Fractions, Ratios, and Rates

Read fractions as division and distinguish part-to-part ratios from part-to-whole fractions. Use units and denominators to interpret rates, averages, and normalized weighted sums.

A fraction can represent a number, a division, or a comparison between two quantities. A ratio names the quantities being compared, and a rate also keeps track of their units. These distinctions become important when we work with percentages, averages, probabilities, and measured change.

Fractions Are Division

The fraction

34\frac{3}{4}

means 33 divided by 44, so its decimal value is 0.750.75. The number above the bar is the numerator. The number below it is the denominator:

numeratordenominator.\frac{\text{numerator}}{\text{denominator}}.

The denominator cannot be zero because division by zero is not defined. For example, 62=3\frac{6}{2}=3 asks how many groups of size 22 make 66. There is no corresponding number of zero-sized groups that makes 66.

Exercise: Read a fraction as division

What is 72\frac{7}{2} as a decimal?

Compute it first, then check your number.

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HintUse the fraction bar

Read the bar as division: 7÷27\div2.

SolutionDivide numerator by denominator

The fraction means:

72=7÷2=3.5\frac{7}{2}=7\div2=3.5

The result is larger than 11 because the numerator is larger than the denominator.

Ratios Compare Quantities

Suppose a batch contains 88 correct predictions and 22 incorrect predictions. The ratio of correct to incorrect predictions is:

8:2=4:18:2 = 4:1

The fraction of all predictions that are correct uses a different denominator:

88+2=810=0.8\frac{8}{8+2}=\frac{8}{10}=0.8

The first expression compares correct predictions with incorrect predictions. The second compares correct predictions with the whole batch. A ratio is not fully understood until both quantities and their order are named. The ratio 4:14:1 could describe four correct predictions for every one incorrect prediction, but those numbers alone do not say what was counted.

A percentage is a part-to-whole fraction expressed per hundred. Because 0.8×100=800.8\times100=80, the correct predictions make up 80%80\% of the batch.

Exercise: Choose the relevant denominator

A dataset contains 3030 images, of which 1212 show cats. What fraction of the dataset shows cats?

Compute it first, then check your number.

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HintCompare the part with the whole

The numerator is the number of cat images. The denominator is the total number of images.

SolutionForm the part-to-whole fraction

The required fraction is:

1230=25=0.4\frac{12}{30}=\frac{2}{5}=0.4

The numerator counts the cat images, while the denominator counts every image in the dataset. Thus 40%40\% of the dataset shows cats.

Rates Include a Unit

A rate is a ratio between quantities measured in different units. If a program processes 600600 examples in 33 seconds, its average processing rate is:

600 examples3 seconds=200 examples per second\frac{600\ \text{examples}}{3\ \text{seconds}} =200\ \text{examples per second}

The denominator is reduced to one second, so 200200 examples per second is a unit rate. Reversing the ratio gives:

3 seconds600 examples=0.005 seconds per example.\frac{3\ \text{seconds}}{600\ \text{examples}} =0.005\ \text{seconds per example}.

This reciprocal rate is also correct, but it answers a different question. Keeping the units beside the numbers helps us notice which quantity is in the numerator and which is in the denominator.

Exercise: Compute a unit rate

A sensor records 480480 measurements in 66 seconds. What is the average number of measurements recorded per second?

Compute it first, then check your number.

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HintKeep the requested unit

The answer must have measurements in the numerator and seconds in the denominator.

SolutionDivide by the elapsed time

The requested rate is:

480 measurements6 seconds=80 measurements per second.\frac{480\ \text{measurements}}{6\ \text{seconds}} =80\ \text{measurements per second}.

Dividing in the opposite order would give seconds per measurement, which is a different rate.

Equivalent Fractions and Proportions

Multiplying the numerator and denominator by the same nonzero number is equivalent to multiplying the fraction by 11, so its value does not change:

25=25⋅33=2⋅35⋅3=615.\frac{2}{5} =\frac{2}{5}\cdot\frac{3}{3} =\frac{2\cdot3}{5\cdot3} =\frac{6}{15}.

The two quantities scale together, so their relative size stays the same. Dividing the numerator and denominator by a shared nonzero factor simplifies the fraction for the same reason. These are equivalent fractions: their written forms differ, but their values are equal.

An equation stating that two ratios are equal is called a proportion. For example,

38=x40\frac{3}{8}=\frac{x}{40}

is a proportion. Here the denominator is multiplied by 55, so the numerator must be multiplied by 55 as well. This gives x=15x=15. The same reasoning can be used whenever one scale factor connects corresponding quantities.

Exercise: Complete a proportion

Complete the equality:

38= ? 40.\frac{3}{8}=\frac{\,?\,}{40}.

Compute it first, then check your number.

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HintUse the same scale factor

The denominator was multiplied by 55. An equivalent fraction requires the same change in the numerator.

SolutionScale both parts equally

Since 8⋅5=408\cdot5=40, multiply the numerator by 55 as well:

38=3⋅58⋅5=1540.\frac{3}{8} =\frac{3\cdot5}{8\cdot5} =\frac{15}{40}.

The missing numerator is 1515.

Weighted Averages Are Ratios

The arithmetic mean of nn values x1,…,xnx_1,\ldots,x_n is their sum divided by the number of values:

xˉ=x1+⋯+xnn\bar{x}=\frac{x_1+\cdots+x_n}{n}

A weighted average lets some values contribute more than others. It divides a weighted sum by the total weight:

∑iwixi∑iwi\frac{\sum_i w_i x_i}{\sum_i w_i}

For the usual weighted-average interpretation, the weights are nonnegative and their sum is greater than zero. The denominator matters when the weights do not already sum to 11. It normalizes the weighted sum so that multiplying every weight by the same positive number does not change the average.

When the weights are nonnegative, the weighted average lies between the smallest and largest values. This gives a quick check on the calculation.

Exercise: Compute a weighted average

Two scores are 44 and 1010, with weights 11 and 22. What is their weighted average?

Compute it first, then check your number.

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HintSeparate the two totals

First find the weighted sum. Then divide by the sum of the weights.

SolutionNormalize the weighted sum

The weighted sum is:

1⋅4+2⋅10=241\cdot4+2\cdot10=24

The weights sum to 33, so:

243=8\frac{24}{3}=8

The larger weight on 1010 pulls the average closer to 1010.

Read the Denominator First

The denominator tells us what the numerator is being compared with: the whole, an elapsed time, a number of observations, or a total weight. Before computing a fraction, ratio, rate, or average, name both quantities and state the unit or comparison that the quotient will represent. This small habit prevents many otherwise plausible calculations from answering the wrong question.

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