Pre-Algebra Workbook 8: Ratios, Rates, and Proportions

In a class, there are $8$ boys and $12$ girls. What is the ratio of boys to girls in simplest form?

Choices

A

4

6

B

2

3

C

3

2

D

8

20

Strategy

Write the ratio as $8:12$ then simplify by dividing both numbers by their greatest common factor (GCF).

Solution

Start with the ratio boys : girls = $8:12$

The greatest common factor of $8$ and $12$ is $4$

$8 \div 4 = 2, \quad 12 \div 4 = 3$

So the simplified ratio is $2:3$

The correct choice is B. $2:3$ .

A pet store has $15$ cats and $10$ dogs. What is the ratio of dogs to total animals in simplest form?

Choices

A

10

25

B

3

5

C

2

5

D

5

2

Strategy

First find the total number of animals, then write the ratio dogs : total and simplify.

Solution

Total animals = $15 + 10 = 25$

The ratio of dogs to total animals is $10:25$

Simplify by dividing both numbers by $5$

$10 \div 5 = 2, \quad 25 \div 5 = 5$

So the simplified ratio is $2:5$

The correct choice is C. $2:5$ .

A car travels $180$ miles in $3$ hours. What is the car's speed in miles per hour?

Choices

A

30

miles per hour

B

60

miles per hour

C

90

miles per hour

D

120

miles per hour

Strategy

A unit rate has “ $1$ ” in the denominator. Divide total miles by total hours to find miles per $1$ hour.

Solution

Speed as a rate is

$\dfrac{180 \text{ miles}}{3 \text{ hours}}$

Divide numerator and denominator by $3$

$\dfrac{180}{3} = 60$

So the car travels $60$ miles in $1$ hour.

The speed is $60$ miles per hour.

The correct choice is B. $60$ miles per hour.

A package of $6$ energy bars costs $ $9$ What is the cost per bar?

Choices

A $

0.60

B $

1.25

C $

1.50

D $

2.00

Strategy

To find unit price, divide total cost by number of items.

Solution

Write the unit rate as a fraction of cost over number of bars:

$\dfrac{9}{6}$

Divide $9$ by $6$

$9 \div 6 = 1.5$

So the unit price is $1.5$ dollars per bar.

So each bar costs $ $1.50$

The correct choice is C. $ $1.50$ .

Solve the proportion: $\dfrac{2}{5} = \dfrac{6}{x}$

Choices

A

5

B

6

C

10

D

15

Strategy

Use cross multiplication: $2 \cdot x = 5 \cdot 6$ Then solve the resulting equation for $x$

Solution

Use cross multiplication:

$2 \cdot x = 5 \cdot 6$

Compute the right side:

$2x = 30$

Solve for $x$ by dividing both sides by $2$

$x = \dfrac{30}{2} = 15$

So $x = 15$

The correct choice is D. $15$ .

A map uses a scale of $1$ inch $=$ $50$ miles. If two cities are $3.5$ inches apart on the map, how far apart are they in miles?

Choices

A

150

miles

B

175

miles

C

200

miles

D

250

miles

Strategy

Use the scale as a rate: $50$ miles per inch. Multiply $3.5$ inches by $50$ miles per inch.

Solution

The scale means:

$1 \text{ inch} = 50 \text{ miles}$

For $3.5$ inches:

$3.5 \times 50 = 175$

So the cities are $175$ miles apart.

The correct choice is B. $175$ miles.

A recipe uses $3$ cups of flour to make $12$ cookies. At the same rate, how many cookies can be made with $5$ cups of flour?

Choices

A

15

cookies

B

18

cookies

C

19

cookies

D

20

cookies

Strategy

Set up a proportion: $\dfrac{3 \text{ cups}}{12 \text{ cookies}} = \dfrac{5 \text{ cups}}{x \text{ cookies}}$ and solve for $x$

Solution

Set up the proportion:

$\dfrac{3}{12} = \dfrac{5}{x}$

Use cross multiplication:

$3x = 12 \cdot 5 = 60$

Solve for $x$

$x = \dfrac{60}{3} = 20$

So $5$ cups of flour can make $20$ cookies.

The correct choice is D. $20$ cookies.

Two different brands of cereal are on sale:

Brand A: $12$ ounces for $ $3.60$
Brand B: $18$ ounces for $ $4.68$

Which brand is the better buy (lower cost per ounce)?

Choices

A Brand A (more expensive per ounce)

B Both are the same price per ounce

C Brand B (cheaper per ounce)

D Not enough information

Strategy

Find the unit price for each brand by dividing cost by ounces. Compare the cost per ounce.

Solution

Find the cost per ounce for each brand.

Brand A:

$\dfrac{3.60}{12 \text{ oz}} = 0.30 \text{ dollars per ounce}$

Brand B:

$\dfrac{4.68}{18 \text{ oz}} = 0.26 \text{ dollars per ounce}$

(since $4.68 \div 18 = 0.26$ ).

Brand B has the lower cost per ounce, so it is the better buy.

The correct choice is C. Brand B.

The ratio of red marbles to blue marbles in a bag is $4:7$ If there are $28$ blue marbles, how many red marbles are there?

Choices

A

16

B

21

C

32

D

49

Strategy

Use the ratio $4:7$ = red : blue. Set up a proportion $\dfrac{4}{7} = \dfrac{x}{28}$ and solve for $x$

Solution

Set up the proportion using red : blue:

$\dfrac{4}{7} = \dfrac{x}{28}$

Use cross multiplication:

$7x = 4 \cdot 28 = 112$

Solve for $x$

$x = \dfrac{112}{7} = 16$

So there are $16$ red marbles.

The correct choice is A. $16$ .

Kurt drives $150$ miles in $2.5$ hours. What is his average speed in miles per hour?

Choices

A

50

miles per hour

B

55

miles per hour

C

75

miles per hour

D

60

miles per hour

Strategy

Average speed is total distance divided by total time. Compute $\dfrac{150}{2.5}$ to find miles per hour.

Solution

Write the rate as

$\dfrac{150 \text{ miles}}{2.5 \text{ hours}}$

Compute $150 \div 2.5$

One way is to multiply numerator and denominator by $10$ to clear the decimal:

$\dfrac{150}{2.5} = \dfrac{150 \times 10}{2.5 \times 10} = \dfrac{1500}{25} = 60$

So his average speed is $60$ miles per hour.

The correct choice is D. $60$ miles per hour.