Ratio Concepts

Grade 6 · ratios · 100 practice problems · read aloud

🔊 Listen to this explanation

Ratio Concepts: Comparing Quantities

What is a Ratio? 🤔

A ratio is a way to compare two or more quantities. It shows the relative size of one thing to another. We use ratios in real life for recipes, maps, and comparing speeds! Ratios can be written in three ways: using the word "to" (3 to 2), a colon (3:2), or as a fraction (3/2).

How to Work with Ratios: Step-by-Step

  1. Identify the two quantities you are comparing.
  2. Write the ratio in the order asked.
  3. Simplify the ratio by dividing both numbers by their greatest common factor (GCF), just like simplifying a fraction.
  4. Keep the units the same for both parts of the ratio.

Visual Examples

Example 1: Simplifying a Ratio

A classroom has 12 girls and 16 boys. What is the ratio of girls to boys? Simplify it.

  1. Identify: Girls = 12, Boys = 16.
  2. Write: 12 to 16, or 12:16.
  3. Simplify: The GCF of 12 and 16 is 4. 12 ÷ 4 = 3, 16 ÷ 4 = 4.
  4. Simplified Ratio: 3:4.

Example 2: Equivalent Ratios

A recipe requires a ratio of 2 cups of flour to 3 cups of milk. How much milk is needed for 6 cups of flour?

  1. Set up equivalent ratios: 2/3 = 6/m
  2. Notice that 2 was multiplied by 3 to get 6 (2 × 3 = 6).
  3. Do the same to 3: 3 × 3 = 9.
  4. Answer: 9 cups of milk are needed.

Common Mistakes to Avoid 🚫

  • Wrong Order: The ratio of cats to dogs is different from dogs to cats! Always check what is being compared first.
  • Forgetting to Simplify: Always write ratios in simplest form, just like fractions.
  • Mixing Units: Don't compare inches to feet! Convert to the same unit first.

Tips & Tricks

  • GCF is Your Friend: Use the greatest common factor to simplify ratios quickly.
  • Double-Check: After finding an equivalent ratio, make sure you can multiply or divide both parts by the same number to get back to the original.
  • Word Clues: The word "per" often means a ratio (like miles per hour).

How to Practice

To master ratios, try these activities:

  • Find ratios in your daily life (e.g., the ratio of windows to doors in your house).
  • Use online math games that focus on ratios and proportions.
  • Practice with worksheets that ask you to simplify ratios and find missing values in equivalent ratios.

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 3:4 = 15:?

Hint: Remember that equivalent ratios maintain the same proportional relationship. Think about what number you need to multiply the first term by to get the second term.

Show the answer

Answer: 20

  1. Write the proportion as fractions. 3/4 = 15/x where x is the unknown number.
  2. Cross-multiply to solve for x. 3 * x = 4 * 15
  3. Perform the multiplication on the right side. 3 * x = 60
  4. Divide both sides by 3 to isolate x. x = 60 / 3
  5. Calculate the division. x = 20
  6. Conclusion. The missing term is 20, so 3 : 4 = 15 : 20. Final answer: 20

We are given the proportion: 3 : 4 = 15 : ?

2 3:5 = 18:?

Hint: Remember that equivalent ratios maintain the same proportional relationship. If you have a ratio like 2:3 = 8:x, you can find x by determining what number maintains the same relationship.

Show the answer

Answer: 30

  1. Write the proportion as fractions. 3:5 means 3/5, and 18:? means 18/x, where x is the unknown number. So we have: 3/5 = 18/x
  2. Cross-multiply to solve for x. 3 * x = 18 * 5
  3. Perform the multiplication. 3x = 90
  4. Divide both sides by 3 to isolate x. x = 90 / 3 x = 30
  5. Check the answer. 3:5 = 18:30 3/5 = 0.6, 18/30 = 0.6, so it matches. Final answer: 30

We are given the proportion: 3:5 = 18:?

3 2/3 ÷ 4/9 = ?

Hint: When dividing fractions, remember to multiply by the reciprocal of the second fraction. For example, when dividing 1/2 by 1/4, you would multiply 1/2 by 4/1.

Show the answer

Answer: 1.5

  1. Write down the problem. 2/3 ÷ 4/9
  2. Change the division to multiplication by the reciprocal of the second fraction. The reciprocal of 4/9 is 9/4. So the problem becomes: 2/3 × 9/4
  3. Multiply the fractions. Multiply the numerators: 2 × 9 = 18 Multiply the denominators: 3 × 4 = 12 This gives us: 18/12
  4. Simplify the fraction. Both 18 and 12 can be divided by 6. 18 ÷ 6 = 3 12 ÷ 6 = 2 So we get 3/2
  5. Convert the fraction to a decimal. 3 divided by 2 equals 1.5 Therefore, the final answer is 1.5.

To divide fractions, we use the rule: dividing by a fraction is the same as multiplying by its reciprocal.

4 (2/3) ÷ (4/9) = ?

Hint: Remember that dividing by a fraction is the same as multiplying by its reciprocal.

Show the answer

Answer: 3/2

  1. Write the division of fractions as multiplication by the reciprocal: (2/3) ÷ (4/9) = (2/3) × (9/4)
  2. Multiply the numerators: 2 × 9 = 18
  3. Multiply the denominators: 3 × 4 = 12
  4. Write the resulting fraction: 18/12
  5. Simplify the fraction by dividing numerator and denominator by their greatest common factor (6): 18 ÷ 6 = 3, 12 ÷ 6 = 2
  6. The simplified fraction is 3/2

The answer is 3/2.

5 (2/3) ÷ (5/6) = ?

Hint: When dividing fractions, multiply by the reciprocal of the second fraction. For example, to divide 1/2 by 3/4, you would multiply 1/2 by 4/3.

Show the answer

Answer: 4/5

  1. Write the division of fractions as multiplication by the reciprocal: (2/3) ÷ (5/6) = (2/3) × (6/5)
  2. Multiply the numerators: 2 × 6 = 12
  3. Multiply the denominators: 3 × 5 = 15
  4. Simplify the fraction 12/15 by dividing numerator and denominator by 3: 12 ÷ 3 = 4, 15 ÷ 3 = 5
  5. The simplified fraction is 4/5

The answer is 4/5.

6 If 2/3 = x/15, then x = ?

Hint: When two ratios are equal, you can use cross-multiplication to find the missing value. For example, if a/b = c/d, then a × d = b × c.

Show the answer

Answer: 10

  1. The equation means that the two fractions are equal. To solve for x, we can cross-multiply. That means: 2 * 15 = 3 * x
  2. Perform the multiplication on the left side: 2 * 15 = 30 So we have: 30 = 3 * x
  3. Divide both sides by 3 to solve for x: 30 / 3 = x
  4. Calculate the division: 30 ÷ 3 = 10 So x = 10
  5. Check the answer: 2/3 = 10/15 Simplify 10/15 by dividing numerator and denominator by 5: 10 ÷ 5 = 2, 15 ÷ 5 = 3, so 10/15 = 2/3. This matches the original equation. Final answer: x = 10

We start with the equation: 2/3 = x/15

Practise this topic — 10 free problems, no signup →