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.
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CK-12 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.
Cross-Multiplication Theorem: Let \begin{align*}a, b, c,\end{align*} and \begin{align*}d\end{align*} be real numbers, with \begin{align*}b \neq 0\end{align*} and \begin{align*}d \neq 0\end{align*}. If \begin{align*}\frac{a}{b}=\frac{c}{d}\end{align*}, then \begin{align*}ad=bc\end{align*}.
The proof of the Cross-Multiplication Theorem is an algebraic proof. Recall that multiplying by \begin{align*}\frac{2}{2}, \frac{b}{b},\end{align*} or \begin{align*}\frac{d}{d}=1\end{align*} because it is the same number divided by itself \begin{align*}(b \div b=1)\end{align*}.
Proof of the Cross-Multiplication Theorem:
\begin{align*}\frac{a}{b} &= \frac{c}{d} \qquad \ \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} \qquad \text{The denominators are the same, so the numerators are equal.}\\ ad &= bc\end{align*}
Think of the Cross-Multiplication Theorem as a shortcut. Without this theorem, you would have to go through all of these steps every time to solve a proportion. The Cross-Multiplication Theorem has several sub-theorems that follow from its proof. The formal term is corollary.
Corollary #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*}.
Corollary #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*}.
Corollary #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{d}{c}\end{align*}.
Corollary #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 #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 A
Solve the proportions.
a) \begin{align*}\frac{4}{5}=\frac{x}{30}\end{align*}
b) \begin{align*}\frac{y+1}{8}=\frac{5}{20}\end{align*}
c) \begin{align*}\frac{6}{5}=\frac{2x+4}{x-2}\end{align*}
To solve a proportion, you need to cross-multiply.
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.
\begin{align*}\frac{30}{36} = \frac{50}{x} \longrightarrow 1800 &= 30x\\ 60 &= x\end{align*}
So, the dimension of your parents’ house is 50 ft by 60 ft.
Example C
Suppose we have the proportion \begin{align*}\frac{2}{5}=\frac{14}{35}\end{align*}. Write down the other three true proportions that follow from this one.
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, we would get:
- \begin{align*}\frac{2}{14}=\frac{5}{35}\end{align*}
- \begin{align*}\frac{35}{5}=\frac{14}{2}\end{align*}
- \begin{align*}\frac{5}{2}=\frac{35}{14}\end{align*}
If you cross-multiply all four of these proportions, you would get \begin{align*}70 = 70\end{align*} for each one.
Watch this video for help with the Examples above.
CK-12 Foundation: Chapter7ProportionPropertiesB
Concept Problem Revisited
The scale model of a python is 0.75 ft long and in the ratio 1:24. If \begin{align*} x\end{align*} is the length of the real python in ft:
\begin{align*}\frac{1}{24} = \frac{0.75}{x} \longrightarrow x &= 24(0.75)\\ x &= 18 \end{align*}
The real python is 18 ft long.
Vocabulary
A ratio is a way to compare two numbers. Ratios can be written in three ways: \begin{align*}\frac{a}{b}\end{align*}, \begin{align*}a:b\end{align*}, and \begin{align*}a\end{align*} to \begin{align*}b\end{align*}. A proportion is two ratios that are set equal to each other. To solve a proportion you should cross-multiply, which means to set the product of the numerator of the first fraction and the denominator of the second fraction equal to the product of the denominator of the first fraction and the numerator of the second fraction. A corollary is a theorem that follows directly from another theorem.
Guided Practice
1. 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*}.
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}{x-2}=\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{y-3}{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, 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?
- 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 15-18, \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: Cross-multiply and see if it equals \begin{align*}ad=bc\end{align*}.
- Writing Explain why \begin{align*}\frac{a-b}{b}=\frac{c-d}{d}\end{align*} is a valid proportion. HINT: Cross-multiply and see if it equals \begin{align*}ad=bc\end{align*}.