Sec 1 Percentages and Percentage Change: Examples & Guide

Secondary 1 Mathematics · G3 focus · Study guide

Identify the quantity that represents 100%, calculate percentage changes and reverse percentages, and combine successive changes accurately.

Concept → worked examples → practice16 min read · practice linked below

Start by identifying what counts as 100%

A percentage means “out of 100”. Thus $35\%=\frac{35}{100}=0.35$ . To convert a percentage to a decimal, divide by 100. To convert a decimal or fraction to a percentage, multiply by 100%. Percentages can be smaller than 1% or larger than 100%: $0.5\%=0.005$ , while $150\%=1.5$ .

The base quantity is the amount representing 100%. It might be the original price, a total class size or the cost price of an item. Always name it before calculating. Percentages compare quantities on the same scale, so convert to the same units first.

Worked example 1: a part of a whole and a value above 100%

A class has 48 students, and 18 of them belong to a club. What percentage of the class is in the club? Also find 150% of 48.

The whole class is the base: $\frac{18}{48}\times100\%=\frac38\times100\%=\boxed{37.5\%}.$
Convert 150% to 1.5, then multiply: $150\%\text{ of }48=1.5\times48=\boxed{72}.$

Check: $0.375\times48=18$ . Also, 150% is one whole plus one half, so $48+24=72$ . A percentage above 100% gives an amount larger than the base; it does not mean that more than all 48 students belong to this particular club.

Compare amounts and calculate percentage change

To express $A$ as a percentage of $B$ , calculate $\frac AB\times100\%$ , with $B\ne0$ . To find a percentage increase or decrease from a positive original amount, compare the change with the original:

\text{percentage change}=\frac{\text{new}-\text{original}}{\text{original}}\times100\%.

A positive result means an increase; a negative result means a decrease. If the original amount is zero, this percentage-change formula is undefined.

Worked example 2: the denominator depends on the comparison

A bag increases in price from S$80 to S$100. Find (a) the new price as a percentage of the old price, (b) the percentage increase, and (c) the percentage decrease needed to return to S$80.

The new price as a percentage of the old is $\frac{100}{80}\times100\%=\boxed{125\%}$ .
The increase is S$20, measured against S$80: $\frac{20}{80}\times100\%=\boxed{25\%}$ .
For the return journey, the starting price is now S$100. The S$20 decrease is $\frac{20}{100}\times100\%=\boxed{20\%}$ .

Check: $80(1.25)=100$ and $100(0.8)=80$ . “125% of the original” means a 25% increase, not a 125% increase. Equal dollar changes need not be equal percentage changes.

The longer bar represents the new base when the price falls back from S$100.

Use multipliers for forward and reverse percentages

An increase of $r\%$ keeps the original 100% and adds $r\%$ , so multiply by $1+\frac r{100}$ . A decrease of $r\%$ leaves $(100-r)\%$ , so multiply by $1-\frac r{100}$ . A 15% increase uses 1.15; a 15% discount uses 0.85.

For a reverse percentage, you know the new amount and need the original. Undo a non-zero multiplier by division:

\text{original amount}=\frac{\text{new amount}}{\text{multiplier}}.

A 100% decrease leaves zero, so the original amount cannot be recovered from the final amount alone.

You can also use a unitary method: if 80% is S$68, then 1% is $68\div80$ dollars and 100% is $68\div80\times100$ dollars.

Worked example 3: recover the price before a discount

After a 20% discount, a jacket costs S$68. Find its original price.

The sale price represents $100\%-20\%=80\%$ of the original.
If the original price is $P$ dollars, $0.8P=68$ .
Divide by 0.8: $P=68\div0.8=\boxed{85}.$

Answer: the original price was S$85. Check: 20% of S$85 is S$17, and $85-17=68$ . Increasing S$68 by 20% would use the wrong base and give S$81.60.

Successive changes act on the latest amount

After the first change, the second percentage applies to a new base. Multiply the multipliers; do not simply add the percentage changes. A 20% increase followed by a 20% decrease gives $1.2\times0.8=0.96$ , leaving 96% of the original: a 4% decrease.

Explore two successive changes

Start with 100 units. Set two signed percentage changes: positive means an increase and negative means a decrease. Predict whether the final amount will be above or below 100.

The second percentage acts on the amount after the first change. A 100% decrease leaves zero.

100 × 1.20 × 0.80 = 96. The overall change is a 4% decrease.

Worked example 4: two changes in a bill

A bill for S$240 receives a 15% discount. A service charge of 10% is then added to the discounted amount. Find the final bill and its overall percentage change from S$240.

The discounted amount is $240(0.85)=204$ dollars.
The service charge acts on S$204, so the final bill is $204(1.10)=\boxed{224.40}$ dollars.
The combined multiplier is $0.85\times1.10=0.935$ . The bill is 93.5% of its original value, an overall 6.5% decrease.

Check: $240-224.40=15.60$ , and $15.60\div240\times100\%=6.5\%$ . Subtracting 15% and adding 10% as if the bases were equal would incorrectly give a 5% decrease.

Profit, prices and other percentage contexts

Profit is selling price minus cost price; loss is cost price minus selling price. A percentage profit or loss normally uses the cost price as its base. A discount uses the marked price. If a question states another base, use that stated base.

Worked example 5: profit after a discount

A shop buys a bag for S$60. It wants a 20% profit after offering customers a 10% discount on the marked price. Find the marked price.

The required selling price is $60(1.2)=72$ dollars.
After the 10% discount, the selling price is 90% of the marked price $M$ . Thus $0.9M=72$ .
Divide: $M=72\div0.9=\boxed{80}.$

Answer: mark the bag at S$80. The discount is S$8, leaving S$72. The profit is S$12, and $12\div60=0.2=20\%$ .

When more than one factor determines a total, apply the changes to each factor. For example, if a price increases by 10% but the number sold falls by 20%, revenue is multiplied by $1.10\times0.80=0.88$ : it decreases by 12%. Similarly, increasing both length and breadth of a rectangle by 10% multiplies its area by $1.1^2=1.21$ , a 21% increase.

Worked example 6: simple and repeated interest

S$800 earns interest at 3% per year for two years. Compare (a) simple interest, calculated each year on the original S$800, and (b) interest compounded annually, calculated on each year's new balance.

Simple interest is $I=Prt$ , where $r$ is the yearly rate as a decimal and $t$ is the number of years. Here $I=800(0.03)(2)=48$ , giving a final amount of $\boxed{\text{S\$}848}$ .
With annual compounding, after one year the balance is $800(1.03)=824$ dollars. The second year's interest is 3% of S$824.
The final compounded amount is $800(1.03)^2=\boxed{\text{S\$}848.72}$ , which is S$0.72 more than the simple-interest amount.

Check: the second year's compound interest is S$24.72, compared with S$24 under simple interest. The difference is exactly S$0.72. For a rate per year, convert months into years before using $I=Prt$ .

Percentage points compare two percentage rates

A difference between two percentages can be reported in percentage points. Subtract the two rates directly. A percentage change in the rate is different: divide that difference by the original rate, then multiply by 100%. State which comparison you mean.

Worked example 7: percentage points or percentage increase?

The proportion of students who cycle to school rises from 30% to 36%. Describe the rise in percentage points and as a percentage increase in the proportion.

The difference between the rates is $36\%-30\%=\boxed{6\text{ percentage points}}$ .
For the relative increase, use the original rate as the base: $\frac{36-30}{30}\times100\%=\boxed{20\%}.$

Check: multiplying the original rate by 1.20 gives $30\%\times1.20=36\%$ . The same change is a rise of 6 percentage points and a 20% relative increase. The percentages alone do not tell us the change in the number of cyclists unless we also know the number of students in each year.

Common mistakes and next steps

Dividing by the new amount for percentage change: use the original amount.
Confusing “of” with “more than”: 130% of a quantity means 30% more than it.
Reversing a decrease by adding the same percentage: divide by the original multiplier.
Adding successive changes: multiply their multipliers because the base changes.
Confusing percentage points with percentage change: a rate rising from 20% to 25% rises by 5 percentage points, but by $5\div20\times100\%=25\%$ relative to the old rate.
Rounding money too early: keep the calculation's precision and round the final payment to cents unless instructed otherwise.

Build fluency using the percentage question bank. Begin with percentage change and reverse percentages, then attempt combined changes and commercial problems.