Ratios, Rates, and Proportions

Turn a ratio into equal-sized parts

A mixture has red and blue beads in the ratio $3:5$. There are 8 equal parts in all. If there are 40 beads, each part contains 5 beads, giving 15 red and 25 blue.

The fraction of red beads is $\frac{3}{8}$, not $\frac{3}{5}$. The latter compares red to blue; the former compares red to the total. Decide which comparison the question requests.

Find a unit rate before scaling

If 6 kilograms cost 27 dollars, the unit price is $\frac{27}{6}=4.5$ dollars per kilogram. At the same rate, 10 kilograms cost 45 dollars. This assumes a constant unit price; a fixed fee would require a different model.

A proportional relationship has the form $y=kx$ and passes through $(0,0)$. A model such as $y=4x+7$ is linear but not proportional because of the added constant.

Cancel units through conversions

To convert 72 kilometers per hour into meters per second, compute $72\cdot \frac{1000}{3600}=20$. The factor 1000 converts kilometers to meters, while 3600 converts hours to seconds in the denominator.

Write the unit cancellation beside the arithmetic. If converting square units, square the conversion factor: one square meter equals 10,000 square centimeters, because each of its two dimensions contains 100 centimeters.

Check whether the proportion keeps the same order

If 4 notebooks cost 10 dollars, then $\frac{4}{10}=\frac{n}{25}$ compares notebooks to dollars on both sides. Solving gives $n=10$. Writing $\frac{4}{10}=\frac{25}{n}$ swaps the comparison and answers a different equation.

For a miss, record whether the error was total parts, unit order, fixed fee, or conversion power. Repeat that setup before adding time pressure.

Check your understanding

Sample ratios, rates, and proportions questions

Practice this skill

1. Label the numerator and denominator of every ratio.
2. Solve one unit rate, one part-to-total problem, and one conversion.
3. Check whether doubling the input should double the output before using a proportion.

Print the ratios, rates, and proportions worksheet and worked answers.

Then use a mixed practice module to check whether you can choose this method among other question types.