Practice Maths

L43 — Adding and Subtracting Fractions

Key Terms

LCD (lowest common denominator)
The smallest number that is a multiple of all the denominators involved. Required before adding or subtracting fractions. E.g. LCD of 3 and 4 is 12.
equivalent fraction
A fraction that represents the same value with a different numerator and denominator. E.g. 23 = 812.
improper fraction
A fraction where the numerator is greater than or equal to the denominator. E.g. 74 = 134.
mixed number
A number written as a whole number plus a proper fraction. E.g. 212.
simplify
Reduce a fraction to its lowest terms by dividing numerator and denominator by the HCF. E.g. 68 = 34.

Adding and Subtracting Fractions

  1. Find the lowest common denominator (LCD).
  2. Convert both fractions to equivalent fractions with the LCD.
  3. Add or subtract the numerators only — the denominator stays the same.
  4. Simplify the result.

Mixed numbers: Either convert to improper fractions first, or add/subtract the whole number and fraction parts separately.

Hot Tip: NEVER add or subtract the denominators! Only the numerators change. 13 + 13 = 23, NOT 26.

Worked Example 1 — 23 + 34

Step 1: LCD of 3 and 4 = 12.

Step 2: 23 = 812    34 = 912

Step 3: 812 + 912 = 1712 = 1512

Worked Example 2 — 2½ − 1¾

Step 1: Convert to improper fractions: 2½ = 52    1¾ = 74

Step 2: LCD = 4: 52 = 104

Step 3: 104 − 74 = 34

Building on What You Know

You’ve already seen fraction addition in earlier lessons. This lesson extends those skills to include mixed numbers and more complex denominators. The core rule never changes: same denominator before adding or subtracting numerators.

Think of fractions as different sized cups. Adding 23 + 34 is like pouring from two different measuring cups — you need to convert everything to the same unit first. The LCD is that common unit.

Adding Mixed Numbers: Two Approaches

For 2½ + 1¾, you have a choice:

Approach 1 — Separate the whole and fraction parts:

  • Add the whole numbers: 2 + 1 = 3
  • Add the fractions: LCD of 2 and 4 = 4. 12 = 24. So 24 + 34 = 54 = 1¼
  • Combine: 3 + 1¼ = 4¼

Approach 2 — Convert to improper fractions first:

  • 2½ = 52, 1¾ = 74. LCD = 4. So 104 + 74 = 174 = 4¼

Same answer! Approach 2 is safer when subtracting (avoids borrowing headaches).

Remember: The golden rule: denominators must match before you add or subtract. NEVER add denominators together. 13 + 13 = 23, NOT 26. The denominator stays the same because the size of each piece hasn’t changed.

Subtracting Mixed Numbers: The Borrowing Problem

The subtraction 5½ − 2¾ looks straightforward, but notice: 12 = 24 < 34, so you can’t subtract directly! You need to “borrow” from the whole number part, or use improper fractions:

  • 5½ = 112 = 224, 2¾ = 114. So 224 − 114 = 114 = 2¾.
Common Mistake: Subtracting the larger fraction from the smaller and ignoring the borrow. For 5½ − 2¾, some students write ¾ − ½ = ¼ and get 3¼ — which is wrong. Always check: if the fraction being subtracted is larger, borrow or convert to improper fractions first.

  1. Add fractions with the same denominator

    1. 15 + 35
    2. 38 + 18
    3. 29 + 59
    4. 512 + 712
    5. 411 + 611
    6. 715 + 415
    7. 310 + 910
    8. 57 + 47

  2. Add fractions with different denominators

    1. 12 + 14
    2. 13 + 16
    3. 23 + 14
    4. 34 + 16
    5. 25 + 310
    6. 14 + 23
    7. 56 + 38
    8. 35 + 56

  3. Subtract fractions

    1. 34 − 14
    2. 56 − 13
    3. 78 − 12
    4. 35 − 14
    5. 56 − 34
    6. 910 − 25
    7. 712 − 14
    8. 56 − 79

  4. Add mixed numbers

    1. 112 + 214
    2. 213 + 116
    3. 325 + 1310
    4. 234 + 158
    5. 423 + 256
    6. 179 + 323

  5. Subtract mixed numbers

    1. 334 − 112
    2. 456 − 213
    3. 512 − 334
    4. 614 − 258
    5. 413 − 125
    6. 716 − 434

  6. Check the working — is it correct? If not, fix it.

    1. Student writes: 13 + 14 = 27. Correct or not?
    2. Student writes: 35 + 15 = 45. Correct or not?
    3. Find the LCD for 56 + 38 and explain each conversion step.
    4. Is 34 + 23 greater than or less than 2? Estimate first, then calculate to check.

  7. Word problems with fractions

    1. A recipe uses 23 cup of milk and 34 cup of cream. How much liquid in total?
    2. Tom jogged 234 km on Monday and 158 km on Tuesday. What is the total distance?
    3. A plank of wood is 312 m long. A piece of 134 m is cut off. How long is the remaining piece?
    4. In a class, 25 of students play football and 14 play tennis. What fraction play either sport? What fraction play neither?
    5. Lily worked 412 hours on Saturday and 323 hours on Sunday. How many more hours did she work on Saturday?

  8. Add and subtract three fractions

    1. 12 + 13 + 16
    2. 34 + 18 − 38
    3. 1 − 25 − 14
    4. 56 − 14 + 13
    5. 23 + 34 − 56
    6. 710 + 25 − 34

  9. Fraction chains and equations

    1. Find the missing fraction: 34 + ☐ = 114
    2. Find the missing fraction: 56 − ☐ = 12
    3. Find the missing fraction: ☐ + 23 = 116
    4. Find the missing fraction: 2 − ☐ = 138
    5. Is it possible for ab + cd = a+cb+d? Explain with an example or counterexample.

  10. Multi-step fraction word problems

    1. A jug contains 312 litres of juice. Amy pours out 34 of a litre and then Ben adds 114 litres. How much juice is now in the jug?
    2. A builder has three pieces of timber: 234 m, 158 m, and 312 m. What is the total length? If a piece of 414 m is needed, how much extra must be purchased?
    3. In a relay race, the four legs are 38 km, 14 km, 512 km and 724 km. What is the total race distance? Which two legs have the same combined length as the other two?
    4. A water tank starts 56 full. During the day, 13 of the full tank is used, and then it rains and the tank gains 14 of its full capacity. What fraction of the full tank does it contain at the end of the day?
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