Ratios and rates: ratios and rates

Compare 8 miles and 3 miles by subtraction.

$\text{8 mile}-\text{3 miles}=\text{5 miles}$

This means that 8 miles is 5 miles more than 3 miles.

Examples of use : I can now jog 8 miles whereas I used to jog only 3 miles. So, I can now jog 5 miles more than I used to.

Compare 12 and 5 by subtraction.

$\text{12}-5=7$

This means that 12 is 7 more than 5.

Comparing 8 miles and 5 gallons by subtraction makes no sense.

$\text{8 miles}-\text{5 gallons}=?$

Compare 36 and 4 by division.

$\text{36}÷4=9$

This means that 36 is 9 times as large as 4. Recall that $\text{36}÷4=9$ can be expressed as $\frac{\text{36}}{4}=9$ .

Compare 8 miles and 2 miles by division.

$\frac{\text{8 miles}}{\text{2 miles}}=4$

This means that 8 miles is 4 times as large as 2 miles.

Example of use : I can jog 8 miles to your 2 miles. Or, for every 2 miles that you jog, I jog 8. So, I jog 4 times as many miles as you jog.

Notice that when like quantities are being compared by division, we drop the units. Another way of looking at this is that the units divide out (cancel).

Compare 30 miles and 2 gallons by division.

$\frac{\text{30 miles}}{\text{2 gallons}}=\frac{\text{15 miles}}{\text{1 gallon}}$

Example of use : A particular car goes 30 miles on 2 gallons of gasoline. This is the same as getting 15 miles to 1 gallon of gasoline.

Notice that when the quantities being compared by division are unlike quantities, we do not drop the units.