Rational Multiplication/Division

Grade 7 · ratios · 74 practice problems · read aloud

🔊 Listen to this explanation

Multiplying and Dividing Ratios

🧠 What is it and Why is it Useful?

Ratios show the relationship between two quantities. Multiplying and dividing ratios helps us scale recipes, compare different situations, and solve real-world problems like adjusting serving sizes or calculating unit rates.

📝 Step-by-Step Guide

  1. Write the ratios as fractions. For example, 3:4 becomes 3/4.
  2. For Multiplication: Multiply the numerators together and the denominators together. Simplify if needed.
  3. For Division: Multiply the first ratio by the reciprocal (flipped version) of the second ratio. Then simplify.

🔍 Visual Examples

Example 1: Multiplication

Multiply 2:3 and 5:7

Step 1: Write as fractions: (2/3) × (5/7)

Step 2: Multiply across: (2 × 5) / (3 × 7) = 10/21

Answer: 10:21

Example 2: Division

Divide 4:5 by 2:3

Step 1: Write as fractions: (4/5) ÷ (2/3)

Step 2: Multiply by the reciprocal: (4/5) × (3/2)

Step 3: Multiply across: (4 × 3) / (5 × 2) = 12/10

Step 4: Simplify: 6/5 or 6:5

⚠️ Common Mistakes

Adding instead of multiplying: Remember, you don't find a common denominator like in addition.

Forgetting to simplify: Always reduce your final answer to its simplest form.

Mixing up division: Division means multiplying by the reciprocal. Don't just divide the numbers straight across.

💡 Tips & Tricks

KCF for Division: Keep the first fraction, Change the sign to multiplication, Flip the second fraction.

Cancel early: Before multiplying, see if any numbers in the numerator and denominator can be canceled out. This makes the math easier!

Double-check with units: Does your answer make sense? If you're scaling a recipe, your final quantity should be reasonable.

🎯 Practice Suggestions

  • Start with simple whole number ratios before trying ratios with fractions or decimals.
  • Create your own "recipe problems." For example, if a cookie recipe for 2 dozen uses 3 cups of flour, how much flour is needed for 5 dozen?
  • Use online practice websites or flashcards to build speed and accuracy.
  • Always write out each step instead of trying to do it all in your head.

Practice problems

6 of the 74, worked through step by step — try them before opening the answer.

1 -3/4 × 2/5 ÷ (-1/2) = ?

Hint: When multiplying and dividing signed fractions, work step by step and pay attention to the signs. Remember that dividing by a fraction is the same as multiplying by its reciprocal.

Show the answer

Answer: 3/5

  1. Understand the division of fractions** Dividing by (-1/2) is the same as multiplying by its reciprocal. Reciprocal of (-1/2) is -2/1, which is -2. So: -3/4 × 2/5 ÷ (-1/2) = -3/4 × 2/5 × (-2) --- **
  2. Multiply the numerators together** Numerators: (-3) × 2 × (-2) First: (-3) × 2 = -6 Then: (-6) × (-2) = 12 --- **
  3. Multiply the denominators together** Denominators: 4 × 5 × 1 = 20 So we have: 12/20 --- **
  4. Simplify** 12/20 = (12 ÷ 4)/(20 ÷ 4) = 3/5 --- **Final Answer:** 3/5

Let's solve step-by-step. We have: -3/4 × 2/5 ÷ (-1/2) --- **

2 -2/3 × (5/4 ÷ -3/8) = ?

Hint: Remember to follow the order of operations and pay attention to the signs when multiplying and dividing fractions.

Show the answer

Answer: 20/9

  1. Start with the division inside the parentheses: 5/4 ÷ -3/8
  2. Dividing by a fraction is the same as multiplying by its reciprocal: 5/4 × -8/3
  3. Multiply the fractions: (5 × -8)/(4 × 3) = -40/12
  4. Simplify the fraction: -40/12 = -10/3
  5. Now multiply by -2/3: -2/3 × -10/3
  6. Multiply the fractions: (-2 × -10)/(3 × 3) = 20/9
  7. The final answer is 20/9.

3 -2/3 × (5/6 ÷ -3/4) = ?

Hint: Remember to simplify fractions before multiplying and pay attention to the signs when dividing negative numbers.

Show the answer

Answer: 20/27

  1. Start with the division inside the parentheses: 5/6 ÷ -3/4
  2. Dividing by a fraction is the same as multiplying by its reciprocal: 5/6 × (-4/3)
  3. Multiply the fractions: (5 × -4)/(6 × 3) = -20/18
  4. Simplify -20/18 by dividing numerator and denominator by 2: -10/9
  5. Now multiply by -2/3: -2/3 × -10/9
  6. Multiply the fractions: (-2 × -10)/(3 × 9) = 20/27
  7. The positive result comes from multiplying two negative numbers

The answer is 20/27.

4 -3/4 × (2/5 ÷ -1/10) = ?

Hint: When dividing fractions, multiply by the reciprocal. Remember the rules for multiplying positive and negative numbers.

Show the answer

Answer: 3

  1. Handle the division inside parentheses** 2/5 ÷ -1/10 Dividing by a fraction is the same as multiplying by its reciprocal: = 2/5 × (-10/1) Multiply numerators: 2 × (-10) = -20 Multiply denominators: 5 × 1 = 5 So: -20/5 = -4 Now the expression becomes: -3/4 × (-4) --- **
  2. Multiply** -3/4 × (-4) Write -4 as -4/1: Numerators: (-3) × (-4) = 12 Denominators: 4 × 1 = 4 So: 12/4 = 3 --- **
  3. Final answer** The result is 3.

Let's solve step-by-step. We have: -3/4 × (2/5 ÷ -1/10) --- **

5 -3.5 × (2.4 ÷ (-0.6)) = ?

Hint: When working with signed decimals, remember to handle the division inside the parentheses first, then apply the multiplication while carefully tracking the signs.

Show the answer

Answer: 14

  1. Handle the division inside the parentheses** 2.4 ÷ (-0.6) Dividing a positive number by a negative number gives a negative result. 2.4 ÷ 0.6 = 4, so 2.4 ÷ (-0.6) = -4. So now the expression becomes: -3.5 × (-4) **
  2. Multiply the two numbers** -3.5 × (-4) Multiplying two negative numbers gives a positive result. 3.5 × 4 = 14. So the final answer is: 14

Let's solve step-by-step. We have: -3.5 × (2.4 ÷ (-0.6)) **

6 -2.5 × (3.6 ÷ (-0.9)) = ?

Hint: Remember to follow the order of operations and pay attention to the signs when multiplying and dividing positive and negative numbers.

Show the answer

Answer: 10

  1. Start with the expression: -2.5 × (3.6 ÷ (-0.9))
  2. First, solve the division inside the parentheses: 3.6 ÷ (-0.9) = -4
  3. Now the expression becomes: -2.5 × (-4)
  4. Multiply the two numbers: -2.5 × (-4) = 10
  5. Since a negative times a negative equals a positive, the result is positive 10.

The answer is 10.

Practise this topic — 10 free problems, no signup →