Rational Addition/Subtraction

Grade 7 · ratios · 69 practice problems · read aloud

🔊 Listen to this explanation

Adding and Subtracting Rational Numbers

🧠 What Is It & Why Is It Useful?

This operation involves adding or subtracting numbers that can be written as fractions (ratios). We use this all the time in real life, like when combining ingredients in a recipe or figuring out how much time you've spent on homework!

📝 Step-by-Step Guide

  1. Find a Common Denominator: The bottom numbers (denominators) must be the same.
  2. Rewrite the Fractions: Change each fraction to an equivalent one with the common denominator.
  3. Add/Subtract the Numerators: Add or subtract the top numbers (numerators). Keep the denominator the same.
  4. Simplify: Always reduce your answer to its simplest form.

🔢 Worked Examples

Example 1: 1/4 + 1/2

  1. Common Denominator: 4
  2. Rewrite: 1/4 + (1×2)/(2×2) = 1/4 + 2/4
  3. Add Numerators: 1 + 2 = 3
  4. Answer: 3/4

Example 2: 2/3 - 1/5

  1. Common Denominator: 15
  2. Rewrite: (2×5)/(3×5) - (1×3)/(5×3) = 10/15 - 3/15
  3. Subtract Numerators: 10 - 3 = 7
  4. Answer: 7/15

⚠️ Common Mistakes to Avoid

Don't Add Denominators! When adding 1/4 + 2/4, the denominator stays 4. You do not add 4+4 to get 8. Only the numerators get added.

Check Your Signs! When subtracting, make sure you subtract the entire second numerator. For example, in 5/8 - 3/8, you calculate 5 - 3, not 5 - -3.

💡 Tips & Tricks

Butterfly Method (Shortcut): To quickly add/subtract fractions without finding the LCD, you can cross-multiply and add, then multiply the denominators. For a/b + c/d, the answer is (ad + bc)/(bd). Always simplify after!

Double-Check with Decimals: Convert your fractions to decimals to quickly check if your final answer makes sense.

🎯 How to Practice

  • Start with problems that have the same denominator.
  • Move on to problems where one denominator is a multiple of the other.
  • Finally, practice with problems that have unrelated denominators.
  • Create your own real-world problems, like splitting a pizza or a chocolate bar!

Practice problems

6 of the 69, worked through step by step — try them before opening the answer.

1 -3/4 + 2/3 - 1/6 = ?

Hint: When adding and subtracting fractions with different denominators, first find a common denominator for all terms. Remember to maintain the correct signs for negative fractions throughout the calculation.

Show the answer

Answer: -1/4

  1. Find a common denominator for all fractions. The denominators are 4, 3, and 6. The least common multiple of 4, 3, and 6 is 12.
  2. Convert each fraction to have denominator 12. -3/4 = ( -3 × 3 ) / (4 × 3) = -9/12 2/3 = ( 2 × 4 ) / (3 × 4) = 8/12 -1/6 = ( -1 × 2 ) / (6 × 2) = -2/12
  3. Rewrite the expression with the common denominator. -9/12 + 8/12 - 2/12
  4. Combine the numerators over the common denominator. Numerator: -9 + 8 - 2 First: -9 + 8 = -1 Then: -1 - 2 = -3 So we have: -3/12
  5. Simplify the fraction. -3/12 = -1/4 (dividing numerator and denominator by 3) Final answer: -1/4

Let's solve step-by-step: We have: -3/4 + 2/3 - 1/6

2 -2/3 + 5/4 - 1/6 = ?

Hint: When adding and subtracting fractions with different denominators, first find a common denominator for all terms. Remember to handle negative signs carefully when combining the numerators.

Show the answer

Answer: 5/12

  1. Find a common denominator for the fractions 3, 4, and 6. The least common multiple of 3, 4, and 6 is 12.
  2. Rewrite each fraction with denominator 12. - For -2/3: Multiply numerator and denominator by 4 → (-2 × 4)/(3 × 4) = -8/12 - For 5/4: Multiply numerator and denominator by 3 → (5 × 3)/(4 × 3) = 15/12 - For -1/6: Multiply numerator and denominator by 2 → (-1 × 2)/(6 × 2) = -2/12
  3. Now the expression is: -8/12 + 15/12 - 2/12
  4. Combine the numerators over the common denominator: (-8 + 15 - 2) / 12
  5. Simplify the numerator: -8 + 15 = 7 7 - 2 = 5 So we have: 5/12 Final answer: 5/12

Let's solve step by step: -2/3 + 5/4 - 1/6

3 (-3/4) + (7/8) - (-1/2) = ?

Hint: Remember to find a common denominator when adding or subtracting fractions, and be careful with negative signs.

Show the answer

Answer: 5/8

  1. Write the expression: (-3/4) + (7/8) - (-1/2)
  2. Subtracting a negative is the same as adding: (-3/4) + (7/8) + (1/2)
  3. Find a common denominator for 4, 8, and 2. The least common denominator is 8.
  4. Convert each fraction: -3/4 = -6/8, 7/8 = 7/8, 1/2 = 4/8
  5. Add the fractions: (-6/8) + (7/8) + (4/8) = (-6 + 7 + 4)/8 = 5/8
  6. The answer is 5/8.

4 (-5/6 + 2/3) - (3/4 - 7/12) = ?

Hint: First simplify each expression inside parentheses by finding common denominators, then perform the subtraction between the results.

Show the answer

Answer: -1/3

  1. Simplify (-5/6 + 2/3) Find common denominator for 6 and 3: 6 -5/6 = -5/6 2/3 = 4/6 -5/6 + 4/6 = -1/6
  2. Simplify (3/4 - 7/12) Find common denominator for 4 and 12: 12 3/4 = 9/12 7/12 = 7/12 9/12 - 7/12 = 2/12 = 1/6
  3. Perform the final subtraction: (-1/6) - (1/6) -1/6 - 1/6 = -2/6 = -1/3

The answer is -1/3.

5 (-3/5 + 7/10) - (2/3 - 5/6) = ?

Hint: First simplify each expression inside the parentheses by finding common denominators, then combine the results

Show the answer

Answer: 4/5

  1. Simplify (-3/5 + 7/10) Find common denominator: 10 -3/5 = -6/10 -6/10 + 7/10 = 1/10
  2. Simplify (2/3 - 5/6) Find common denominator: 6 2/3 = 4/6 4/6 - 5/6 = -1/6
  3. Combine the results: 1/10 - (-1/6) = 1/10 + 1/6
  4. Find common denominator for 1/10 + 1/6 Common denominator: 30 1/10 = 3/30 1/6 = 5/30 3/30 + 5/30 = 8/30
  5. Simplify 8/30 = 4/15

The answer is 4/15.

6 (-7/8 + 3/4) - (-5/6 + 2/3) = ?

Hint: When working with multiple operations on fractions, find a common denominator for each set of parentheses before combining the results.

Show the answer

Answer: 1/8

  1. Calculate the first parentheses: -7/8 + 3/4 Find common denominator for 8 and 4: 8 -7/8 = -7/8 3/4 = 6/8 -7/8 + 6/8 = -1/8
  2. Calculate the second parentheses: -5/6 + 2/3 Find common denominator for 6 and 3: 6 -5/6 = -5/6 2/3 = 4/6 -5/6 + 4/6 = -1/6
  3. Subtract the results: (-1/8) - (-1/6) Subtracting a negative is adding: -1/8 + 1/6
  4. Find common denominator for 8 and 6: 24 -1/8 = -3/24 1/6 = 4/24 -3/24 + 4/24 = 1/24

The answer is 1/24.

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