Practice Maths

L27 — Adding & Subtracting Mixed Numbers

Key Terms

mixed number
A whole number and a proper fraction written together, e.g., 234. It means “2 wholes and three-quarters more.”
improper fraction
A fraction where the numerator is greater than the denominator, e.g., 114. Convert: whole × denominator + numerator over denominator.
lowest common denominator (LCD)
The smallest number that is a multiple of both denominators. Required before adding or subtracting fractions.

Two Methods

Method 1 (recommended): Convert all mixed numbers to improper fractions, find the LCD, add or subtract, then convert back.

Method 2: Add/subtract the whole-number parts and fraction parts separately. Useful for addition; can get complicated when subtracting requires borrowing.

Converting mixed → improper: 234 = (2 × 4) + 34 = 114

Converting improper → mixed: 114: 11 ÷ 4 = 2 remainder 3, so 234

Worked Example

Calculate 213 + 134

Step 1 — LCD of 3 and 4 is 12: rewrite as 2412 + 1912

Step 2 — Add: whole parts 2 + 1 = 3; fraction parts 412 + 912 = 1312 = 1112

Step 3 — Combine: 3 + 1112 = 4112

What Is a Mixed Number?

A mixed number combines a whole number and a proper fraction — like 234 cups of flour, or 112 hours of sport. They show up constantly in cooking, carpentry, and everyday measurement.

Converting Between Forms

Mixed to improper: 234 means “2 whole things plus 34 of another.” Since each whole has 4 quarters, 2 wholes = 8 quarters. So 234 = 84 + 34 = 114. Shortcut: (2 × 4 + 3) ÷ 4 = 114.

Improper to mixed: 114 — divide 11 by 4. 4 goes in 2 times (2 wholes), with 3 left over. So 114 = 234.

Method 1: Convert to Improper Fractions

This is the most reliable method, especially for subtraction. Let’s calculate 213 + 134:

  1. Convert: 213 = 73,   134 = 74
  2. LCD of 3 and 4 is 12: 73 = 2812   and   74 = 2112
  3. Add: 2812 + 2112 = 4912
  4. Convert back: 49 ÷ 12 = 4 remainder 1, so 4112

Method 2: Separate Whole and Fraction Parts

For addition, this is often quicker. 213 + 134: whole parts: 2 + 1 = 3. Fraction parts: 13 + 34 = 412 + 912 = 1312 = 1112. Total: 3 + 1112 = 4112. Same answer!

Method 2 can get tricky in subtraction when the fraction part being subtracted is larger than the fraction part you have — you need to “borrow” from the whole number. Method 1 (convert to improper fractions) avoids this complication entirely.

Real-Life Context: The Carpenter’s Problem

A carpenter has 312 m of timber and uses 134 m. How much is left?

  • Convert: 312 = 72 = 144.   134 = 74.
  • Subtract: 144 − 74 = 74 = 134 m remaining.
Common Mistake: When converting a mixed number, the formula is (whole × denominator + numerator) over denominator. A very common error is writing 234 as (2+3)/4 = 54 instead of the correct 114. Always multiply the whole number by the denominator first.

  1. Convert to improper fractions

    1. 112
    2. 234
    3. 325
    4. 413
    5. 538

  2. Convert to mixed numbers

    1. 73
    2. 114
    3. 175
    4. 136
    5. 238

  3. Add

    1. 112 + 214
    2. 223 + 116
    3. 314 + 112
    4. 125 + 2310
    5. 213 + 134

  4. Subtract

    1. 334 − 112
    2. 413 − 223
    3. 512 − 234
    4. 616 − 356
    5. 438 − 234

  5. Calculate

    1. 212 + 134 − 114
    2. 423 − 112 + 56
    3. 313 + 216 − 112

  6. Carpenter Timber

    A carpenter has 312 m of timber. He uses 134 m. How much is left?

  7. Jogging Distance

    Sarah jogged 213 km on Monday and 156 km on Tuesday. How far did she jog in total?

  8. Jug of Water

    A jug holds 234 litres. 118 litres are poured out. How much remains?

  9. Multi-Step Calculation

    Calculate: 325 + 234 − 112

  10. Enough Rope?

    Tom needs 5 m of rope. He has two pieces: 213 m and 156 m. Does he have enough? Show your working.

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