Lesson 3: Decimals and Percentages

Decimals and percentages run Singapore's commercial life: prices in dollars and cents, GST at 9%, bank interest, discounts at the Great Singapore Sale, statistics in the news. Mathematically, decimals extend place value to the right of the decimal point, and percentages are simply fractions of 100. This lesson builds both from place-value foundations, with plenty of S$ practice — because in Singapore, decimal fluency is financial fluency.

Decimal Place Value

Decimals continue the place-value pattern past the ones column: tenths (1/10), hundredths (1/100), thousandths (1/1000). In 4.672, the 4 is ones, 6 is tenths (6/10), 7 is hundredths (7/100), 2 is thousandths (2/1000). Singaporean pupils learn this with decimal discs and place-value charts — concrete and pictorial before abstract — so that 0.6 is understood as six tenths, not "zero point six" as a meaningless chant.

Comparing decimals requires aligning place values: 0.65 versus 0.7 — write 0.70 and compare hundredths: 65 against 70, so 0.7 is larger. The common error is judging by digit count ("0.125 must exceed 0.9 because it has more digits"); place-value alignment kills this mistake. Money helps: S$0.90 is ninety cents, S$0.125 is twelve and a half cents — the money picture makes the comparison obvious.

Rounding decimals follows the familiar rule with new precision: S$4.678 to the nearest cent (hundredth) looks at the thousandths digit (8), rounding up to S$4.68. Petrol at S$2.834 per litre, rounded to cents, is S$2.83. Singapore's cash economy rounds to the nearest 5 cents (no 1-cent coins circulate), so S$4.67 becomes S$4.65 — a daily encounter with rounding.

Operating with Decimals

Adding and subtracting decimals demands aligned decimal points — which is really aligned place values. S$12.45 + S$3.80: write 12.45 + 3.80, align the points, add hundredths (5 + 0), tenths (4 + 8 = 12, carry), ones, tens: S$16.25. The pictorial version — decimal squares shaded to show tenths and hundredths — reveals why misaligned points give nonsense.

Multiplying decimals: 2.5 × 1.2. Ignore the points: 25 × 12 = 300. Count decimal places: one in 2.5, one in 1.2, two total — so 3.00. Estimation checks it: 2.5 × 1.2 is near 2.5 × 1 = 2.5 and near 3 × 1 = 3, so 3.00 is sensible. A hawker example: 2.5 kg of chicken at S$8.40/kg costs 2.5 × 8.4 = S$21.00 — estimate 2 × 8 = 16 to 3 × 9 = 27, and 21 sits comfortably inside.

Dividing decimals: 4.8 ÷ 0.6. Multiply both by 10 to clear the decimal: 48 ÷ 6 = 8. The principle — scaling dividend and divisor equally preserves the quotient — is the same as equivalent fractions. A useful check: 4.8 ÷ 0.6 must exceed 4.8 (dividing by less than 1 gives more), and 8 does.

Percentages — Fractions of 100

A percentage is a fraction with denominator 100: 35% = 35/100 = 7/20. The key translations — 50% = 1/2, 25% = 1/4, 75% = 3/4, 20% = 1/5, 10% = 1/10 — should be instant, because they turn percentage problems into fraction problems. Finding 25% of S$80 is finding 1/4 of S$80: S$20.

The three percentage problem types: find the part (15% of S$200 = 0.15 × 200 = S$30); find the percentage (S$30 is what percentage of S$200? 30/200 = 15%); find the whole (S$30 is 15% of what? 30 ÷ 0.15 = S$200). Singaporean pupils master all three with bar models: draw the whole as a bar, shade the percentage, label the known and unknown. The bar makes "15% of what number is 30?" visual instead of mysterious.

Percentage change: a S$50 shirt marked up 20% becomes S$50 × 1.20 = S$60; then discounted 20%, it becomes S$60 × 0.80 = S$48 — not S$50! Percentages compound asymmetrically because each applies to a different base. This is the classic trap, and the bar model (draw the new whole each time) exposes it.

GST and Discounts — Singapore Arithmetic

Goods and Services Tax (GST) in Singapore is 9% (as of 2024). To add GST: multiply by 1.09. A S$100 meal becomes S$109.00; a S$45 book becomes S$49.05. To find the pre-GST price from a GST-inclusive S$109: divide by 1.09 to get S$100. Shops display GST-inclusive prices by law — but computing backwards matters for business accounts and for checking bills.

Discounts stack in a specific order: a S$200 jacket at 30% off, then 9% GST. First the discount: S$200 × 0.70 = S$140. Then GST: S$140 × 1.09 = S$152.60. Reversing the order (GST first, then discount) gives S$200 × 1.09 × 0.70 = S$152.60 — the same, because multiplication commutes. But a "30% off then additional 10% off" is not 40% off: S$200 × 0.70 × 0.90 = S$126, a 37% total discount.

The PSLE loves these layered problems: "A shop sells a watch for S$350 after a 30% discount. What was the original price?" The bar model: 70% corresponds to S$350, so 10% is S$50, and 100% is S$500. Drawing the bar turns algebra into arithmetic.

Decimals in Measurement and Science

Singapore uses the metric system, so decimals are the language of measurement: 1.75 m tall, 2.5 litres of petrol, 0.5 kg of rice, 36.6°C body temperature. Converting units is decimal shifting: 1.75 m = 175 cm (×100), 2.5 litres = 2500 ml (×1000), 0.5 kg = 500 g. The metric system's decimal structure is why Singaporean pupils handle decimals fluently — measurement gives them daily practice.

Precision matters in science and engineering: a 0.1 mm error in a semiconductor fab ruins chips; a 0.5°C fever threshold decides medical treatment. Significant figures and decimal places are not pedantry but the vocabulary of precision. Singapore's manufacturing economy runs on measurements accurate to thousandths.

Estimation remains the guardian: a recipe calling for 0.75 litres of coconut milk — about three-quarters of a 1-litre carton — is checked pictorially, not just computed. The Singapore habit is always: estimate first, compute second, check third.

Conclusion — Why This Matters

Decimals and percentages are the mathematics of money, measurement, and modern life. In Singapore's commercial, metric, GST-charging economy, they are not school topics but life skills — and the PSLE and O-levels test them relentlessly because adult life does.

The deeper lesson is representational fluency: fractions, decimals, and percentages are one idea in three costumes. Students who translate freely between them — 3/8 = 0.375 = 37.5% — think flexibly; students locked in one form struggle. Practice translation until it is reflex.

And always draw the bar model for percentage problems. It turns the three problem types (find the part, find the rate, find the whole) into one visual routine — the Singapore way.

Common Decimal Errors — Diagnosis and Cure

Error 1: "0.5 is greater than 0.75 because 5 is greater than 75." The cure is place-value alignment: write 0.50 vs 0.75, or better, think money — S$0.50 (fifty cents) vs S$0.75 (seventy-five cents). The money picture never lies. Pupils who anchor decimals in dollars and cents rarely make this mistake.

Error 2: misaligned addition — computing 12.5 + 3.75 as 12.5 + 3.75 misaligned to get 15.125 or similar. The cure: always write decimals in a column with points aligned, and estimate first (12.5 + 3.75 ≈ 16). The estimate exposes misalignment instantly. Singaporean teachers insist on the column format precisely to prevent this.

Error 3: decimal-point slips in multiplication — 2.5 × 4 = 100 (instead of 10). The cure is estimation: 2.5 × 4 is near 2 × 4 = 8 and near 3 × 4 = 12, so 100 is absurd. The rule "count decimal places" is the procedure; estimation is the safety net. Every decimal computation deserves both.

Error 4: percentage confusion — "30% off then 10% off is 40% off." The cure is the bar model: draw the whole, shade 30%, then shade 10% of the remainder — the picture shows 37% total, not 40%. Each error has a visual antidote; the Singapore method is to prescribe the picture, not just the rule.

Worked Example — The Bar Model Beats the "Obvious" Method

Problem: "After a 20% discount, a bag costs S$96. What was its original price?" The tempting method — add 20% of S$96 — gives 96 × 1.20 = S$115.20, and it is wrong, because the discount was taken from the original price, not from S$96. Bar model: the sale price is 80% of the original. Draw the whole bar as 100%; the known part is 80% = S$96. Then 10% = 96 ÷ 8 = S$12, so 100% = S$120. Algebra: let the original price be x; 0.80x = 96, so x = 96 ÷ 0.80 = S$120. Check: 20% of 120 is 24, and 120 − 24 = 96 ✓.

Reverse-percentage problems are where the bar model earns its keep: the picture forces you to ask "percentage *of what*?" before computing.

Exam Experience

Question 1 [2 marks]. A shirt costs S$64. In a sale, its price is reduced by 15%. Calculate the sale price.

Question 2 [3 marks]. S$500 is invested in a bank account paying simple interest at 4% per year. Calculate (a) the interest earned after 3 years, (b) the total amount in the account after 3 years.

Question 3 [6 marks]. The price of a phone before GST is S$1,200. The GST rate is 9%. (a) Calculate the price of the phone including GST. A shop offers a 10% discount on the phone, and then charges GST on the discounted price. (b) Calculate the final price in this shop. Another shop sells the same phone for S$1,250 including GST, with no discount. (c) Which shop is cheaper, and by how much?

Marking scheme. Q1: M1 for multiplying by 0.85 (or finding 15% = S$9.60 and subtracting); A1 for S$54.40. Q2: (a) M1 for I = (500 × 4 × 3)/100; A1 for S$60. (b) B1 for S$560. Q3: (a) M1 for 1,200 × 1.09; A1 for S$1,308. (b) M1 for 1,200 × 0.90 = 1,080 followed by × 1.09 (either order accepted); A1 for S$1,177.20. (c) M1 for comparing 1,250 with 1,177.20; A1 for "the first shop, by S$72.80".