Fractions, Decimals and Approximations

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your NECO SSCE Mathematics paper.

A fraction a/b (b ≠ 0) expresses a part of a whole or a quotient, a decimal is the base-10 positional form of a rational number, and approximation rounds a value to a required degree of accuracy using the digit ≥ 5 → round up rule. NECO tests three skills here: converting between fractions, decimals and percentages; performing the four operations on fractions using BODMAS; and rounding to a stated number of decimal places (d.p.) or significant figures (s.f.). Recurring decimals like 0.·3 = 1/3 and 0.·16 = 16/99 appear almost every year.

To divide fractions, invert the divisor: a/b ÷ c/d = ad/bc.
To round to n d.p., inspect the (n+1)th digit; to n s.f., count from the first non-zero digit, then apply the same ≥5 rule.
To convert a percentage, divide by 100: x% = 0.01x.

Skill	NECO trigger phrase
Convert fraction ↔ decimal	”Express 3/8 as a decimal”
Recurring → fraction	”Convert 0.·24 to a fraction”
Approximation	”Correct to 3 significant figures”

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Fraction Types and Equivalence

A proper fraction has numerator smaller than denominator (3/7); an improper fraction has numerator ≥ denominator (9/4); a mixed number combines a whole with a proper fraction (2¼). Two fractions are equivalent when multiplying or dividing both numerator and denominator by the same non-zero integer yields each other: 2/5 = 4/10 = 6/15. To compare or order fractions, find the LCM of the denominators or convert each to a decimal first.

The Four Operations with BODMAS

When a calculation mixes fractions, decimals and percentages, apply BODMAS strictly: Brackets → Orders (powers, roots) → Division → Multiplication → Addition → Subtraction. A frequent NECO trap evaluates ½ + ⅓ × ¼ as (½ + ⅓) × ¼; the correct order gives ½ + 1/12 = 7/12. For division of fractions, invert the divisor: a/b ÷ c/d = ad/bc.

Conversions Between Fractions, Decimals and Percentages

The link fraction ↔ decimal uses division: a/b = a ÷ b. Percentages convert by dividing by 100, so 35% = 0.35 = 35/100 = 7/20. A terminating decimal has a finite digit string (1/8 = 0.125); a recurring decimal repeats a digit block forever (1/3 = 0.·3, 5/6 = 0.8·3).

Conversion	Rule
Fraction → Decimal	Divide numerator by denominator
Recurring 0.·ab → Fraction	(ab − a) / 99
Percentage → Decimal	Divide by 100
Decimal → Percentage	Multiply by 100

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Recurring-Decimal Conversions — The Two Patterns

The repetend length determines the denominator of the converted fraction. A single-digit repetend uses 9: 0.·3 = 3/9 = 1/3; 0.·7 = 7/9. A two-digit repetend uses 99: 0.·16 = 16/99; 0.·45 = 45/99 = 5/11. For a non-repeating digit before the repetend, isolate and subtract: 0.8·3 = (83 − 8)/90 = 75/90 = 5/6. NECO often sets these as 2–3 mark Paper 2 questions, expecting the fraction in lowest terms.

Rounding: d.p. vs s.f. — The Critical Distinction

Decimal places are counted from the decimal point to the right: 4.7263 to 2 d.p. is 4.73. Significant figures are counted from the first non-zero digit moving inward, regardless of where the decimal sits. Round 0.00459 to 1 s.f. by locating the first non-zero digit (4), keeping one digit of it, then rounding the next digit (5 → up): 0.00459 ≈ 0.005. Many candidates wrongly write 0.0 because they start from the decimal point. For very large or very small numbers, NECO also accepts standard form (a × 10ⁿ, where 1 ≤ a < 10).

Common Mistakes and Practice Prompts

Inverting only one of two fractions when dividing a chain of three terms.
Dropping the dot notation so 0.·3 is treated as the terminating 0.333.
Misreading “correct to 3 s.f.” as “3 d.p.” — different answers follow.
Treating 125% as 125/1000 instead of 1.25.

Mistake	Correct handling
Round 0.00459 to 1 s.f. as 0.0	First non-zero digit is 4 → answer is 0.005
Evaluate ½ + ⅓ × ¼ as (½ + ⅓) × ¼	BODMAS gives ½ + 1/12 = 7/12
Write 0.·3 as 333/1000	Use 3/9 = 1/3

Practice 1: Convert 0.2·45 to a fraction in lowest terms. (Answer: (245 − 2)/990 = 243/990 = 27/110.) Practice 2: Round 0.07286 to (a) 3 d.p., (b) 2 s.f. (Answers: 0.073; 0.072.)

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Fractions, Decimals and Approximations