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➗ Math · Numbers and Algebra · Lesson 7 of 8 · 7 min

Ratios, proportions and unit rates

🚗 Miles per hour, drops per minute, cups of flour per batch: if you can set up a proportion, you can solve them all the same way.

💡 Key idea

Set up matching units on both sides, cross-multiply, and divide to find a unit rate.

🧠 The idea

A RATIO compares two quantities: 3 nurses to 12 patients is 3:12, which simplifies to 1:4. A ratio can also split a total into parts: in a 2:3 ratio, the total has 2 + 3 = 5 parts, so 40 items split 16 and 24. A PROPORTION says two ratios are equal, such as 3/12 = x/20. Solve it by CROSS-MULTIPLYING: 3 × 20 = 12x, so 60 = 12x and x = 5. Keep the units in the same positions on both sides (patients over nurses on the left means patients over nurses on the right). A UNIT RATE is an amount per one unit, found by dividing: 150 miles in 3 hours is 150 ÷ 3 = 50 miles per hour. Unit rates make price comparisons easy: $4.80 for 12 oz is $0.40 per oz, which beats $6.30 for 14 oz at $0.45 per oz.

✏️ Worked example

A recipe uses 3 cups of rice for 8 servings. How many cups are needed for 20 servings?
Step 1: Set up the proportion with cups over servings: 3/8 = x/20.
Step 2: Cross-multiply: 8x = 3 × 20 = 60.
Step 3: Divide both sides by 8: x = 60 ÷ 8 = 7.5.
Step 4: Check the rate: 3 ÷ 8 = 0.375 cup per serving, and 0.375 × 20 = 7.5. ✓
Answer: 7 1/2 cups.

📌 Takeaways

  • In a ratio a:b, the total is a + b parts.
  • Proportion: keep units in matching positions, then cross-multiply.
  • Unit rate = total ÷ number of units.

📖 Terms in this lesson

Ratio: A comparison of two quantities by division; for example, 3 nurses to 12 patients is a ratio of 1:4.

Proportion: An equation stating that two ratios are equal, often solved by cross-multiplying; for example, 2/5 = x/20, so x = 8.

Rate: A ratio comparing quantities with different units; for example, 120 mL per hour.

✅ Check yourself

Question 1 of 3

A car travels 225 miles on 9 gallons of gas. At this rate, how many miles can it travel on 14 gallons?

  1. 350 miles
  2. 230 miles
  3. 375 miles
  4. 126 miles
Show the answer and rationale

Answer: A

Unit rate: 225 ÷ 9 = 25 miles per gallon. Then 25 × 14 = 350 miles. The answer 126 multiplies the gallons together (9 × 14), 230 adds the extra 5 gallons as if they were 5 miles, and 375 uses 15 gallons instead of 14.

Question 2 of 3

Red and blue beads are in a ratio of 3:5. There are 48 beads in all. How many are blue?

  1. 30
  2. 40
  3. 18
  4. 16
Show the answer and rationale

Answer: A

The total has 3 + 5 = 8 parts, so each part is 48 ÷ 8 = 6 beads. Blue is 5 parts: 5 × 6 = 30. The answer 18 is the red beads, and 16 divides 48 by 3 as if there were only 3 parts.

Question 3 of 3

Which is the best buy?

  1. 16 oz for $4.96
  2. 12 oz for $3.30
  3. 10 oz for $3.20
  4. 8 oz for $2.40
Show the answer and rationale

Answer: B

Find each unit price: $2.40 ÷ 8 = $0.30, $3.20 ÷ 10 = $0.32, $3.30 ÷ 12 = $0.275, $4.96 ÷ 16 = $0.31 per ounce. The 12-ounce size has the lowest price per ounce. The 8-ounce size has the lowest total price, but that is not the same as the best rate.

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