A recurring decimal is secretly a fraction in disguise — there is a fixed, reliable method to convert any repeating decimal back into a fraction.
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Adding, subtracting, multiplying and dividing decimals follows whole-number rules, with one extra step: placing the decimal point correctly.
📌 The decimal point rules
Add/subtract: line up the decimal points first, padding with zeros if needed.
Multiply: ignore the decimal points, multiply as whole numbers, then count the TOTAL decimal places from both numbers for the answer.
Divide by a decimal: multiply both numbers by a power of 10 until the divisor is a whole number.
✨ Worked example
Multiply
2.5 × 1.2
25 × 12 = 300, then place 2 decimal places → 3.00 = 3
Divide
6 ÷ 0.4
Multiply both by 10: 60 ÷ 4 = 15
Every recurring decimal can be converted to an exact fraction using a simple algebraic trick.
📊 Converting recurring decimals
| Recurring decimal | Method | Fraction |
|---|---|---|
| 0.333… (0.3) | Let x = 0.333…; 10x = 3.333…; 10x − x = 3; x = 3/9 | 1/3 |
| 0.666… | Same method: 10x − x = 6; x = 6/9 | 2/3 |
| 0.1666… (0.16) | One non-repeating digit needs a different power of 10 (100x − 10x) | 1/6 |
Shortcut for purely recurring decimals: put the repeating digits over as many 9s as there are repeating digits. 0.454545… = 45/99 = 5/11.
Rounding rules never change, no matter how many decimal places are involved.
Rounding rule: look ONLY at the digit right after the place you are rounding to. 5 or more rounds up; 4 or less rounds down. 3.847 rounded to 2 decimal places is 3.85, not 3.84 — the digit after is 7, which rounds up.
12 questions on decimal operations, recurring decimals and rounding.
