Ratios, Proportions, and Percents
Ratios, Proportions, and Percents
Ratios and Proportions
A ratio is a comparison of two quantities. For example, if a class consists of 15 women and 10 men, the ratio of women to men is 15 to 10. This ratio can also be written as 15:10 or \(\frac{15}{10}\)
Ratios, like fractions, can be reduced by dividing by common factors.
A proportion is a statement that two ratios are equal. For example, the proportion \(\frac{5}{10}=\frac{7}{14}\) is true because both ratios are equal to \(\frac{1}{2}\)
The cross product is found by multiplying the numerator of one fraction by the denominator of the other (across the equal sign).
\(\frac{a}{b}=\frac{c}{d}\rightarrow ad=bc\)
The fact that the cross products of proportions are equal can be used to solve proportions in which one of the values is missing. Use x to represent the missing value, then cross multiply and solve.
\(\frac{5}{x}=\frac{7}{14}\)
5(14) = x(7)
70 = 7x
x = 10
Percents
A percent (or percentage) means per hundred and is expressed with the percent symbol (%). For example, 54% means 54 out of 100. Percentages are converted to decimals by moving the decimal point two places to the left.
54% = 0.54
Percentages can be solved by setting up a proportion.
\(\frac{part}{whole}=\frac{\%}{100}\)
Percent change involves a change from an original amount. Often percent change problems appear as word problems that include discounts, growth, or markups.
In order to solve percent change problems, it is necessary to identify the percent change (as a decimal), the amount of change, and the original amount. (Keep in mind that one of these will be the value being solved for.) These values can then be substituted in the equations below.
\(amount\ of\ change=original\ amount\ ×\ percent\ change\)
\(percent\ change = \frac{amount\ of\ change}{original\ amount}\)
\(original\ amount = \frac{amount\ of\ change}{percent\ change}\)