Practice Maths

Adding & Subtracting Fractions

The Golden Rule

To add or subtract fractions, the denominators (the bottom numbers) must be exactly the same.

Key Terms

denominator
the bottom number in a fraction; shows how many equal parts make a whole
numerator
the top number in a fraction; shows how many of those parts are being counted
common denominator
a shared denominator for two or more fractions, needed before adding or subtracting
lowest common multiple (LCM)
the smallest number that is a multiple of two or more numbers; used to find the common denominator
equivalent fraction
a fraction with the same value but a different numerator and denominator (e.g. 12 = 48)
simplify
to write a fraction in its lowest terms by dividing the numerator and denominator by their greatest common factor
Hot Tip Whatever you do to the bottom, you MUST do to the top! If you multiply the denominator by 3, you must multiply the numerator by 3.

Worked Example

Calculate 13 + 14

Step 1: Find the Lowest Common Multiple (LCM) of 3 and 4.
Multiples of 3: 3, 6, 9, 12 …
Multiples of 4: 4, 8, 12 …
LCM = 12. This is the common denominator.

Step 2: Convert both fractions to have denominator 12.
13 = 412    and    14 = 312

Step 3: Add the numerators, keep the denominator.
412 + 312 = 712

Why You Can't Add Apples and Oranges

Imagine you have 1 quarter of a pizza and 1 third of a pizza — but they're different sized fractions. Can you just say "2 somethings"? No! Adding 14 + 13 is like adding 1 apple + 1 orange and calling it "2 appleoranges." It makes no sense until you convert them to the same unit.

The common denominator is that shared unit. Once both fractions are in the same "language" (same denominator), you can add the numerators freely, because now you're counting the same-sized pieces.

Finding the Lowest Common Multiple (LCM)

The LCM is the smallest number that both denominators divide into evenly. For denominators 3 and 4:

  • Multiples of 3: 3, 6, 9, 12, 15…
  • Multiples of 4: 4, 8, 12, 16…
  • LCM = 12. So we convert both fractions to have denominator 12.

13 becomes 412 (multiply top and bottom by 4). 14 becomes 312 (multiply top and bottom by 3). Now they speak the same language!

So: 13 + 14 = 412 + 312 = 712.

The Golden Rule: Same to Top AND Bottom

When you convert 13 to 412, you multiply the denominator by 4. You MUST also multiply the numerator by 4. Why? Because multiplying top and bottom by the same number is the same as multiplying by 1 (since 44 = 1). Multiplying by 1 doesn't change the value — just the appearance.

Think of it like exchanging money: $1 = 100 cents. The value is the same, just expressed differently. Similarly, 13 = 412. Same amount of pizza, just cut into different-sized pieces.

Remember: Whatever you multiply (or divide) the denominator by, you MUST do the same to the numerator. If you forget the top, you're changing the value of the fraction, not just the way it's written.

Subtraction: Same Process, Different Operation

Subtracting fractions follows exactly the same steps — find the common denominator, convert, then subtract the numerators. The denominator never changes in the final answer!

Example: 34 − 13. LCM = 12. So: 912 − 412 = 512.

Always simplify your answer if possible. For example, 612 = 12. Check by dividing top and bottom by their greatest common factor.

Common Mistake: Adding or subtracting the denominators. NEVER do this! 14 + 14 = 24 (not 28). The denominator stays the same because the size of the pieces doesn't change — only the number of pieces does.

Practice Questions

  1. Simplifying Fractions Fluency

    Simplify these fractions to their lowest terms:

    1. 412
    2. 1520
    3. 1421
    4. 25100

  2. Same Denominator Addition Fluency

    Calculate and simplify where possible:

    1. 15 + 25
    2. 38 + 18
    3. 49 + 29
    4. 712 + 112

  3. Same Denominator Subtraction Fluency

    Calculate and simplify:

    1. 78 − 38
    2. 56 − 16
    3. 911 − 411
    4. 1015 − 515

  4. Finding the LCM Fluency

    Find the Lowest Common Multiple (LCM) for each pair of numbers:

    1. 3 and 4
    2. 2 and 5
    3. 6 and 8
    4. 10 and 15

  5. Equivalent Fractions Understanding

    Find the missing numerator to make each pair of fractions equal:

    1. 12 = ?10
    2. 23 = ?12
    3. 35 = ?20
    4. 56 = ?18

  6. Easy Different Denominators Understanding

    One denominator is a multiple of the other. Solve and simplify:

    1. 12 + 16
    2. 13 + 19
    3. 15 + 110
    4. 14 + 18

  7. Medium Different Denominators Understanding

    Find a common denominator, then solve and simplify:

    1. 23 + 14
    2. 35 + 12
    3. 16 + 14
    4. 27 + 13

  8. Subtraction Challenge Understanding

    Solve and simplify:

    1. 34 − 13
    2. 45 − 12
    3. 56 − 14
    4. 710 − 25

  9. The Pizza & Party Problems Problem Solving

    1. Aussie Jack eats 18 of a pizza for lunch and 12 for dinner. What total fraction of the pizza is eaten?
    2. A jug of juice is 34 full. Sarah drinks 13 of a jug. How much juice remains?

  10. Trail Mix Challenge Problem Solving

    Aaliyah is making trail mix. She uses 14 cup of nuts, 13 cup of dried fruit, and 38 cup of chocolate chips.

    1. What is the total amount of trail mix she makes? Express your answer as a fraction of a cup in simplest form.
    2. How much more would she need to add to completely fill a 1-cup bowl?
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