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7.1: Ratios and Proportions

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Write and solve ratios and proportions.
  • Use ratios and proportions in problem solving.

Review Queue

  1. Are the two triangles congruent? If so, how do you know?
  2. If \begin{align*}AC = 5\end{align*}AC=5, what is \begin{align*}GI\end{align*}GI? What is the reason?
  3. How many inches are in a:
    1. foot?
    2. yard?
    3. 3 yards?
    4. 5 feet?

Know What? You want to make a scale drawing of your room and furniture for a little redecorating. Your room measures 12 feet by 12 feet. In your room are a twin bed (36 in by 75 in) and a desk (4 feet by 2 feet). You scale down your room to 8 in by 8 in, so it fits on a piece of paper. What size are the bed and desk in the drawing?

Using Ratios

Ratio: A way to compare two numbers. Ratios can be written: \begin{align*}\frac{a}{b}\end{align*}ab, \begin{align*}a:b\end{align*}a:b, and \begin{align*}a\end{align*}a to \begin{align*}b\end{align*}b.

Example 1: There are 14 girls and 18 boys in your math class. What is the ratio of girls to boys?

Solution: The ratio would be 14:18. This can be simplified to 7:9.

Example 2: The total bagel sales at a bagel shop for Monday is in the table below. What is the ratio of cinnamon raisin bagels to plain bagels?

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

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

Reduce a ratio just like a fraction. Always reduce ratios.

Example 3: What is the ratio of honey wheat bagels to total bagels sold?

Solution: 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, \begin{align*}80 + 30 + 25 + 20 + 45 + 50 = 250\end{align*}80+30+25+20+45+50=250.

The ratio is \begin{align*}\frac{50}{250} = \frac{1}{5}\end{align*}50250=15.

Equivalent Ratios: When two or more ratios reduce to the same ratio.

50:250 and 2:10 are equivalent because they both reduce to 1:5.

Example 4: What is the ratio of cinnamon raisin bagels to sesame bagels to jalapeno cheddar bagels?

Solution: 30:25:20, which reduces to 6:5:4.

Converting Measurements

How many feet are in 2 miles? How many inches are in 4 feet? Ratios are used to convert these measurements.

Example 5: Simplify the following ratios.

a) \begin{align*}\frac{7 \ ft}{14 \ in}\end{align*}7 ft14 in

b) 9m:900cm

c) \begin{align*}\frac{4 \ gal}{16 \ gal}\end{align*}4 gal16 gal

Solution: Change these so that they are in the same units. There are 12 inches in a foot.

a) \begin{align*}\frac{7 \ \bcancel{ft}}{14 \ \cancel{in}} \cdot \frac{12 \ \cancel{in}}{1 \ \bcancel{ft}} = \frac{84}{14} = \frac{6}{1}\end{align*}7 ft14 in12 in1 ft=8414=61

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

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

\begin{align*}\frac{9 \ \bcancel{m}}{900 \ \cancel{cm}} \cdot \frac{100 \ \cancel{cm}}{1 \ \bcancel{m}} = \frac{900}{900} = \frac{1}{1}\end{align*}9 m900 cm100 cm1 m=900900=11

c) \begin{align*}\frac{4 \ \bcancel{gal}}{16 \ \bcancel{gal}} = \frac{1}{4}\end{align*}4 gal16 gal=14

Example 6: A talent show has dancers and singers. The ratio of dances to singers is 3:2. There are 30 performers total, how many of each are there?

Solution: 3:2 is a reduced ratio, so there is a number, \begin{align*}n\end{align*}n, that we can multiply both by to find the total number in each group.

\begin{align*}\text{dancers} = 3n, \ \text{singers} = 2n \ \longrightarrow \ 3n+2n &= 30\\ 5n &= 30\\ n &= 6\end{align*}dancers=3n, singers=2n  3n+2n5nn=30=30=6

There are \begin{align*}3 \cdot 6 = 18\end{align*}36=18 dancers and \begin{align*}2 \cdot 6 = 12\end{align*}26=12 singers.

Solving Proportions

Proportion: Two ratios that are set equal to each other.

Example 7: Solve the proportions.

a) \begin{align*}\frac{4}{5} = \frac{x}{30}\end{align*}45=x30

b) \begin{align*}\frac{y+1}{8} = \frac{5}{20}\end{align*}y+18=520

c) \begin{align*}\frac{6}{5} = \frac{2x+5}{x-2}\end{align*}65=2x+5x2

Solution: To solve a proportion, you need to cross-multiply.




Cross-Multiplication Theorem: \begin{align*}a, b, c,\end{align*}a,b,c, and \begin{align*}d\end{align*}d are real numbers, with \begin{align*}b \neq 0\end{align*}b0 and \begin{align*}d \neq 0\end{align*}d0. If \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}ab=cd, then \begin{align*}ad=bc\end{align*}ad=bc.

The proof of the Cross-Multiplication Theorem is an algebraic proof. Recall that multiplying by the same number over itself is 1 \begin{align*}(b \div b=1)\end{align*}(b÷b=1).

Proof of the Cross-Multiplication Theorem

\begin{align*}\frac{a}{b} &= \frac{c}{d} && \text{Multiply the left side by} \ \frac{d}{d} \ \text{and the right side by} \ \frac{b}{b}.\\ \frac{a}{b} \cdot \frac{d}{d} &= \frac{c}{d} \cdot \frac{b}{b}\\ \frac{ad}{bd} &= \frac{bc}{bd} && \text{The denominators are the same, so the tops are equal}.\\ ad &= bc\end{align*}ababddadbdad=cd=cdbb=bcbd=bcMultiply the left side by dd and the right side by bb.The denominators are the same, so the tops are equal.

Example 8: Your parents have an architect’s drawing of their home. On the paper, the house’s dimensions are 36 in by 30 in. If the shorter length of the house is actually 50 feet, what is the longer length?

Solution: Set up a proportion. If the shorter length is 50 feet, then it lines up with 30 in, the shorter length of the paper dimensions.

\begin{align*}\frac{30}{36} = \frac{50}{x} \longrightarrow \ 1800 &=30x\\ 60 &=x \quad \quad \text{The longer length is 60 feet.}\end{align*}3036=50x 180060=30x=xThe longer length is 60 feet.

Properties of Proportions

The Cross-Multiplication Theorem has several sub-theorems, called corollaries.

Corollary: A theorem that follows directly from another theorem.

Below are three corollaries that are immediate results of the Cross Multiplication Theorem.

Corollary 7-1: If \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*} are nonzero and \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}\frac{a}{c} = \frac{b}{d}\end{align*}. Switch \begin{align*}b\end{align*} and \begin{align*}c\end{align*}.

Corollary 7-2: If \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*} are nonzero and \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}\frac{d}{b} = \frac{c}{a}\end{align*}. Switch \begin{align*}a\end{align*} and \begin{align*}d\end{align*}.

Corollary 7-3: If \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*} are nonzero and \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}\frac{b}{a} = \frac{c}{d}\end{align*}. Flip each ratio upside down.

In each corollary, you will still end up with \begin{align*}ad=bc\end{align*} after cross-multiplying.

Example 9: Suppose we have the proportion \begin{align*}\frac{2}{5} = \frac{14}{35}\end{align*}. Write three true proportions that follow.

Solution: First of all, we know this is a true proportion because you would multiply \begin{align*}\frac{2}{5}\end{align*} by \begin{align*}\frac{7}{7}\end{align*} to get \begin{align*}\frac{14}{35}\end{align*}. Using the three corollaries:

  1. \begin{align*}\frac{2}{14} = \frac{5}{35}\end{align*}
  2. \begin{align*}\frac{35}{5} = \frac{14}{2}\end{align*}
  3. \begin{align*}\frac{5}{2} = \frac{35}{14}\end{align*}

Corollary 7-4: If \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*} are nonzero and \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}\frac{a+b}{b} = \frac{c+d}{d}\end{align*}.

Corollary 7-5: If \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*} are nonzero and \begin{align*}\frac{a}{b} = \frac{c}{d}\end{align*}, then \begin{align*}\frac{a-b}{b} = \frac{c-d}{d}\end{align*}.

Example 10: In the picture, \begin{align*}\frac{AB}{XY} = \frac{BC}{YZ} = \frac{AC}{XZ}\end{align*}.

Find the measures of \begin{align*}AC\end{align*} and \begin{align*}XY\end{align*}.

Solution: Plug in the lengths of the sides we know.

Example 11: In the picture, \begin{align*}\frac{ED}{AD} = \frac{BC}{AC}\end{align*}. Find \begin{align*}y\end{align*}.

Solution: Substitute in the lengths of the sides we know.

\begin{align*}\frac{6}{y} = \frac{8}{12+8} \ \longrightarrow \ 8y &= 6(20)\\ y &= 15\end{align*}

Example 12: If \begin{align*}\frac{AB}{BE} = \frac{AC}{CD}\end{align*} in the picture above, find \begin{align*}BE\end{align*}.


\begin{align*}\frac{12}{BE} = \frac{20}{25} \ \longrightarrow \ {20(BE)} & = 12(25)\\ BE & = 15\end{align*}

Know What? Revisited Everything needs to be scaled down by a factor of \begin{align*}\frac{1}{18} (144 \ in \div 8 \ in)\end{align*}. Change everything into inches and then multiply by the scale factor.

\begin{align*}\underline{\text{Bed}}: \ 36 \ \text{in by}\ 75 \ \text{in} \ \longrightarrow \ 2 \ \text{in by}\ 4.167 \ \text{in}\!\\ \underline{\text{Desk}}: \ 48 \ \text{in by}\ 24 \ \text{in} \ \longrightarrow 2.67 \ \text{in by}\ 1.33 \ \text{in}\end{align*}

Review Questions

  • Questions 1 and 2 are similar to Examples 1-4.
  • Questions 7-13 are similar to Example 5.
  • Questions 14-19, 26, and 27 are similar to Example 6 and 8.
  • Questions 20-25 are similar to Example 7.
  • Questions 28-31 are similar to Example 9
  • Questions 32-35 are similar to Examples 10-12.
  1. The votes for president in a club election were: \begin{align*}\text{Smith}:24 \qquad \text{Munoz}:32 \qquad \text{Park}:20\end{align*} Find the following ratios and write in simplest form.
    1. Votes for Munoz to Smith
    2. Votes for Park to Munoz
    3. Votes for Smith to total votes
    4. Votes for Smith to Munoz to Park

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

\begin{align*}AEFD \ \text{is a square} \qquad ABCD \ \text{is a rectangle}\end{align*}

  1. \begin{align*}AE:EF\end{align*}
  2. \begin{align*}EB:AB\end{align*}
  3. \begin{align*}DF:FC\end{align*}
  4. \begin{align*}EF:BC\end{align*}
  5. Perimeter \begin{align*}ABCD\end{align*}:Perimeter \begin{align*}AEFD\end{align*}:Perimeter \begin{align*}EBCF\end{align*}

Convert the following measurements.

  1. 16 cups to gallons
  2. 8 yards to feet
  3. 6 meters to centimeters

Simplify the following ratios.

  1. \begin{align*}\frac{25 \ in}{5 \ ft}\end{align*}
  2. \begin{align*}\frac{8 \ pt}{2 \ gal}\end{align*}
  3. \begin{align*}\frac{9 \ ft}{3 \ yd}\end{align*}
  4. \begin{align*}\frac{95 \ cm}{1.5 \ m}\end{align*}
  1. The measures of the angles of a triangle are have the ratio 3:3:4. What are the measures of the angles?
  2. 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?
  3. 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?
  4. A math class has 36 students. The ratio of boys to girls is 4:5. How many girls are in the class?
  5. The senior class has 450 students in it. The ratio of boys to girls is 8:7. How many boys are in the senior class?
  6. The varsity football team has 50 players. The ratio of seniors to juniors is 3:2. How many seniors are on the team?

Solve each proportion.

  1. \begin{align*}\frac{x}{10} = \frac{42}{35}\end{align*}
  2. \begin{align*}\frac{x}{x-2} = \frac{5}{7}\end{align*}
  3. \begin{align*}\frac{6}{9} = \frac{y}{24}\end{align*}
  4. \begin{align*}\frac{x}{9} = \frac{16}{x}\end{align*}
  5. \begin{align*}\frac{y-3}{8} = \frac{y+6}{5}\end{align*}
  6. \begin{align*}\frac{20}{z+5} = \frac{16}{7}\end{align*}
  7. Shawna drove 245 miles and used 8.2 gallons of gas. At the same rate, if she drove 416 miles, how many gallons of gas will she need? Round to the nearest tenth.
  8. The president, vice-president, and financial officer of a company divide the profits is a 4:3:2 ratio. If the company made $1,800,000 last year, how much did each person receive?

Given the true proportion, \begin{align*}\frac{10}{6}= \frac{15}{d} = \frac{x}{y}\end{align*} and \begin{align*}d, x,\end{align*} and \begin{align*}y\end{align*} are nonzero, determine if the following proportions are also true.

  1. \begin{align*}\frac{10}{y} = \frac{x}{6}\end{align*}
  2. \begin{align*}\frac{15}{10} = \frac{d}{6}\end{align*}
  3. \begin{align*}\frac{6+10}{10} = \frac{y+x}{x}\end{align*}
  4. \begin{align*}\frac{15}{x} = \frac{y}{d}\end{align*}

For questions 32-35, \begin{align*}\frac{AE}{ED} = \frac{BC}{CD}\end{align*} and \begin{align*}\frac{ED}{AD} = \frac{CD}{DB} = \frac{EC}{AB}\end{align*}.

  1. Find \begin{align*}DB\end{align*}.
  2. Find \begin{align*}EC\end{align*}.
  3. Find \begin{align*}CB\end{align*}.
  4. Find \begin{align*}AD\end{align*}.

Review Queue Answers

  1. Yes, they are congruent by SAS.
  2. \begin{align*}GI = 5\end{align*} by CPCTC
    1. 12 in = 1 ft
    2. 36 in = 3 ft
    3. 108 in = 3 yd
    4. 60 in = 5 ft.

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