NumberFoundation + HigherCore topic
Decimals.
Calculations & conversions.
A decimal extends place value to the right of the decimal point. Tenths, hundredths, thousandths. Any decimal can be written as a fraction — and any fraction as a decimal. Even the recurring ones.
I · Key definitions
Decimal point
Separates the whole from the fractional part.
3.45 — 3 is whole, .45 is fractional
Terminating
A decimal that ends.
0.5, 0.375, 2.8
Recurring
Digits that repeat forever. Shown with a dot or bar.
0.3̇ = 0.333…
Tenths
1st digit after decimal point.
in 4.27, the 2 = 0.2
Hundredths
2nd digit after decimal point.
in 4.27, the 7 = 0.07
Thousandths
3rd digit after decimal point.
in 4.275, the 5 = 0.005
II · Decimal place value
Hundreds
Tens
Ones
Tenths
Hund'ths
Thous'ths
Reads: Two hundred and forty-seven point three eight five. The green 7 = 7 ones. The gold 3 = 3 tenths = 0.3. Every place is 10 times the one to its right.
III · Conversions · decimal ↔ fraction
Terminating decimal → fraction
Write as digits / power of 10, simplify
0.375 = 375/1000 = 3/8
Fraction → decimal
Divide top by bottom
3/8 = 3 ÷ 8 = 0.375
Recurring → fraction (algebra trick)
Let x = 0.3̇
10x = 3.3̇
Subtract: 9x = 3 → x = 3/9 = 1/3
Use 100x if two digits recur, etc.
Which fractions terminate?
Only if denominator = 2ᵃ × 5ᵇ
otherwise it recurs
1/8 (=0.125) terminates · 1/3 (=0.333…) recurs
IV · Multiplying & dividing decimals
Multiplying decimals
1. Ignore decimal points, multiply as whole numbers
2. Count total decimal places in both numbers
3. Put that many d.p. back in the answer
0.3 × 0.04 → 3 × 4 = 12 → 3 d.p. → 0.012
Dividing by a decimal
Multiply both by 10, 100 … until divisor is whole
Then divide normally
4.8 ÷ 0.3 = 48 ÷ 3 = 16
V · Worked examples
Convert decimal to fraction
2 marks
Convert 0.68 to a fraction in its simplest form.
i.Write as: 68/100
ii.Find HCF(68, 100) = 4
iii.Divide top and bottom by 4: 17/25
0.68 = 17/25
Convert recurring to fraction
3 marks
Show that 0.2̇7̇ = 3/11 (where 27 recurs).
i.Let x = 0.27272727…
ii.Two digits recur, so multiply by 100: 100x = 27.272727…
iii.Subtract: 100x − x = 27.272… − 0.272… → 99x = 27
iv.x = 27/99 = 3/11 (dividing by 9)
0.2̇7̇ = 3/11 as required
Multiply decimals (non-calc)
2 marks
Work out 0.04 × 0.7 without a calculator.
i.Ignore decimal points: 4 × 7 = 28
ii.Count d.p. in question: 0.04 has 2, 0.7 has 1 → total 3
iii.Put 3 d.p. back: 0.028
0.04 × 0.7 = 0.028
VI · Common mistakes & examiner tips
Common mistakes
0.5 < 0.45? No — 0.5 = 0.50 is bigger.
Forgetting zeros when counting d.p. 0.040 × 0.2 = 0.008, not 0.08.
Line up decimal points when adding/subtracting. Place value matters.
Cancelling the wrong digits. 68/100 ÷ 4 = 17/25, not 17/24.
Losing the recurring dot in the answer. Use 0.3̇ (or 0.3̄) — it matters.
Examiner tips
Always simplify fractions at the end — examiners deduct marks.
Use powers of 10 trick for recurring decimals: 10, 100, 1000 depending on repeat length.
Check terminating vs recurring by looking at denominator — only 2s and 5s terminate.
Pad with zeros when comparing decimals: 0.5 vs 0.45 → 0.50 vs 0.45.