Try each one on paper first, then check a single step. That is worth far more than reading a finished solution.
Example 1 — HCF and LCM by prime factorisation
Find the HCF and the LCM of 24 and 36.
- 124=2×2×2×3=23×3Divide by the smallest prime repeatedly until you reach 1.
- 236=2×2×3×3=22×32Same process.
- 3HCF: shared primes at the lowest power — 22×3=12Both numbers contain at least two 2s and at least one 3.
- 4LCM: every prime at the highest power — 23×32=72Enough of each prime to cover both numbers.
- 5Check: 12×72=864=24×36 ✓HCF times LCM always equals the product of the two numbers.
Example 2 — order of operations
Evaluate 3+4×22−(6−2).
- 1Brackets first: 6−2=4Nothing outranks a bracket.
- 2Then indices: 22=4Order comes before multiplication.
- 33+4×4−4Rewrite before continuing — it stops mistakes.
- 4Multiplication: 4×4=16Before any adding or subtracting.
- 53+16−4=15Left to right for the last two. Working left to right from the start gives 24, which is wrong.
Example 3 — standard form, both directions
Write 0.000472 in standard form, and write 3.6×104 as an ordinary number.
- 1For 0.000472, place the point after the first non-zero digit: 4.72Standard form needs exactly one non-zero digit before the point.
- 2The point moved 4 places to the right, and the number is less than 1, so the index is −4.Small numbers take a negative index.
- 30.000472=4.72×10−4
- 4For 3.6×104, move the point 4 places right: 36000A positive index means a large number.