Compare ratios: word problems

key notes :

🔢 What is a Ratio?

  • A ratio compares two quantities using :`, to, or fraction form
    👉 Example: 2 : 3, 2 to 3, or 2/3

🧠 Understanding Ratio Word Problems

Ratios can compare:

  • 👦👧 People (boys : girls)
  • 🍎🍌 Objects (apples : bananas)
  • ⏱️ Time, distance, money, or quantities

🔍 How to Compare Ratios (Step-by-Step)

  1. 📖 Read carefully and identify the two ratios.
  2. 🔄 Write ratios in the same form (preferably fractions).
  3. ⚖️ Make denominators the same (find equivalent ratios).
  4. ➗ Compare the numbers to decide which ratio is larger, smaller, or equal.

✏️ Example

  • Ratio A: 3 boys : 5 girls → 3/5
  • Ratio B: 4 boys : 6 girls → 4/6 = 2/3
    👉 Compare 3/5 and 2/3
    👉 2/3 is greater, so Ratio B is larger ✔️

🔁 Using Equivalent Ratios

  • Multiply or divide both terms by the same number
    👉 2 : 3 = 4 : 6 = 6 : 9
  • Helps in easy comparison 👍

📈 Cross-Multiplication Trick

For fractions a/b and c/d:

  • Multiply 👉 a × d and b × c
  • Compare the results

Bigger product = bigger ratio ⭐

🧩 Real-Life Applications

  • 🏫 Classroom strength (boys vs girls)
  • 🧃 Mixing juice (water : syrup)
  • 💰 Comparing prices and discounts
  • 🚗 Speed and distance problems

⚠️ Common Mistakes to Avoid

  • ❌ Comparing ratios with different units
  • ❌ Not simplifying ratios
  • ❌ Ignoring the context of the problem

🌟 Quick Tip

Always compare ratios in the same form and unit before deciding! ✅

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! 🖊️