7.2: Proportion Properties
What if you were told that a scale model of a python is in the ratio of 1:24? If the model measures 0.75 feet long, how long is the real python? After completing this Concept, you'll be able to solve problems like this one by using a proportion.
Watch This
CK12 Foundation: Chapter7ProportionPropertiesA
James Sousa: Applications of Proportions
James Sousa: Using Similar Triangles to Determine Unknown Values
Guidance
A proportion is when two ratios are set equal to each other.
CrossMultiplication Theorem: Let
The proof of the CrossMultiplication Theorem is an algebraic proof. Recall that multiplying by
Proof of the CrossMultiplication Theorem:
Think of the CrossMultiplication Theorem as a shortcut. Without this theorem, you would have to go through all of these steps every time to solve a proportion. The CrossMultiplication Theorem has several subtheorems that follow from its proof. The formal term is corollary.
Corollary #1: If
Corollary #2: If
Corollary #3: If
Corollary #4: If
Corollary #5: If
Example A
Solve the proportions.
a)
b)
c)
To solve a proportion, you need to crossmultiply.
a)
b)
c)
Example B
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 your parents’ house is actually 50 feet, what is the longer length?
Set up a proportion. If the shorter length is 50 feet, then it will line up with 30 in. It does not matter which numbers you put in the numerators of the fractions, as long as they line up correctly.
So, the dimension of your parents’ house is 50 ft by 60 ft.
Example C
Suppose we have the proportion
First of all, we know this is a true proportion because you would multiply

214=535 
355=142 
52=3514
If you crossmultiply all four of these proportions, you would get
Watch this video for help with the Examples above.
CK12 Foundation: Chapter7ProportionPropertiesB
Concept Problem Revisited
The scale model of a python is 0.75 ft long and in the ratio 1:24. If
The real python is 18 ft long.
Vocabulary
A ratio is a way to compare two numbers. Ratios can be written in three ways:
Guided Practice
1. In the picture,
Find the measures of
2. In the picture, \begin{align*}\frac{ED}{AD}=\frac{BC}{AC}\end{align*}. Find \begin{align*}y\end{align*}.
3. If \begin{align*}\frac{AB}{BE}=\frac{AC}{CD}\end{align*} in the picture above, find \begin{align*}BE\end{align*}.
Answers:
1. This is an example of an extended proportion. Substituting in the numbers for the sides we know, we have \begin{align*}\frac{4}{XY}=\frac{3}{9}=\frac{AC}{15}\end{align*}. Separate this into two different proportions and solve for \begin{align*}XY\end{align*} and \begin{align*}AC\end{align*}.
\begin{align*}\frac{4}{XY} &= \frac{3}{9} && \quad \ \ \ \frac{3}{9}=\frac{AC}{15}\\ 36 &= 3(XY) && 9(AC)=45\\ XY &= 12 && \quad \ AC=5\end{align*}
2. Substituting in the numbers for the sides we know, we have
\begin{align*}\frac{6}{y} =\frac{8}{12+8}. \longrightarrow 8y &= 6(20)\\ y &= 15\end{align*}
3. \begin{align*}\frac{12}{BE}=\frac{20}{25} \longrightarrow 20(BE) &= 12(25)\\ BE &= 15\end{align*}
Practice
Solve each proportion.
 \begin{align*}\frac{x}{10}=\frac{42}{35}\end{align*}
 \begin{align*}\frac{x}{x2}=\frac{5}{7}\end{align*}
 \begin{align*}\frac{6}{9}=\frac{y}{24}\end{align*}
 \begin{align*}\frac{x}{9}=\frac{16}{x}\end{align*}
 \begin{align*}\frac{y3}{8}=\frac{y+6}{5}\end{align*}
 \begin{align*}\frac{20}{z+5}=\frac{16}{7}\end{align*}
 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.
 The president, vicepresident, 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?
 Many recipes describe ratios between ingredients. For example, one recipe for paper mache paste suggests 3 parts flour to 5 parts water. If we have one cup of flour, how much water should we add to make the paste?
 A recipe for krispy rice treats calls for 6 cups of rice cereal and 40 large marshmallows. You want to make a larger batch of goodies and have 9 cups of rice cereal. How many large marshmallows do you need? However, you only have the miniature marshmallows at your house. You find a list of substitution quantities on the internet that suggests 10 large marshmallows are equivalent to 1 cup miniatures. How many cups of miniatures do you need?
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.
 \begin{align*}\frac{10}{y}=\frac{x}{6}\end{align*}
 \begin{align*}\frac{15}{10}=\frac{d}{6}\end{align*}
 \begin{align*}\frac{6+10}{10}=\frac{y+x}{x}\end{align*}
 \begin{align*}\frac{15}{x}=\frac{y}{d}\end{align*}
For questions 1518, \begin{align*}\frac{AE}{ED} = \frac{BC}{CD}\end{align*} and \begin{align*}\frac{ED}{AD}=\frac{CD}{DB}=\frac{EC}{AB}\end{align*}.
 Find \begin{align*}DB\end{align*}.
 Find \begin{align*}EC\end{align*}.
 Find \begin{align*}CB\end{align*}.
 Find \begin{align*}AD\end{align*}.
 Writing Explain why \begin{align*}\frac{a+b}{b}=\frac{c+d}{d}\end{align*} is a valid proportion. HINT: Crossmultiply and see if it equals \begin{align*}ad=bc\end{align*}.
 Writing Explain why \begin{align*}\frac{ab}{b}=\frac{cd}{d}\end{align*} is a valid proportion. HINT: Crossmultiply and see if it equals \begin{align*}ad=bc\end{align*}.
Cross Products
To simplify a proportion using cross products, multiply the diagonals of each ratio.CrossMultiplication Theorem
The CrossMultiplication theorem states that if and are real numbers, with and and if , then .Image Attributions
Here you'll learn how to set up and solve proportions.