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# Forms of Ratios

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What if you wanted to make a scale drawing of your room and furniture for a little redecorating? Your room measures 12 feet by 12 feet. Also in your room is a twin bed (36 in by 75 in), a desk (4 feet by 2 feet), and a chair (3 feet by 3 feet). You decide to scale down your room to 8 in by 8 in, so the drawing fits on a piece of paper. What size should the bed, desk and chair be? Draw an appropriate layout for the furniture within the room. Do not round your answers. After completing this Concept, you'll be able to use ratios to answer this question.

### Guidance

A ratio is a way to compare two numbers. Ratios can be written: $\frac{a}{b}, a:b,$ and $a$ to $b$ . Let's look at some examples

#### Example A

There are 12 girls and 15 boys in your math class. What is the ratio of girls to boys?

Remember that order matters. The question asked for the ratio of girls to boys . The ratio would be 12:15. This can be simplified to 4:5.

#### Example B

Simplify the following ratios.

a) $\frac{7 \ ft}{14 \ in}$

b) 9m:900cm

c) $\frac{4 \ gal}{16 \ gal}$

First, change each ratio so that each part is in the same units. Remember that there are 12 inches in a foot.

a) $\frac{7 \ \bcancel{ft}}{14 \ \cancel{in}} \cdot \frac{12 \ \cancel{in}}{1 \ \bcancel{ft}} = \frac{84}{14} = \frac{6}{1}$

The inches and feet cancel each other out. Simplified ratios do not have units.

b) It is easier to simplify a ratio when written as a fraction.

$\frac{9 \ \bcancel{m}}{900 \ \cancel{cm}} \cdot \frac{100 \ \cancel{cm}}{1 \ \bcancel{m}} = \frac{900}{900} = \frac{1}{1}$

c) $\frac{4 \ \bcancel{gal}}{16 \ \bcancel{gal}} = \frac{1}{4}$

#### Example C

A talent show has dancers and singers. The ratio of dancers to singers is 3:2. There are 30 performers total, how many of each are there?

To solve, notice that 3:2 is a reduced ratio, so there is a number, $n$ , that we can multiply both by to find the total number in each group. Represent dancers and singers as expressions in terms of $n$ . Then set up and solve an equation.

$\text{dancers} = 3n, \ \text{singers} = 2n \ \longrightarrow \ 3n+2n &= 30\\5n &= 30\\n &= 6$

There are $3 \cdot 6 = 18$ dancers and $2 \cdot 6 = 12$ singers.

Watch this video for help with the Examples above.

#### Concept Problem Revisited

Everything needs to be scaled down by a factor of $\frac{1}{18} \ (144 \ in \div 8 \ in)$ . Change everything into inches and then multiply by the scale factor.

Bed : 36 in by 75 in $\longrightarrow$ 2 in by 4.167 in

Desk : 48 in by 24 in $\longrightarrow$ 2.67 in by 1.33 in

Chair : 36 in by 36 in $\longrightarrow$ 2 in by 2 in

There are several layout options for these three pieces of furniture. Draw an 8 in by 8 in square and then the appropriate rectangles for the furniture. Then, cut out the rectangles and place inside the square.

### Vocabulary

A ratio is a way to compare two numbers. Ratios can be written in three ways: $\frac{a}{b}$ , $a:b$ , and $a$ to $b$ .

### Guided Practice

The total bagel sales at a bagel shop for Monday is in the table below.

Type of Bagel Number Sold
Plain 80
Cinnamon Raisin 30
Sesame 25
Jalapeno Cheddar 20
Everything 45
Honey Wheat 50

1. What is the ratio of cinnamon raisin bagels to plain bagels?

2. What is the ratio of honey wheat bagels to total bagels sold?

3. What is the ratio of cinnamon raisin bagels to sesame bagels to jalapeno cheddar bagels?

1. The ratio is 30:80. Reducing the ratio by 10, we get 3:8.

2. Order matters. Honey wheat is listed first, so that number comes first in the ratio (or on the top of the fraction). Find the total number of bagels sold, $80 + 30 + 25 + 20 + 45 + 50 = 250$ .

The ratio is $\frac{50}{250} = \frac{1}{5}$ .

3. You can have ratios that compare more than two numbers and they work just the same way. The ratio for this problem is 30:25:20, which reduces to 6:5:4.

### Practice

1. The votes for president in a club election were: $\text{Smith}: 24 \qquad \text{Munoz}: 32 \qquad \text{Park}: 20$ Find the following ratios and write in simplest form.
1. Votes for Munoz to Smith
2. Votes for Park to Munoz
4. Votes for Smith to Munoz to Park

Use the picture to write the following ratios for questions 2-6.

$AEFD$ is a square

$ABCD$ is a rectangle

1. $AE:EF$
2. $EB:AB$
3. $DF:FC$
4. $EF:BC$
5. Perimeter $ABCD$ : Perimeter $AEFD$ : Perimeter $EBCF$
6. The measures of the angles of a triangle are have the ratio 3:3:4. What are the measures of the angles?
7. The lengths of the sides in a triangle are in a 3:4:5 ratio. The perimeter of the triangle is 36. What are the lengths of the sides?
8. The length and width of a rectangle are in a 3:5 ratio. The perimeter of the rectangle is 64. What are the length and width?
9. The length and width of a rectangle are in a 4:7 ratio. The perimeter of the rectangle is 352. What are the length and width?
10. The ratio of the short side to the long side in a parallelogram is 5:9. The perimeter of the parallelogram is 112. What are the lengths of the sides?
11. The length and width of a rectangle are in a 3:11 ratio. The area of the rectangle is 528. What are the length and width of the rectangle?
12. The length and width of a rectangle are in a 2:15 ratio. The area of the rectangle is 3630. What are the length and width of the rectangle?
13. Two squares have areas of $4cm^2$ and $16cm^2$ . What is the ratio of the side length of the smaller square to the side length of the larger square?
14. Two triangles are congruent. What is the ratio between their side lengths?