Revision notes — the five strands of Unit 1
1Recurring decimals → fractions
Dot notation: a dot over a digit means that digit repeats; dots over the first and last digit of a block
mean the whole block repeats. 0.7 = 0.777… 0.45 = 0.454545… 2.165 = 2.165165165…
Instant rules: one 9 for each digit in the repeating block → 1 digit: 0.7 = 7/9
· 2 digits: 0.45 = 45/99 = 5/11 · 3 digits: 0.123 = 123/999 = 41/333
then add one 0 for each digit sitting after the point but before the block →
0.12 345… has block 345 and “12” in front, so the denominator is 999 then 00 = 99900,
and the top is 12345 − 12 = 12333 → 12333/99900 = 4111/33300.
No digits in front → no 0s → that’s the plain “over 9 / 99 / 999” case.
Algebra method (use for any mixed decimal — this is the full-marks method):
- Let x = the decimal.
- Multiply by 10a so the repeating part starts straight after the point.
- Multiply by 10b so one whole repeating block sits before the point (b > a).
- Subtract the two equations — the repeating tails cancel.
- Solve for x and simplify.
0.4515151… x = 0.4515151… 10x = 4.515151… 1000x = 451.515151…
1000x − 10x = 990x = 447 → x = 447/990 = 149/330
Traps: give the fraction in its simplest form; convert to a mixed number if the question says so;
a rounded decimal scores zero.
2Percentages
Multipliers are the fastest route.
| Job | Do this | Example |
|---|
| Increase by r% | × (1 + r/100) | +37.4% → ×1.374 |
| Decrease by r% | × (1 − r/100) | −16% → ×0.84 |
| Percentage change | (difference ÷ original) × 100 | 70 → 98: 28/70 = 40% |
| Reverse (find original) | final ÷ multiplier | £420 after 20% off → 420 ÷ 0.8 = £525 |
Simple interest: interest = P × r × n (same interest every year, on the original only)
Compound interest / depreciation: amount = P × (1 ± r)n (interest earns interest)
Profit & loss are always a percentage of the cost price. “Sells at 80% above cost” means the
selling price is 180% of cost; a further 25% off gives 180 × 0.75 = 135% → 35% profit.
Traps: round only at the end; “how much interest” is not the total in the account;
VAT, deposits and “per year” rates all use the same multiplier idea.
3Degree of accuracy & bounds
Bounds = the value ± half the stated accuracy. Error interval: lower bound ≤ x < upper bound.
| Statement | Lower bound | Upper bound |
|---|
| 105 cm (nearest cm) | 104.5 | 105.5 |
| 4.2 (1 d.p.) | 4.15 | 4.25 |
| 0.08 (1 s.f.) | 0.075 | 0.085 |
| 70 (1 s.f.) | 65 | 75 |
| 40 (nearest 10) | 35 | 45 |
Calculations: for the biggest answer use the biggest top and the smallest bottom
(max: UB×UB, UB÷LB | min: LB×LB, LB÷UB).
For a sum, max uses both UBs and min uses both LBs. For a subtraction, max = UB − LB and min = LB − UB.
Truncation cuts off rather than rounds: 159 000 truncated to 3 s.f. lies in 159 000 ≤ x < 160 000.
“Suitable degree of accuracy”: work out both bounds, then quote the value at the accuracy where the
bounds agree, and say so: “x = 4.2 to 1 d.p. as both bounds round to 4.2.”
4Ratio
Simplifying: divide every part by the HCF — and convert units first (2 km : 3400 m = 2000 : 3400 = 10 : 17).
With algebra: 12a² : 8a = 3a : 2.
Sharing into a ratio: add the parts, 1 part = total ÷ number of parts, then multiply up.
The exam favourite: one part is given → find 1 part first, then scale.
children : adults = 2 : 3 with 80 children → 1 part = 40 → 120 adults.
If a difference is given, divide the difference by the difference in parts (7 : 9 with 40 more = 2 parts = 40 → 1 part = 20).
Checklist: find 1 part → work out exactly what is asked (other part / total / difference / money). Don’t stop at 1 part.
5Direct & inverse proportion
Always three steps: write the equation with k → find k from the given pair → substitute.
| Statement | Equation | k = |
|---|
| y ∝ x | y = kx | y ÷ x |
| y ∝ x² | y = kx² | y ÷ x² |
| y ∝ x³ | y = kx³ | y ÷ x³ |
| y ∝ √x | y = k√x | y ÷ √x |
| y ∝ 1/x | y = k/x | y × x |
| y ∝ 1/x² | y = k/x² | y × x² |
Worked: y ∝ x² with y = 24 when x = 2 → k = 24/4 = 6 → y = 6x² → x = 4 gives y = 96.
“What happens when … is halved?” reason from the power: F ∝ 1/d², so halving d
makes d² a quarter as big and F four times as big.
“Which relationship fits the table?” compute y ÷ x, y ÷ x², y ÷ x³, y ÷ √x (or y × x, y × x²)
and pick the one that stays constant.
What the unit covers
Higher tier, five strands: decimals, fractions, percentages,
degree of accuracy and ratio & proportion.
Tonight’s order, highest marks per minute:
bounds calculations → percentage multipliers (compound & reverse) → proportion (find k first)
→ recurring decimals.
Past-paper practice — 4 real exam questions on recurring decimals
P1
Show that 0.254 = 14/55
2 marks
2019 Paper 1H, Q15 · real past paper, spec 4MA1
Show answer & working
Front: 2 (1 digit) →
0 |
Block: 54 (2 digits) →
99 → denominator
990Top: 254 − 2 =
252x = 0.25454…
100x = 25.454…
100x − x = 99x = 25.2
x = 25.2/99 = 252/990 = 14/55 (÷18)
Mark scheme: M1 for two recurring decimals that subtract to a whole number or terminating decimal (25.2 or 252), A1 for completion to 14/55 — algebra must be shown. P2
Show that 0.2813 = 557/1980
2 marks
2021 Paper 1H, Q16 · real past paper, spec 4MA1
Show answer & working
Dots over 1 and 3 →
block = 13; in front =
28 → 0s:
00, 9s:
99 → denominator
9900Top: 2813 − 28 =
2785x = 0.281313…
100x = 28.1313…
10000x = 2813.13…
10000x − 100x = 9900x = 2785
x = 2785/9900 = 557/1980 (÷5)
Mark scheme also accepts 1000x = 281.313… with 10x = 2.81313… → 990x = 278.5 → 557/1980 — it only needs two recurring decimals that subtract to a whole number or terminating decimal.P3
0.4ẋ as a fraction, in terms of x
3 marks
2021 Oct Paper 1H, Q18 · real past paper, spec 4MA1
Show answer & working
The dot is over the
x only, so 0.4ẋ = 0.4xxxx… (only x repeats).
y = 0.4xxxx…
10y = 4.xxxx…
100y = 4x.xxxx… ← 4x here is the two-digit number 40 + x
100y − 10y = 90y = (40 + x) − 4 = 36 + x
y = (36 + x)/90
Check with x = 5: (36 + 5)/90 = 41/90 = 0.45555… = 0.4
5 ✓
Mark scheme: A1 (36 + x)/90 oe, dependent on two correct recurring-decimal expressions being subtracted; working required.P4
Show that 0.612 = 101/165
2 marks
2025 Paper 1HR, Q16 · real past paper, spec 4MA1
Show answer & working
Dots over 1 and 2 →
block = 12; in front =
6 → 0s:
0, 9s:
99 → denominator
990Top: 612 − 6 =
606x = 0.6121212…
10x = 6.121212…
1000x = 612.121212…
1000x − 10x = 990x = 606
x = 606/990 = 101/165 (÷6)
Mark scheme: M1 for two recurring decimals that subtract to a whole number or terminating decimal (606), A1 for completion to 101/165 using algebra.
