Ratios, Proportions, and Percentages

Ratios also follow the rules of fractions. Performing arithmetic operations on ratios follows the same procedures as on fractions. Ratios should also generally appear in lowest terms. Therefore, the constituent numbers in a ratio represent the relative quantities of each side, not absolute quantities. For example, because the ratio 1:2 is equal to 2:4, 5:10, and 600:1,200, ratios are insufficient to determine the absolute number of entities in a problem.

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BE CAREFUL!

Avoid confusing standard ratios with odds (such as “3:1 odds”). Both may use a colon, but their meanings differ. In general, a ratio is the same as a fraction containing the same numbers.

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KEY POINT

Mathematically, ratios act just like fractions.
For example, the ratio 8:13 is mathematically the same as the fraction \(\frac{8}{13}\).

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Ratios Review

Proportions

A proportion is an equation of two ratios. An illustrative case is two equivalent fractions:

\(\frac{21}{28}\)=\(\frac{3}{4}\)

This example of a proportion should be familiar: going left to right, it is the conversion of one fraction to an equivalent fraction in lowest terms by dividing the numerator and denominator by the same number (7, in this case).

Equating fractions in this way is correct, but it provides little information. Proportions are more informative when one of the numbers is unknown. Using a question mark (?) to represent an unknown number, setting up a proportion can aid in solving problems involving different scales. For instance, if the ratio of maple saplings to oak saplings in an acre of young forest is 7:5 and that acre contains 65 oaks, the number of maples in that acre can be determined using a proportion: \(\frac{7}{5}=\frac{?}{65}\).

Note that to equate two ratios in this manner, the numerators must contain numbers that represent the same entity or type, and so must the denominators. In this example, the numerators represent maples and the denominators represent oaks.

\(\frac{7 \hspace{.5mm}maples}{5 \hspace{.5mm}oaks}=\frac{?\hspace{.5mm} maples}{65 \hspace{.5mm}oaks}\)

Recall from the properties of fractions that if you multiply the numerator and denominator by the same number, the result is an equivalent fraction.

Therefore, to find the unknown in this proportion, first divide the denominator on the right by the denominator on the left.
65 ÷ 5 = 13

Then, multiply the quotient by the numerator on the left.
\(\frac{7\times 13}{5\times 13}=\frac{?}{65}\)

The unknown (?) is 7 × 13 = 91. In the example, the acre of forest has 91 maple saplings.

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DID YOU KNOW?

When taking the reciprocal of both sides of a proportion, the proportion still holds. When setting up a proportion, ensure that the numerators represent the same type and the denominators represent the same type.

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When solving for a proportion, they may be written as a ratio expression, such as 25:50 :: x:100. In this example, 25:50 is an expression of \(\frac{25}{50}\) and is equivalent to the value of \(\frac{x}{100}\). Solving for the proportion determines that x = 50.

Proportions Review

Rates of Change

Numbers that describe current quantities can be informative, but how they change over time can provide even greater insight into a problem. The rate of change for some quantity is the ratio of the quantity’s difference over a specific time period to the length of that period. For example, if an automobile increases its speed from 50 mph to 100 mph in 10 seconds, the rate of change of its speed (its acceleration) is

where \(t_f\) is the “final” (or ending) time and \(t_i\) is the “initial” (or starting) time. Also, \(x_f\) is the (final) quantity at (final) time\(t_f\), and \(x_i\) is the (initial) quantity at (initial) time \(t_i\). In this formula, the numerator is the difference between the two quantities and the denominator is the difference in time.

In the example above, the final time is 10 seconds and the initial time is 0 seconds—hence the omission of the initial time from the calculation.

Consider that the final quantity occurs at the final time, and the initial quantity occurs at the initial time. As long as both quantities stay consistent in comparison, like with proportions, the order of the terms in the formula can be reversed. 

\(\frac{x_f-x_i}{t_f-t_i}=\frac{x_i-x_f}{t_i-t_f}\)

This concept stays consistent with the rules of fractions, because multiplying the equation by -1 will result in the reversible formula.

The key to getting the correct rate of change is to ensure that the first number in the numerator and the first number in the denominator correspond to each other (that is, the quantity from the numerator corresponds to the time from the denominator). This must also be true for the second number.

Rates of Change Review

Let’s Review!

• A percent—meaning “per hundred”—represents a relative quantity as a fraction or decimal. It is the absolute number multiplied by 100 and followed by the % symbol.
• A ratio is a relationship between two numbers expressed using fraction or colon notation (for example, \(\frac{3}{2}\) or 3:2). Ratios behave mathematically just like fractions.
• An equation of two ratios is called a proportion. Proportions are used to solve problems involving scale.
• Rates of change are the speeds at which quantities increase or decrease. The formula \(\frac{x_f-x_i}{t_f-t_i}\) provides the rate of change of quantity x over the period between some initial (i) time and final (f) time.