C1.13; E1.13 - Percentages of quantities | Cambridge IGCSE Mathematics

Percentages describe amounts relative to a chosen whole. You will calculate percentages of quantities, express one amount as a percentage of another, and choose the correct base for deposits, discounts, earnings, profit and loss.

Finding a percentage of a quantity

Can a percentage of an amount be greater than the amount itself? If $100\%$ means one whole, $120\%$ means one whole and another fifth. A percentage is not restricted to the range from zero to one hundred.

Percentage

A percentage is a number of parts per hundred. Thus $12\%=12/100=0.12$ , while $120\%=120/100=1.2$ . The base quantity is the amount represented by $100\%$ .

To find $12\%$ of $450$ , first find one hundredth of $450$ , then take twelve of those parts. Multiplying by $12/100$ performs both steps together.

Percentage of a quantity

p\%\text{ of an amount}=\frac{p}{100}\times\text{amount}

Use the amount to which the percentage refers as the whole.

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A deposit is an initial payment towards a total price. A $12\%$ deposit on an item means pay $12\%$ of its full price now.

Worked example 1: a deposit

Question. A desk costs $\$ 450 $. Calculate a deposit of$ 12%$ of the price.

Plan. Take twelve hundredths of the full price.

\begin{aligned} 450\div100&=4.50 &&\text{one per cent of the price}\\ 4.50\times12&=54 &&\text{twelve per cent} \end{aligned}

Answer. The deposit is $\$ 54$.

Check. $10\%$ is $\$ 45 $, and$ 2% $is$ $9 $. Together they give$ $54$.

Change only the percentage from $12\%$ to $120\%$ . The same calculation now gives more than the whole price; it would no longer describe an ordinary deposit.

Worked example 2: more than one whole

Question. Calculate $120\%$ of $\$ 450$.

Plan. Multiply by $120/100$ , which is greater than $1$ .

\begin{aligned} \frac{120}{100}&=1.2 &&\text{convert the percentage}\\ 1.2\times450&=540 &&\text{multiply the base amount} \end{aligned}

Answer. $\$ 540$.

Check. $100\%$ is $\$ 450 $and$ 20% $is$ $90 $; their sum is$ $540$.

The calculation finds an amount; the context tells you how to use it. A $12\%$ discount on the desk saves $\$ 54 $, leaving$ 450-54=$396 $to pay. A salesperson earning$ 5% $**commission** on$ $800 $of sales receives$ 0.05\times800=$40 $commission. If their fixed pay is$ $300 $, total earnings are$ 300+40=$340$.

Watch out

$12\%$ is $0.12$ , not $12$ and not $0.012$ . Also distinguish the percentage amount from the final payment: a $12\%$ discount is the saving, not the price after the saving.

Expressing a proportion as a percentage

To express one quantity as a percentage of another, first form their fraction, then scale it to a number of hundredths. The quantity after “of” supplies the denominator. Both quantities must use common units.

One quantity as a percentage of another

\text{percentage}=\frac{\text{quantity being compared}}{\text{base quantity}}\times100\%

The base must be non-zero. Use consistent units before dividing.

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For example, $250\,\mathrm{ml}$ as a percentage of $2\,\mathrm{litres}$ compares $250$ with $2000$ , not with $2$ . After conversion, the fraction is $250/2000=1/8$ , or $12.5\%$ .

Worked example 3: a score as a percentage

Question. A student scores $36$ out of $48$ . Express the score as a percentage.

Plan. Divide the score by the total available, then multiply by one hundred.

\begin{aligned} \frac{36}{48}&=\frac34 &&\text{fraction of available marks}\\ \frac34\times100\%&=75\% &&\text{convert to hundredths} \end{aligned}

Answer. $75\%$ .

Check. $75\%$ of $48$ is $\frac34\times48=36$ , recovering the score.

Keep the base of $48$ , but compare $60$ with it. This is no longer a score out of $48$ ; the numerator is larger than the base, so expect more than $100\%$ .

Worked example 4: a quantity above the base

Question. Express $60$ as a percentage of $48$ .

Plan. Keep $48$ as the denominator because it follows “of”.

\begin{aligned} \frac{60}{48}&=1.25 &&\text{compare with the base}\\ 1.25\times100\%&=125\% &&\text{express as a percentage} \end{aligned}

Answer. $125\%$ .

Check. $100\%$ of $48$ is $48$ and $25\%$ is $12$ , giving $60$ .

Watch out

“ $36$ as a percentage of $48$ ” is $36/48\times100\%$ , not $48/36\times100\%$ . Reversing the fraction changes the question.

Choosing the base quantity

Before dividing, say what represents $100\%$ . A deposit or discount is normally a percentage of the full listed price; commission is a percentage of the sales amount stated in the question. Profit and loss percentages are normally based on the cost price, the amount originally paid for the item.

Profit and loss

If the selling price exceeds the cost price, their difference is the profit. If the selling price is lower, the difference is the loss. Divide the profit or loss by the cost price to find the corresponding profit or loss percentage, unless another base is explicitly requested.

The profit itself is not the selling price. Selling a $\$ 80 $item for$ $100 $brings in$ $100 $, but only$ $20 $is profit: the other$ $80$ covers its cost.

Worked example 5: profit compared with cost

Question. An item costs $\$ 80 $and is sold for$ $100$. Calculate the profit and the percentage profit.

Plan. Find the extra amount, then compare it with the cost price.

\begin{aligned} 100-80&=20 &&\text{profit in dollars}\\ \frac{20}{80}\times100\%&=25\% &&\text{cost price is the base} \end{aligned}

Answer. Profit $\$ 20 $; percentage profit$ 25%$.

Check. $25\%$ of $\$ 80 $is$ $20 $, and$ 80+20=100$, the selling price.

Change only the selling price to $\$ 60$. The result becomes a loss, but the cost price remains the base.

Worked example 6: loss compared with cost

Question. An item costs $\$ 80 $and is sold for$ $60$. Calculate the loss and the percentage loss.

Plan. Find how much of the cost was not recovered, then divide by the cost.

\begin{aligned} 80-60&=20 &&\text{loss in dollars}\\ \frac{20}{80}\times100\%&=25\% &&\text{use the original cost} \end{aligned}

Answer. Loss $\$ 20 $; percentage loss$ 25%$.

Check. Subtracting $25\%$ of $\$ 80 $from$ $80 $gives$ $60$, the actual selling price.

The same $\$ 20 $profit in the first example is$ 20% $of the$ $100$ selling price. That calculation answers a different question. Always read the requested base rather than assuming that any available total belongs in the denominator.

Watch out

A percentage requires a reference amount. “The deposit is $\$ 50 $” is not enough to find its percentage unless the full price is known. Likewise, a$ $20$ profit alone does not determine a percentage profit.

Quick Recap

A percentage means parts per hundred; multiply by $p/100$ to find $p\%$ of an amount.
To express one quantity as a percentage of another, divide by the stated base and multiply by $100\%$ .
Put comparable quantities into common units, and identify the non-zero amount representing $100\%$ .
Profit or loss percentages normally use cost price; deposits, discounts and commission use their stated reference amount.

The opening answer is yes: $120\%$ of an amount is $1.2$ times that amount, so it is greater than the whole.

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C1.13; E1.13 - Percentages of quantities

Finding a percentage of a quantity

Expressing a proportion as a percentage

Choosing the base quantity

Quick Recap

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