Number, fractions and standard form
Choose a useful representation, calculate accurately and respect the requested answer form.
What you will learn
- Use common factors to solve grouping problems.
- Calculate with fractions and powers of ten.
Factors describe equal groups
A factor divides a whole number without a remainder. The highest common factor is useful when you must divide several quantities into the greatest number of identical groups without leftovers. The lowest common multiple answers a different question: when repeated cycles first coincide.
Prime factorisation makes the distinction clear. For the HCF, take common primes to the smaller power. For the LCM, take every prime appearing, to its larger power.
Keep fractions exact
Add or subtract fractions using a common denominator. Multiply numerators and denominators when multiplying. To divide by a non-zero fraction, multiply by its reciprocal. Cancel common factors to give the final fraction in simplest form.
A decimal may have the same value as a fraction but still fail an instruction asking for a simplified fraction. Always check both the numerical value and the requested format.
Work with powers of ten
Standard form writes a non-zero positive number as A × 10ⁿ with 1 ≤ A < 10 and integer n. Multiply coefficients and add exponents when multiplying; divide coefficients and subtract exponents when dividing. Adjust the coefficient afterwards if it lies outside the required interval.
Worked example
Calculate (3 × 10⁴)(8 × 10⁻²) in standard form.
Show the worked solution
- Multiply coefficients: 3 × 8 = 24.
- Add exponents: 4 + (−2) = 2, so the intermediate result is 24 × 10².
- Move the decimal point one place left and increase the exponent by one.
Answer 2.4 × 10³
Common mistakes
- Do not add denominators when adding fractions.
- 24 × 10² is equivalent in value but is not standard form.
What can you explain now?
Calculate (5/6 − 1/3) ÷ 3/4 as a simplified fraction.
Compare with the explanation
2/3
The difference is 1/2. Multiplying by 4/3 gives 4/6 = 2/3.
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