x = -32
How to Apply Proportions to Real-World Problems
Proportions can be used to solve many real-world problems involving rates, ratios, and percentages. For example, you can use proportions to find out how much of an ingredient you need for a recipe, how much time it takes to travel a certain distance, how much interest you earn on an investment, or how much tax you pay on a purchase.
To apply proportions to real-world problems, follow these steps:
- Read the problem carefully and identify the given information and the unknown quantity.
- Write a ratio that compares two quantities with the same units in the problem.
- Write another ratio that compares two quantities with the same units in the problem, one of which is the unknown quantity.
- Write a proportion by setting the two ratios equal to each other.
- Solve the proportion using cross-multiplication or any other method that works for you.
- Check your answer by plugging it back into the original proportion and simplifying.
- Write your answer in a complete sentence with appropriate units.
Here are some examples of how to apply proportions to real-world problems:
Example 1: A recipe for chocolate chip cookies calls for 2 cups of flour and 1 cup of chocolate chips. How many cups of chocolate chips do you need if you use 3 cups of flour?
- The given information is 2 cups of flour and 1 cup of chocolate chips. The unknown quantity is how many cups of chocolate chips you need if you use 3 cups of flour.
- A ratio that compares two quantities with the same units in the problem is $$\frac{\text{flour}}{\text{chocolate chips}}=\frac{2}{1}$$.
- Another ratio that compares two quantities with the same units in the problem, one of which is the unknown quantity, is $$\frac{\text{flour}}{\text{chocolate chips}}=\frac{3}{x}$$, where x is the unknown quantity.
- A proportion by setting the two ratios equal to each other is $$\frac{2}{1}=\frac{3}{x}$$.
- To solve the proportion using cross-multiplication, we multiply the numerator of one ratio by the denominator of the other ratio, and write the product on one side of an equation. Then we multiply the denominator of one ratio by the numerator of the other ratio, and write the product on the other side of the equation. Then we solve for x by isolating it on one side of the equation: $$2\times x=1\times 3 \Rightarrow 2x=3 \Rightarrow \frac{2x}{2}=\frac{3}{2} \Rightarrow x=\frac{3}{2}$$.
- To check our answer by plugging it back into the original proportion and simplifying, we substitute x with $$\frac{3}{2}$$ and see if both sides are equal: $$\frac{2}{1}=\frac{3}{\frac{3}{2}} \Rightarrow \frac{2}{1}=\frac{3\times 2}{3} \Rightarrow \frac{2}{1}=\frac{6}{3} \Rightarrow \frac{2}{1}=\frac{6 \div 3}{3 \div 3} \Rightarrow \frac{2}{1}=\frac{2}{1} \Rightarrow \text{True}$$.
- To write our answer in a complete sentence with appropriate units, we say: You need $$\frac{3}{2}$$ cups of chocolate chips if you use 3 cups of flour.
Example 2: A car travels 120 miles in 2 hours. How long will it take to travel 180 miles at the same speed?
- The given information is 120 miles in 2 hours and 180 miles. The unknown quantity is how long it will take to travel 180 miles at the same speed.
- A ratio that compares two quantities with the same units in the problem is $$\frac{\text{miles}}{\text{hours}}=\frac{120}{2}$$.
- Another ratio that compares two quantities with the same units in the problem, one of which is the unknown quantity, is $$\frac{\text{miles}}{\text{hours}}=\frac{180}{x}$$, where x is the unknown quantity.
- A proportion by setting the two ratios equal to each other is $$\frac{120}{2}=\frac{180}{x}$$.
- To solve the proportion using cross-multiplication, we multiply
the numerator of one ratio by
the denominator of
the other ratio,
and write
the product on
one side
of an equation.
Then we multiply
the denominator
of one ratio by
the numerator
of
the other ratio,
and write
the product on
the other side
of
the equation.
Then we solve
for x by isolating
it on one side
of
the equation:
$$120\times x=2\times 180 \Rightarrow
120x=360 \Rightarrow
\frac{120x}{120}=\frac{360}{120}
\Rightarrow
x=3$$.
- To check our answer by plugging it back into
the original proportion and simplifying,
we substitute x with
3 and see if both sides are equal:
$$\frac{120}{2}=\frac{180}{3}
\Rightarrow
\frac{120 \div 60}{2 \div 60}=\frac{180 \div 60}{3 \div 60}
\Rightarrow
\frac{2}{0.033}=
\frac{3}{0.05}
\Rightarrow
\frac{2\times 0.05}{0.033\times 0.05}=
\frac{3\times 0.033}{0.05\times 0.033}
\Rightarrow
\frac{0.1}{0.00165}=
\frac{0.099}{0.00165}
\Rightarrow
60=60
\Rightarrow
\text{True}$$.
- To write our answer in a complete sentence with appropriate units, we say: It will take 3 hours to travel 180 miles at the same speed.
Conclusion
Proportions are useful tools to solve many real-world problems involving rates, ratios, and percentages. In this article, we have shown you how to solve proportions using cross-multiplication, and provided you with the answer key and tips for Lesson 7 Homework Practice Solving Proportions. We hope this article has helped you understand and apply proportions better. Remember to check your answers by plugging them back into the original proportions and simplifying. Happy solving!
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