5F Solving Proportions

A special type of equation that has many applications is a proportion. A “proportion” is when two fractions are equal to each other. For example $\frac{3}{5}=\frac{6}{10}$. Notice the two fractions are equal to each other. One way to check if two fractions are equal is by looking at the cross products. Notice the numerator of one fraction (3) times the denominator of the other fraction (10) is equal to the other cross product (6x5). Notice both cross products are equal to 30. This is an easy way to check if two fractions are equal.

In general, with d and f nonzero, if $\frac{c}{d}=\frac{e}{f}$ then $c\times f=e\times d$.

Try the following examples with your instructor. Determine if the two fractions are equal.

Often we need to solve a proportion though. Setting the cross products equal is an easy way to simplify a proportion equation and make it much easier to solve.

Let’s look at an example. Solve the following proportion: $\frac{15}{16}=\frac{h}{8}$

First we notice that this is two fractions set equal, so it is a proportion. If it does not have this form, you need to use the applicable equation-solving methods discussed in sections 5C–5E. Since this is a proportion we can set the cross products equal to each other.

$$\begin{array}{l}\frac{15}{16}=\frac{h}{8}\\ h\times 16=15\times 8\end{array}$$

Simplifying gives us 16h = 120. Now we solve by isolating the variable (divide both sides by 16).

$$\begin{array}{l}16h=120\\ \frac{16h}{16}=\frac{120}{16}\\ \frac{{}^{1}16h}{16}=\frac{120}{16}\\ 1h=7.5\\ h=7.5\end{array}$$

At the beginning of this chapter, we talked about the famous z-score formula in statistics.$z=\frac{x-\mu }{\sigma }$. Can we find the x-value (pounds) for a z-score of -2.4? If $\mu =9.5$ and $\sigma =1.5$ can we back solve the formula and figure out what x needs to be?

Plugging in the correct values into the formula gives us the following:

$$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ -2.4=\frac{x-9.5}{1.5}\end{array}$$

Now if we were to write -2.4 as -2.4/1 we would have a proportion! We could then cross multiply and solve. Let’s try it.

$$\begin{array}{l}z=\frac{x-\mu }{\sigma }\\ -2.4=\frac{x-9.5}{1.5}\\ \frac{-2.4}{1}=\frac{x-9.5}{1.5}\end{array}$$

Setting the cross products equal and solving gives us the x value.

$$\begin{array}{rcl}\frac{-2.4}{1}& =& \frac{x-9.5}{1.5}\\ 1(x-9.5)& =& -2.4(1.5)\\ x-9.5& =& -3.6\\ +9.5& & +9.5\\ x+0& =& 5.9\\ x& =& 5.9\end{array}$$

So the x value for a z-score of -2.4 is 5.9 pounds.

Solve the following proportion problems with your instructor:

Practice Problems Section 5F

Solve the following proportion problems. Simplify all fraction answers completely. Exclude any value that makes an original denominator zero, and check each candidate in the original proportion.

1. $$\frac{1}{x}=\frac{3}{5}$$

   Check answer

   $$x=1⁤\frac{2}{3}\text{ or }1.\overline{6}$$

2. $$\frac{w}{-4}=\frac{1}{6}$$

3. $$\frac{-3}{7}=\frac{y}{4}$$

   Check answer

   $$y=-1⁤\frac{5}{7}$$

4. $$\frac{8}{5}=\frac{3}{2d}$$

5. $$\frac{7.5}{x}=\frac{2.5}{9}$$

   Check answer

   $$x=27$$

6. $$\frac{3w}{7}=\frac{2}{3}$$

7. $$\frac{-4}{13}=\frac{-1}{m}$$

   Check answer

   $$m=3⁤\frac{1}{4}\text{ or }3.25$$

8. $$\frac{-11}{x}=\frac{22}{5}$$

9. $$\frac{8}{2.1}=\frac{w}{6.3}$$

   Check answer

   $$w=24$$

10. $$\frac{1}{5w}=\frac{-8}{15}$$

11. $$\frac{0.25}{4}=\frac{0.15}{L}$$

    Check answer

    $L=2.4$or $2⁤\frac{2}{5}$

12. $$\frac{-18}{25}=\frac{9}{-5u}$$

13. $$\frac{0.4}{1.2}=\frac{-0.8d}{2.4}$$

    Check answer

    $$d=-1$$

14. $$\frac{-50}{3h}=\frac{20}{-9}$$

15. $$\frac{3}{2.5}=\frac{1.8f}{7.5}$$

    Check answer

    $$f=5$$

16. $$\frac{2p-1}{5}=\frac{3p}{10}$$

17. $$\frac{7}{x+4}=\frac{2}{x+9}$$

    Check answer

    $$x=-11$$

18. $$\frac{3w}{7}=\frac{2w+5}{4}$$

19. $$\frac{-5}{m-1}=\frac{-2}{m+3}$$

    Check answer

    $$m=-5⁤\frac{2}{3}\text{ or}-5.\overline{6}$$

20. $$\frac{x+4}{6}=\frac{2x}{11}$$

21. $$\frac{w-7}{1.5}=\frac{w}{1.5}$$

    Check answer

    No Solution

22. $$\frac{6}{x+5}=\frac{4}{x-7}$$

23. $$\frac{w-14}{9}=\frac{3w}{17}$$

    Check answer

    $$w=-23.8\text{ or }-23⁤\frac{4}{5}$$

24. $$\frac{3}{1.5+k}=\frac{7}{4.5}$$

Adapted from Preparing for Algebra and Statistics, Third Edition, by Matt Teachout. License: CC BY 4.0. Copyright and sources.