sales tax is 2.28. The total cost is 35.
2.28 or 37.28
Be careful to read problems carefully. The
question may ask for the amount of tax or for the
total cost. Money must always be rounded to the
nearest cent (hundredth), and money must always
be rounded up to the next cent.
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Percents Greater than 100 or less than 1
Text
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Estimating Percents
To estimate percents, replace the percent with
an equivalent fraction. Then round the other
number to an integer that is compatible with
the fraction. Estimate 25 of 64.95 Replace 25
with ¼ Round 64.95 to 64 since ¼ is compatible
with 64. ¼ of 64 ¼ x 64 16