7th Grade Pre-algebra

Chapter 6 Notes

6.1 Ratios and Rates

Vocabulary

Ratio: a comparison of two numbers by division.

Rate: a ratio of two measurements having different kinds of units.

Unit Rate: when a rate is simplified so that it has a denominator of 1. Ex. mph

Writing Ratios as Fractions

Express the ratio 9 goldfish out of 12 goldfish as a fraction in simplest form.

Express the ratio of girls to boys in this class as a fraction in simplest form

Express the ratio of boys to girls in this class as a fraction in simplest form.

12

9

boys

girls

18

15

girls

boys

15

18

Write Ratios as Fractions

Units of measure must be the same in order to simplify!!!

Express 3 feet to 16 inches as a fraction in simplest form.

4

9

16

36

16

3

in

in

in

ft

Find Unit Rates

Since we want the cost of 1 soda, put the soda on the

bottom of our fraction

Hint: Money will almost ALWAYS be the numerator of the fraction!!

canper

cans

129.0$

1

29.

6

6

6

74.1$

Converting Rates

To convert a rate such as miles per hour to feet per second, you can use dimensional analysis. Carry the units throughout the computation.

A Grizzly bear can run 30 miles in 1 hour. How many feet is this per second?

ond

feet

hr

mi

sec?

?

1

30

ond

feet

hr

mi

sec600,3

400,158

1

30

There are 5280 ft in a mile. 30 x 5280 = 158,400

There are 3600 second in an hour (60min x 60sec)

ond

feet

sec600,3

400,158

sec1

44 ft

Simplify to get 1 in the denominator by dividing both the numerator and denominator by 3600

You Try…

You Try…

Problem Solving

6.2 Using Proportions

Vocabulary

Proportion: a statement of equality of two ratios

Cross Products: when you multiply the numerator of the first ratio by the denominator of the second ratios and compare that product to the product of the denominator of the first ratios and the numerator of the second ratio.

168

24

84

12 )24(84)168(12 20162016

Use Cross Products to identify Proportions

Pairs of ratios are only proportional if their cross products are equal.

2x20 = 408x5 = 40

= equal

3x24 = 724x18 = 72

= equal

2.5x6 = 15.02x7.5 = 15.0

= equal

Solving Proportions

We can solve for the unknown value in a proportion by using cross multiplication.

35x = 385

1135

385

35

35

x

x

Solving Proportions

1m = 216 m = 216 2x = 135

x = 67.5

Problem Solving

18

362122

3

x

x

x

6.3 Scale Drawings and Models

Vocabulary

Scale Drawing: a drawing used to represent an object that is too large or too small to be drawn at actual size.

Scale Model: a model used to represent an object that is too large or too small to be built at actual size.

Scale: gives the relationship between the measurements on the drawing or model and the measurements of the real object.

Scale Factor: The ratio of a length on a scale drawing or model to the corresponding length on the real object.

Using Scales

.5/10=.45/x

.5/10=1/x

.5/10=.75/x

6.4 Fractions, Decimals, and Percents

Vocabulary

Percent: a ratio that compares a number to 100

Writing Percents as Decimals

To write a percent as a decimal, divide by 100 and remove the percent symbol

Divide by 100

Remove % sign

62.10062%62

Writing Decimals as Percents

To write a decimal as a percent, multiply by 100 and add the percent symbol

multiply by 100

Add % sign

%4210042.42.

Writing Percents as Fractions

To write a percent as a fraction, express the ratio as a fraction with a denominator of 100

100

68%68

25

17

100

68

Put % over 100

Simplify

Writing Fractions as percentsMethod #1

To write a Fraction as a percent, write an equivalent fraction with a denominator of 100. Write the new numerator as a percent

multiply by 20 to get the denominator

to be 100

Write the numerator

as a percent

%40100

40

205

202

5

2

Writing Fractions as PercentsMethod #2

To write a fraction as a percent, first change the fraction to a decimal, then change the decimal to a percent

change fractions to a decimal by dividing the

numerator by the denominator

Add % sign

%5.87100875.0875.08

7

change decimal to a percent by

multiplying by 100

Practice

= 90%

10

9

100

90

= 135%

20

27

100

135

= 62%

50

31

100

62

Practice

=50%=.50 =75%

=.75=125%=1.25

Problem Solving

5

3

100

60

6.5 Using the percent Proportion

Vocabulary

Percent Proportion: a proportion in which a ratio is being compared to a percent

Part: the number being compared to the whole (always the numerator)

Base: The whole quantity (the denominator)

Find the Percent

Five is what percent of 8?

Step 1: set up a proportion

1008

5 p

5 is the part

8 is the base

percent is the unknown

always 100

Find the PercentFive is what percent of 8?

Step 2: Find the cross products

Step 3: Solve for the variable (p)

1008

5 p 500 = 8p

500 = 8p

8

8

8

500 p

62.5 = p

Divide each side by 8

62.5%62.5%

Add % sign since we were looking for the

percent

Find the Part

What number is 5.5% of 650?

Step 1: set up a proportion

100

5.5

650

athe part is unknown

650 is the base

5.5 is the percent

always 100

Find the PercentFive is what percent of 8?

Step 2: Find the cross products

Step 3: Solve for the variable (p)

100

5.5

650

a100a = 3575

100a = 3575

100

3575

100

100

a

a = 35.75

Divide each side by 100

35.7535.75

Practice

10050

14 x

1400 = 50x

x = 25

100

300

12

x

100x = 3600

x = 36

Apply the Percent Proportion

20.19$

20.19$8.1232$

8.12

1280100100

40

32

x

x

x

20.19$

60.32 or

Since the jeans are 40% off, the remaining price is 60% of the original price