Every operation with fractions and decimals — adding, subtracting, multiplying, dividing, and converting — explained step by step with fully worked examples and the errors to avoid.
A fraction represents a part of a whole. It is written as one number over another: the numerator (top number) shows how many parts you have, and the denominator (bottom number) shows how many equal parts the whole is divided into.
In 3/8, the whole has been divided into 8 equal parts and you have 3 of them. If you eat 3 out of 8 slices of pizza, you have eaten 3/8 of the pizza.
Divide both numerator and denominator by their highest common factor (HCF).
Simplify 18/24: HCF of 18 and 24 is 6. 18 ÷ 6 = 3; 24 ÷ 6 = 4. Answer: 3/4
Mixed to improper: multiply the whole number by the denominator, add the numerator, keep the denominator.
2¾ → (2 × 4) + 3 = 11 → 11/4
Improper to mixed: divide numerator by denominator. Quotient = whole number; remainder = new numerator.
17/5 → 17 ÷ 5 = 3 remainder 2 → 3 and 2/5
You can only add or subtract fractions with the same denominator. If the denominators differ, find the lowest common denominator (LCD) first.
3/8 + 1/8 = (3 + 1)/8 = 4/8 = 1/2 (simplify!)
Example: 1/3 + 1/4
LCD of 3 and 4 is 12.
1/3 = 4/12; 1/4 = 3/12
4/12 + 3/12 = 7/12
Example: 2¾ + 1⅓
Add whole numbers: 2 + 1 = 3
Add fractions: 3/4 + 1/3 → LCD = 12 → 9/12 + 4/12 = 13/12 = 1 and 1/12
Total: 3 + 1 and 1/12 = 4 and 1/12
Multiply numerator by numerator and denominator by denominator. Simplify before or after multiplying — simplifying before (called cross-cancelling) keeps the numbers smaller.
Example: 2/3 × 3/5
(2 × 3) / (3 × 5) = 6/15 = 2/5
With cross-cancelling: The 3 in the numerator and the 3 in the denominator cancel to 1: (2 × 1) / (1 × 5) = 2/5. Same answer, less arithmetic.
Always convert mixed numbers to improper fractions first, then multiply.
Example: 1½ × 2⅔
= 3/2 × 8/3 = 24/6 = 4
To divide by a fraction, multiply by its reciprocal (flip the second fraction). The phrase to remember: Keep, Change, Flip.
Keep the first fraction as it is. Change ÷ to ×. Flip the second fraction.
Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 and 7/8
Example with mixed numbers: 2½ ÷ 1¼ = 5/2 ÷ 5/4 = 5/2 × 4/5 = 20/10 = 2
A decimal uses a point to separate the whole number part from the fractional part. Each position after the decimal point represents a power of ten: tenths (0.1), hundredths (0.01), thousandths (0.001).
In 4.375: the 3 is in the tenths column, the 7 is in the hundredths column, and the 5 is in the thousandths column.
Compare digits column by column from left to right — starting with the whole number part, then tenths, then hundredths. Do not assume more digits means larger value: 0.7 > 0.65 even though 0.65 has more digits.
Useful trick: add trailing zeros so all decimals have the same number of decimal places, then compare as whole numbers. 0.7 becomes 0.70, and 0.70 > 0.65. ✓
Divide the numerator by the denominator.
Recurring decimals are written with a dot over the repeating digit: 0.333… = 0.3̄. If two digits repeat, dots go over the first and last of the repeating block.
Write the decimal as a fraction with a power of 10 as the denominator, then simplify.
Example: Convert 0.7̄ (0.777…) to a fraction.
Let x = 0.777…
Then 10x = 7.777…
Subtract: 10x − x = 7.777… − 0.777… → 9x = 7 → x = 7/9
Ignore the decimal points, multiply as whole numbers, then place the decimal point in the answer so that the total number of decimal places equals the combined decimal places of both original numbers.
Example: 1.4 × 0.6
14 × 6 = 84. Total decimal places = 1 + 1 = 2. Answer: 0.84
Move the decimal point in both numbers the same number of places to the right until the divisor is a whole number.
Example: 3.6 ÷ 0.12
Multiply both by 100: 360 ÷ 12 = 30
Fractions and decimals underpin virtually every other maths topic — from percentages and ratios to probability and algebra. Solid fluency here removes the most common source of dropped marks throughout the entire course.