5.4 Ratios and Proportions

Using the definition of ratio, let $a=1$ U.S. dollar and let $b=0.82$ Euros. Then the ratio can be written as either 1 to 0.82; or 1:0.82; or $\frac{1}{0.82}.$

Using the definition of ratio, let $a=200$ pounds on Earth and let $b=33$ pounds on the moon. Then the ratio can be written as either 200 to 33; or 200:33; or $\frac{200}{33}.$

Step 1: Set up the two ratios into a proportion; let $x$ be the variable that represents the unknown. Notice that U.S. dollar amounts are in both numerators and Euro amounts are in both denominators.

Step 2: Cross multiply, since the ratios $\frac{a}{b}$ and $\frac{c}{d}$ are proportional, then $a\times d=b\times c$.

You should receive $426.4$ Euros $\left(426.4\text{€}\right)$.

Step 1: Set up the two ratios into a proportion. Notice the Earth weights are both in the numerator and the Mars weights are both in the denominator.

Step 2: Cross multiply, and then divide to solve.

So the 1,000-pound horse would weigh about 376.5 pounds on Mars.

Step 1: Set up the two ratios into a proportion. Notice that the cups of flour are both in the numerator and the amounts of cookies are both in the denominator. To make the calculations more efficient, the cups of flour $\left(2\frac{1}{4}\right)$ is converted to a decimal number (2.25).

Step 2: Cross multiply, and then simplify to solve.

Jackie will need 38.25, or $38\frac{1}{4}$, cups of flour to bake 1,020 cookies.

To find the constant of proportionality, divide the pay by hours using the information from any of the four columns. For example, $\frac{87.5}{7}=12.5$. The constant of proportionality is 12.5, or $12.50. This tells you Isabelle's hourly pay: For every hour she works, she gets paid $12.50.

The constant of proportionality in this problem is 4 miles per hour (or 4 miles in 1 hour). Since $\frac{a}{b}=k,$ where $k$ is the constant of proportionality, we have

$\frac{a\text{miles}}{b\text{hours}}=k$

$\frac{a}{4.5}=4$ (30 minutes is $\mathrm{½}$, or $0.5$, hours)

$a=4\left(4.5\right)$, since from the definition we know $a=kb$

$a=18$

Zac ran 18 miles.

The constant of proportionality in this situation is $2.50 per bucket (or $2.50 for 1 bucket). Since $\frac{a}{b}=k,$ where $k$ is the constant of proportionality, we have

$$\begin{array}{ccc}\hfill \frac{a\text{dollars}}{b\text{buckets}}& \hfill =\hfill & k\hfill \\ \hfill \frac{337.50}{b}& \hfill =\hfill & 2.50\hfill \end{array}$$

Since we are solving for $b$, and we know from the definition that $$b=\frac{a}{k}:$$

$$\begin{array}{ccc}\hfill b& \hfill =\hfill & \frac{337.50}{2.50}\hfill \\ \hfill b& \hfill =\hfill & 135\hfill \end{array}$$

Joe filled 135 buckets.

The constant of proportionality in this situation is 0.62 miles per 1 kilometer. Since $\frac{a}{b}=k,$ where $k$ is the constant of proportionality, we have

$$\begin{array}{ccc}\hfill \frac{a\text{miles}}{b\text{kilometers}}& \hfill =\hfill & k\hfill \\ \hfill \frac{a}{104}& \hfill =\hfill & 0.62\hfill \\ \hfill a& \hfill =\hfill & 0.62(104)\hfill \\ \hfill a& \hfill =\hfill & 64.48\hfill \end{array}$$

Rounding the answer to the nearest tenth, Mabel has to drive 64.5 miles.

When the southern border is measured with a ruler, the length is 4 inches. Since the length of the border in real life is 380 miles, our scale is 1 inch $=95$ miles.

The eastern and western borders both measure 3 inches, so their lengths are about 285 miles. The northern border measures the same as the southern border, so it has a length of 380 miles.

The scale tells us that 1 inch of the model car is equal to 24 inches (2 feet) on the real automobile. So set up the two ratios into a proportion. Notice that the model lengths are both in the numerator and the NASCAR automobile lengths are both in the denominator.

This amount (216) is in inches. To convert to feet, divide by 12, because there are 12 inches in a foot (this conversion from inches to feet is really another proportion!). The final answer is:

The NASCAR automobile is 18 feet long.