1.4. Ratios, Rates, and Percent
Ratios and Rates
Ratio: a relationship between two numbers, expressed as a quotient with the same unit in the denominator and the numerator. There are three ways to write a ratio.
The ratio of a and b is: a to b or a : b or
Example: Write the ratio of 5 cents to 9 cents.
5 to 9 or 5 : 9 or
- Write a ratio in lowest terms (simplify):
– Write the ratio in a fractional form.
– Simplify and drop the units if given (as they cancel each other out).
Example:
Example: 0.75 metres to 0.25 metres
Rate: a ratio of two quantities with different units.
Example: teachers to students; money to time; distance to time, etc.
, ,
- Write a rate in lowest terms (simplify the rate):
Example: 80 kilometres per 320 minutes:
÷80
Unit rate: a rate in which the number in the denominator is 1.
- Some unit rates:
– Miles (or kilometres) per hour (or minute).
– Cost (dollars/cents) per item or quantity.
– Earnings (dollars) per hour (or week).
Proportion: an equation with a ratio (or rate) on two sides (), in which the two ratios are equal.
Example: Write the following sentence as a proportion.
3 printers is to 18 computers as 2 printers is to 12 computers.
Solving a proportion:
- Cross multiply: multiply along two diagonals.
- Solve for the unknown.
Example 1.4.1
4 litres of milk cost $4.38. What is the cost of 2 litres?
|
| 4 L milk | 2 L milk |
| $4.38 | $ x = ? |
|
||
|
||
|
Divide both sides by 4. | |
| 2 litres of milk cost $2.19. | ||
|
Replace x with 2.19. | |
| (4) (2.19) = (2) (4.38) | ||
| 8.76 = 8.76 | Correct! |
Example 1.4.2
Tom’s height is 1.75 metres, and his shadow is 1.09 metres long. A building’s shadow is 10 metres long at the same time. How high is the building?
|
| Tom’s height = 1.75 m | Building’s height (x) = ? |
| Tom’s shadow = 1.09 m | Building’s shadow = 10m |
|
||
|
||
|
Divide both sides by 1.09. | |
| The building’s height is 16.055m. | ||
|
Replace x with 16.055. | |
| Correct! |
Example 1.4.3
If 15 mL of medicine must be mixed with 180 mL of water, how many mL of medicine must be mixed in 230 mL of water?
|
||
|
||
|
||
| 19.17 mL of medicine must be mixed in 230 mL of water. |
Percent
Percent (%): one part per hundred.
Converting between percent, decimals and fractions:
| Conversion | Steps | Example |
| Percent ⇒ Decimal | Move the decimal point two places to the left, then remove %. | 31% = 31.% = 0.31 |
| Decimal ⇒ Percent | Move the decimal point two places to the right, then insert %. | 0.317 = 0. 317 = 31.7 % |
| Percent ⇒ Fraction | Remove %, divide by 100, then simplify. | 15% = |
| Fraction ⇒ Percent | Divide, move the decimal point two places to the right, then insert %. | = 1÷4 = 0.25 = 25 % |
| Decimal ⇒ Fraction | Convert the decimal to a percent, then convert the percent to a fraction. | 0.35 = 35 % =
% = per one hundred |
There are two methods to solve percent problems:
- Percent proportion method
- Translation (translate the words into mathematical symbols.)
Percent proportion method:
| or |
|