“Per cent” means “out of 100”, so x% is simply a fraction with denominator 100. These notes cover moving between percentages, fractions and decimals, expressing one quantity as a percentage of another, percentage increase and decrease using the multiplier method, successive changes, percentage points and reverse percentage.
Percentage underpins every money question in Sec 2 WA and EOY papers, including discount, GST and simple interest, and it runs through the financial-literacy questions at O-Level. Most marks are lost on a small set of repeated slips, almost all of which come from not asking “percentage of what?” before calculating.
What these notes cover
- •Percentages, fractions and decimals
- •Percentages greater than 100%
- •One quantity as a percentage of another
- •Comparing quantities by percentage
- •Percentage increase and decrease
- •The multiplier method
- •Successive percentage changes
- •Percentage points
- •Reverse percentage
- •Discount and GST
- •Simple interest
- •Common mistakes and exam technique
Key definitions
- Percentage
- “Per cent” means “out of 100”, so x% is a fraction with denominator 100. A percentage can exceed 100%, which means more than one whole.
- Multiplier
- The percentage of the original after a change, written as a decimal: (100 + r)% for an increase of r% and (100 − r)% for a decrease of r%. An increase of 15% gives 1.15; a decrease of 12% gives 0.88.
- Percentage point
- The difference between two percentages found by simple subtraction. A rate rising from 2% to 5% has risen by 3 percentage points, which is a percentage increase of 150%.
- Marked price
- The price shown on the price tag, before any discount. Also called the usual price or list price. A discount is always calculated on the marked price.
- GST
- Goods and Services Tax, a tax added to the price of goods and services. In Singapore the rate is 9%, so the price with GST is 109% of the price before GST.
Key formulas
Meaning of a percentage
Removing the % sign means dividing by 100; attaching it means multiplying by 100%. For example, 4.5% = 4.5 ÷ 100 = 0.045.
One quantity as a percentage of another
B, the quantity after the word “of”, is the base and goes in the denominator. Both quantities must be in the same units before dividing.
Percentage change
Always divide by the original value. A positive answer is an increase; a negative answer is a decrease, stated without the negative sign.
New value after a percentage change
New value = (100% ± r%) × original value
r is the percentage change. The bracket written as a decimal is the multiplier: greater than 1 for an increase, between 0 and 1 for a decrease. Successive changes multiply, e.g. a 15% discount then 9% GST is 0.85 × 1.09.
Reverse percentage
Original value = new value ÷ multiplier
Use when the value after the change is given. The multiplier is 1 + r ÷ 100 for an increase of r% and 1 − r ÷ 100 for a decrease of r%. Check by multiplying the answer by the multiplier.
Simple interest
P is the principal in dollars, R is the rate in % per annum (use R = 2.4 for 2.4%, not 0.024) and T is the time in years (divide months by 12). Total amount = P + I.
Worked example
After a 20% increase, Mr Tan’s monthly salary is $2640. Find his salary before the increase.
- 1The original salary is 100%, so after a 20% increase the new salary is 100% + 20% = 120% of the original, which means $2640 represents 120%.
- 2Unitary method: 1% → $2640 ÷ 120 = $22.
- 3Then 100% → $22 × 100 = $2200.
- 4Multiplier method: the multiplier is 1 + 20 ÷ 100 = 1.2, so the original salary = $2640 ÷ 1.2 = $2200.
- 5Check by working forward: 1.2 × $2200 = $2640, which matches the value given, so his salary before the increase was $2200.
Watch out: Do not take 20% off the new salary: 20% × $2640 = $528 and $2640 − $528 = $2112 is wrong, because the 20% increase was 20% of the original salary, not of the new salary.
Common mistakes
Using the wrong base in a percentage change
Wrong: A book rises from $40 to $50, so the percentage increase is (10 ÷ 50) × 100% = 20%, dividing by the new price.
Right: (10 ÷ 40) × 100% = 25%. The denominator must be the original value, the value you started from.
Adding successive percentages
Wrong: A value of 100 is increased by 10%, then by 20%: 10% + 20% = 30%, so the final value is 130.
Right: Multiply the multipliers: 100 × 1.1 × 1.2 = 132, so the overall increase is 32%.
Months not converted to years in simple interest
Wrong: $5000 at 3% per annum for 8 months: I = (5000 × 3 × 8) ÷ 100 = 1200, giving $1200.
Right: T = 8 ÷ 12 years, so I = (5000 × 3 × (8 ÷ 12)) ÷ 100 = 100, giving $100. One full year at 3% earns only $150.
Questions students ask
How do you find the original price after a discount?
After a d% discount, the sale price is (100 − d)% of the original price, so divide the sale price by the multiplier. For example, after a 35% discount a sale price of $91 is 65% of the original, and $91 ÷ 0.65 = $140. Check by working forward: 0.65 × $140 = $91.
How do you find the price before GST?
A price inclusive of 9% GST is 109% of the price before GST, so divide by 1.09. For example, $1417 ÷ 1.09 = $1300, and the GST paid is $1417 − $1300 = $117. Taking 9% of $1417 is wrong, because GST is 9% of the price before GST.
What is the difference between percentage points and percentage increase?
Percentage points are the simple difference between two percentages, while a percentage increase compares the change with the original percentage. If a pass rate rises from 80% to 84%, that is an increase of 4 percentage points, but the percentage increase is (84 − 80) ÷ 80 × 100% = 5%.
Is discount or GST applied first?
Apply the discount to the marked price first, add any service charge to the discounted price, then charge GST last on the subtotal. GST is never calculated on the original marked price when a discount has been given. A 15% discount followed by 9% GST gives an overall multiplier of 0.85 × 1.09 = 0.9265.
In the full PDF
- •Conversion table of percentages, decimals and fractions in lowest terms, including 2½%, 0.6% and 135%
- •Further worked examples on conversions, percentages above 100% and expressing one quantity as a percentage of another with different units
- •Worked examples on comparing class pass rates, successive percentage changes, reverse percentage after a decrease, discount then GST, and finding an interest rate
- •The remaining common mistakes, each with a numerical example showing why the wrong method fails
- •Mark-winning exam habits, a model layout and a “choosing the right operation” table for discount, GST and simple interest
- •A five-point summary of the whole topic