Fractions, Decimals and Percentages

Fractions, Decimals and Percentages

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A fraction $\frac{a}{b}$ expresses a part of a whole, with numerator $a$ and denominator $b \neq 0$. A decimal is the base-10 equivalent obtained by dividing $a$ by $b$, and a percentage is the fraction $\frac{x}{100}$ written as $x%$.

• Convert fraction → decimal: divide numerator by denominator.
• Convert decimal → percentage: multiply by 100 (shift point two places right).
• Convert percentage → decimal: divide by 100.
• A fraction terminates iff its lowest-term denominator has only the prime factors 2 and 5.
• Percentage change = $\dfrac{\text{New} - \text{Old}}{\text{Old}} \times 100%$.

NABTEB hot-spots: profit/loss, discount, simple interest, and conversion between the three forms — expect at least 2 objective items.

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Fraction operations

Two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent when $ad = bc$. Adding or subtracting requires a common denominator — usually the LCM of $b$ and $d$. To multiply, multiply numerators and denominators straight across, ideally cancelling common factors first. To divide by $\frac{c}{d}$, invert it and multiply: $\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}$.

Classifying fractions

• Proper fraction: numerator $<$ denominator (e.g. $\frac{3}{8}$).
• Improper fraction: numerator $\geq$ denominator (e.g. $\frac{11}{4}$).
• Mixed number: whole part plus a proper fraction (e.g. $2\frac{3}{4} = \frac{11}{4}$).

Decimal–percentage conversion

Move the decimal point two places: $\times 100%$ to go from decimal to percent, $\div 100$ to go back. A fraction terminates as a decimal if, in lowest terms, its denominator is of the form $2^m 5^n$. Anything else produces a recurring decimal, e.g. $\frac{1}{3} = 0.\overline{3}$.

Percentage problems

Three quantities are in play: part, base (whole), rate. The master relation is $\text{part} = \text{rate} \times \text{whole}$. Reverse the operation to find the whole ($\text{whole} = \text{part} \div \text{rate}$) or the rate. Simple interest $I = \frac{P \times R \times T}{100}$, where $P$ = principal, $R$ = rate per year (%), $T$ = time in years.

Exam patterns in NABTEB

Expect objective items asking to convert $\frac{3}{8}$ to a percentage (37.5%) and essay questions on profit, discount chains, or interest-bearing savings. Always reduce your final fraction to lowest terms.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Terminating vs recurring — the $2^m5^n$ test

Reduce the fraction first. $\frac{14}{25}$ has denominator $5^2$, so it terminates: $0.56$. But $\frac{7}{20} = \frac{7}{2^2 \cdot 5}$ also terminates (denominator’s only prime factors are 2 and 5). The length of the terminating decimal equals $\max(m, n)$; for $\frac{1}{2^3 5^2} = \frac{1}{200}$ the length is 3 (since $\max(3,2)=3$), giving $0.005$. Fractions with any other prime factor — 3, 7, 11, 13, … — recur, with period determined by the multiplicative order of 10 modulo that prime.

Successive percentage changes

A 10% increase followed by a 10% decrease is not zero. Apply them multiplicatively: $1.10 \times 0.90 = 0.99$ — a net 1% loss. NABTEB tests this with discount-and-tax chains, e.g. a 20% discount on an item already marked up by 25%: final multiplier $= 1.25 \times 0.80 = 1.00$ (no change). Master the multiplier method: factor of $(1 \pm r)$.

Common mistakes

• Dividing by the old value in percentage change, not the new one.
• Forgetting that “20% off” means paying 80%, so multiply by 0.80, not subtract 20 from the price.
• Confusing “percentage of” with “percentage change” — one multiplies, the other compares two values.
• Mis-shifting the decimal: $0.07 = 7%$, but $0.7 = 70%$, not 7%.

Worked micro-example

A trader bought an article for ₦8,000 and sold it for ₦9,400. Profit % = $\frac{9400-8000}{8000}\times 100% = \frac{1400}{8000}\times 100% = 17.5%$. If a 5% discount is then offered on the selling price, the new price is $9400 \times 0.95 =$ ₦8,930, and profit becomes $\frac{8930-8000}{8000}\times 100% = 11.625%$.

Practice prompts

1. Express $\frac{7}{16}$ as a percentage and state, with reason, whether its decimal form terminates.
2. A sum of ₦120,000 earns simple interest at 6.75% per annum. Find the total amount after 4 years.

Adjacent topics to revise

Ratio and proportion, direct/inverse variation, and unitary method all rest on the $\text{part} = \text{rate} \times \text{whole}$ template introduced here.

---

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.