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T.1 Add, subtract, multiply, and divide fractions and mixed numbers

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What are fractions and mixed numbers?

Fractions show parts of a whole and have a numerator (top number) and a denominator (bottom number). A mixed number combines a whole number and a fraction (for example, 213).

Step-by-step example: Read and convert a mixed number
  1. Start with the mixed number: 213.
  2. Identify the whole part and the fraction: whole = 2, fraction = 13.
  3. To convert to an improper fraction: multiply the whole by the denominator and add the numerator:
    whole × denominator = 2 × 3 = 6.
    add numerator: 6 + 1 = 7.
  4. Place that sum over the original denominator: improper fraction = 73.
  5. Interpretation: 213 = 2 + 13 = 73.
Note

Converting mixed numbers to improper fractions is often the first step before multiplying or dividing.

Adding fractions and mixed numbers

To add fractions with different denominators, find a common denominator (often the least common denominator), convert each fraction, add the numerators, then simplify. For mixed numbers you may add whole numbers and fractions separately or convert to improper fractions first.

Example 1:
Add fractions with unlike denominators: 34 + 23
  1. Identify denominators: 4 and 3. Least common denominator (LCD) = 12.
  2. Convert each fraction to denominator 12:
    • 34 = 3×34×3 = 912.
    • 23 = 2×43×4 = 812.
  3. Add numerators: 9 + 8 = 17 → result is 1712.
  4. Convert improper fraction to mixed number: 17 ÷ 12 = 1 remainder 5 → 1 512 .
  5. Check for simplification: 5 and 12 have no common factor > 1, so final answer is 1 512 .
Example 2:
Add mixed numbers: 215 + 1310
  1. Find a common denominator for the fractional parts: 5 and 10 → LCD = 10.
  2. Convert the fractional parts:
    • 15 = 210 (because 1/5 = 2/10).
    • So 215 = 2 210.
  3. Add fractional parts: 210 + 310 = 510 → simplify 12.
  4. Add whole-number parts: 2 + 1 = 3. Combine with the fractional sum: 3 + 12 = 3 12.
Note

Either add whole numbers and fraction parts separately or convert mixed numbers to improper fractions first—use whichever is clearer for the problem.

Subtracting fractions and mixed numbers

To subtract fractions, use a common denominator and subtract numerators. For mixed numbers, if the fractional part of the minuend (the first number) is smaller than the subtrahend's fraction, borrow 1 whole and rename it as an equivalent fraction with the needed denominator.

Example 1:
Subtract fractions with unlike denominators: 78 − 13
  1. Denominators: 8 and 3. LCD = 24.
  2. Convert each fraction:
    • 78 = 2124 (×3)
    • 13 = 824 (×8)
  3. Subtract numerators: 21 − 8 = 13 → 1324
  4. Simplify if possible: 13 and 24 have no common factor > 1 → final answer 1324
Example 2:
Subtract mixed numbers with borrowing: 518 − 234
  1. Write the fractions with the same denominator: 18 and 34 = 68
  2. Compare fractional parts: 18 − 68 is not possible without borrowing because 1 < 6.
  3. Borrow 1 whole from 5: 5 becomes 4, and add 88 to the fractional part: 18 + 88 = 98. Now the minuend is 498.
  4. Subtract fractions: 98 − 68 = 38
  5. Subtract whole numbers: 4 − 2 = 2. Combine results: 2 38
Note

When borrowing, rewrite the mixed number so the fractional part has the common denominator; this keeps subtraction straightforward.

Multiplying fractions and mixed numbers

To multiply fractions, multiply numerators and multiply denominators. For mixed numbers, first convert to improper fractions. Factors (the numbers you multiply) can be whole numbers, fractions, or mixed numbers.

Example 1:
Whole number × fraction: 3 × 25
  1. Write the whole number as a fraction: 3 = 31.
  2. Multiply numerators and denominators: (31) × (25) → numerator = 3×2 = 6; denominator = 1×5 = 5 → 65.
  3. Convert to mixed number: 6 ÷ 5 = 1 remainder 1 → 1 15 .
Example 2:
Fraction × fraction (use cancellation): 23 × 34
  1. Multiply across: numerator = 2×3 = 6; denominator = 3×4 = 12 → 612.
  2. Simplify by dividing numerator and denominator by their greatest common factor, 6: 612 = 12 → final answer 12 .
Example 3: Mixed number × fraction: 112 × 23
  1. Convert mixed number to improper fraction: 112 = 32 (because 1×2 + 1 = 3).
  2. Multiply: 32 × 23 → numerator = 3×2 = 6, denominator = 2×3 = 6 → 66.
  3. Simplify: 66 = 1 → final answer 1. (Cancellation before multiplying also shows 3 and 3 cancel, 2 and 2 cancel.)
Note

Always convert mixed numbers to improper fractions before multiplying. Cancel common factors when possible to simplify calculations early.

Dividing fractions and mixed numbers

To divide by a fraction, multiply by its reciprocal (flip numerator and denominator). For mixed numbers, convert to improper fractions first. Remember: Keep — Change — Flip (keep the first fraction, change the ÷ to ×, flip the second fraction).

Example 1:
Fraction ÷ fraction: 34 ÷ 12
  1. Keep, Change, Flip: (34) ÷ (12) → (34) × (21).
  2. Multiply: numerator 3×2 = 6; denominator 4×1 = 4 → 64.
  3. Simplify / convert: 64 = 32 = 1 12.
Example 2:
Mixed number ÷ fraction: 213 ÷ 34
  1. Convert mixed number to improper fraction: 213 = 73 (2×3 + 1 = 7).
  2. Keep, Change, Flip: (73) ÷ (34) → (73) × (43).
  3. Multiply: numerator 7×4 = 28; denominator 3×3 = 9 → 289.
  4. Convert to mixed number: 28 ÷ 9 = 3 remainder 1 → 3 19 .
Example 3:
Fraction ÷ whole number: 25 ÷ 3
  1. Write the whole number as a fraction: 3 = 31.
  2. Keep, Change, Flip: (25) ÷ (31) → (25) × (13).
  3. Multiply: numerator 2×1 = 2; denominator 5×3 = 15 → 215 .
Note

Always convert mixed numbers to improper fractions and then apply Keep–Change–Flip. Simplify as soon as you can, using cancellation before multiplying if possible.

Checking answers and simplifying

After any operation, simplify fractions to lowest terms and convert improper fractions to mixed numbers when appropriate. This makes answers clearer and easier to compare.

Example 1:
Simplify a fraction: 612
  1. Find the greatest common factor (GCF) of 6 and 12: GCF = 6.
  2. Divide numerator and denominator by 6: (6÷612÷6) = 12.
  3. Final simplified answer: 12.
Example 2:
Convert improper fraction to mixed number: 114
  1. Divide numerator by denominator: 11 ÷ 4 = 2 remainder 3.
  2. Write the quotient as the whole number and the remainder over the denominator: 234.
  3. Final answer: 234.
Note

Always give answers in simplest form. If a problem asks for a mixed-number answer, convert improper fractions to mixed numbers; if it asks for an improper fraction, leave it that way.

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