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.

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 $2:7$ .

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 $2:7$ .

Note that it doesn't matter if there are $100$ or $\mathrm{10,000}$ 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 $23:4$ .

Write the rate as a fraction.

$\frac{170\text{wingbeats}}{10\text{seconds}}$

Divide both the numerator and the denominator by ten.

$=\frac{17\text{wingbeats}}{1\text{second}}$

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

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

$\frac{50\text{meters}}{1\text{hour}}\cdot \frac{8\text{hours}}{1\text{day}}=50(8)\frac{\text{meters}}{\text{day}}$

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

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.

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