Compare ratios: word problems

Key Notes :

Understanding Ratios

  • A ratio is a comparison of two quantities using division.
  • It can be written in three ways: a to b, a:b, or a/b.

Types of Ratio Comparisons

  • Equivalent Ratios: Ratios that represent the same relationship (e.g., 2:3 = 4:6).
  • Comparing Ratios: Use cross-multiplication or simplify to determine which ratio is greater.

Solving Ratio Word Problems

  • Read the Problem Carefully: Identify the two quantities being compared.
  • Write the Ratios: Express the given numbers in ratio form.
  • Find a Common Denominator: Convert ratios to equivalent forms if necessary.
  • Use Cross-Multiplication: If needed, compare two ratios by multiplying across.
  • Check for Unit Consistency: Ensure both ratios use the same units.

Proportional Relationships

  • If two ratios are equal, they form a proportion (e.g., 3:4 = 6:8).
  • Use proportions to find missing values in ratio problems.

Real-Life Applications

  • Comparing recipes (e.g., cups of sugar to flour).
  • Finding the best deal in shopping (price per unit).
  • Speed comparisons (miles per hour).
  • Population densities (people per square mile).

Common Mistakes to Avoid

  • Not simplifying ratios before comparing.
  • Ignoring units in real-world problems.
  • Mixing up the order of terms in a ratio.

Learn with an example

✈️ A restaurant critic reviewed restaurants in Lowell and Morristown. In Lowell, the critic gave 18 positive reviews and 12 negative reviews. In Morristown, 19 of the reviews were positive and 13 were negative.

In which city did the restaurant critic give a higher ratio of positive to negative reviews?

  • Lowell
  • Morristown
  • neither; the ratios are equivalent

The ratio of positive to negative reviews by the restaurant critic for Lowell was 18 to 12. The ratio for Morristown was 19 to 13.

We want to figure out which ratio is higher: 18/12 or 19/13 .

We can compare the ratios more easily if we express them as percentages.

First write the ratio 18/12 as a decimal. Then convert the decimal to a percentage.

18/12 = 1.5 = 150%

Do the same thing for 19/13 .

19/13 ≈ 1.46154 = 146.154%

Now compare the percentages.

150% ? 146.154%

150% is larger than 146.154%.

The restaurant critic gave a higher ratio of positive to negative reviews in Lowell.

✈️ Last season, Emily’s soccer team won 6 games and lost 8 games. His cousin Amy’s team won 11 games and lost 20 games.

Which team had a higher ratio of wins to losses?

  • Emily’s team
  • Amy’s team
  • neither; the ratios are equivalent

Last season, the ratio of wins to losses for Emily’s team was 6 to 8. The ratio for Amy’s team was 11 to 20.

We want to figure out which ratio is higher: 6/8 or 11/20 .

We can compare the ratios more easily if we express them as percentages.

First write the ratio 6/8 as a decimal. Then convert the decimal to a percentage.

6/8 = 0.75 = 75%

Do the same thing for 11/20

11/20 = 0.55 = 55%

Now compare the percentages.

75% ? 55%

75% is larger than 55%.

Emily’s team had a higher ratio of wins to losses.

✈️ Tiana and Aaron are solicitors. Tiana averages 12 civil cases and 10 criminal cases annually. Meanwhile, Aaron averages 17 civil cases and 13 criminal ones.

Which solicitor has a higher ratio of civil to criminal cases?

  • Tiana
  • Aaron
  • neither; the ratios are equivalent

For Tiana, the ratio of civil to criminal cases is 12 to 10. For Aaron, the ratio is 17 to 13.

We want to figure out which ratio is higher: 12/10 or 17/13 .

We can compare the ratios more easily if we express them as percentages.

First write the ratio 12/10 as a decimal. Then convert the decimal to a percentage.

12/10 = 1.2 = 120%

Do the same thing for 17/13 .

17 /13 ≈ 1.30769 = 130.769%

Now compare the percentages.

120% ? 130.769%

130.769% is larger than 120%.

Aaron has a higher ratio of civil to criminal cases.

let’s practice! 🖊️