Rates & Ratios

A ratio is a comparison of two numbers. A ratio can be written using a colon, $3:5$ , or as a fraction $\frac{3}{5}$ .

A rate , by contrast, is a comparison of two quantities which can have different units. For example $5$ miles per $3$ hours is a rate, as is $34$ dollars per square foot.

Example 1:

A punch recipe calls for $6$ ounces of lime juice, $21$ ounces of apricot juice, and $21$ ounces of pineapple juice. What is the ratio of lime juice to apricot juice?

Writing the ratio using a colon, we get $6:21$ .

Note that this can be reduced, like a fraction, by dividing both numbers by a common factor -- in this case, $3$ . In simplest form, the ratio is $2:7$ .

Example 2:

In the recipe above, what is the ratio of apricot juice to the total amount of punch?

To find the total amount of punch, add $6+21+21=48$ .

The ratio of apricot juice to the total amount of punch is $21:48$ . But this ratio is probably more clearly written as a fraction, since the apricot juice makes up a fraction of the whole.

$\frac{21}{48}$

To reduce the fraction, divide both the numerator and the denominator by $3$ .

$\frac{7}{16}$

Note that this can be reduced, like a fraction, by dividing both numbers by a common factor -- in this case, $3$ . In simplest form, the ratio is $2:7$ .

Example 3:

An adult scolopendromorph centipede has $46$ legs and $8$ eyes. In a group of $100$ centipedes of the same species, what is the ratio of legs to eyes?

Note that it doesn't matter if there are $100$ or $10,000$ centipedes; the ratio of legs to eyes will remain the same.

Writing the ratio using a colon, we get $46:8$ .

Divide both numbers by $2$ . In simplest form, the ratio of legs to eyes is $23:4$ .

Example 4:

A bat beats its wings $170$ times in $10$ seconds. Write the rate as a fraction in lowest terms.

Write the rate as a fraction.

$\frac{170\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{wingbeats}}{10\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{seconds}}$

Divide both the numerator and the denominator by ten.

$=\frac{17\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{wingbeats}}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{second}}$

So, the rate is $17$ beats per second.

Example 5:

A mountain climber is $3200$ meters from the peak. He climbs $50$ meters per hour for $8$ hours per day. How many days will it be before he reaches the peak?

The first job is to figure out the rate per day.

$\frac{50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{meters}}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{hour}}\cdot \frac{8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{hours}}{1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{day}}=50\left(8\right)\frac{\text{meters}}{\text{day}}$

$=400\frac{\text{meters}}{\text{day}}$

He is climbing at a rate of $400$ meters per day.

Now divide $3200$ by the daily rate to find the number of days it will take him to reach the top.

$\frac{3200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{meters}}{400\frac{\text{meters}}{\text{day}}}=8\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{days}$