0580 · 1.13Percentages

Percentage basics and the whole

0:00A price rises by twenty per cent, then falls by twenty per cent. It sounds as though the changes should cancel. In fact, the final price is lower than the starting price. The reason is that each percentage uses a different amount as its whole. Understanding that whole makes percentages much easier. We will use it to calculate discounts, deposits, earnings, profit and interest. We will also work backwards when the original amount is missing.

0:25Every method comes from the same idea, rather than a collection of unrelated rules. Per cent means out of one hundred. Thirty per cent describes thirty out of every hundred equal parts of a whole. We can write this as 30/100, or zero point three. Multiplying a quantity by this fraction finds thirty per cent of that quantity. You need to be comfortable multiplying, dividing and converting between fractions and decimals.

0:50Fifteen per cent is 15/100, which equals zero point one five. The original quantity represents one hundred per cent. Before calculating, identify exactly which quantity the percentage refers to. Percentages can exceed one hundred because they can describe more than one whole. One hundred and fifty per cent means one whole plus half of another. Its decimal multiplier is one point five. However, an increase of one hundred and fifty per cent produces two hundred and fifty per cent of the original.

Finding percentages of quantities

1:20Here is an exam-style example that works without a calculator. A bicycle costs two hundred and forty pounds, and the deposit is fifteen per cent. A deposit is an initial payment towards the total price. Calculate the deposit and the balance still to pay. Ten per cent of two hundred and forty pounds is twenty-four pounds. Halving that gives five per cent, which is twelve pounds. Adding twenty-four and twelve gives a deposit of thirty-six pounds.

1:46Subtracting thirty-six from two hundred and forty leaves a balance of two hundred and four pounds. This method splits a percentage into convenient parts. One per cent comes from dividing by one hundred, while ten per cent comes from dividing by ten. Alternatively, multiply two hundred and forty by zero point one five to find the same deposit. Both approaches work because they calculate fifteen hundredths of the price. Suppose someone earns one hundred and twenty per cent of their basic weekly pay after a bonus.

2:14Their basic pay is two hundred and fifty pounds. Converting one hundred and twenty per cent to a decimal gives one point two. Multiplying two hundred and fifty by one point two gives total earnings of three hundred pounds. The wording matters here because the total earnings are one hundred and twenty per cent of basic pay. That means the bonus is twenty per cent, not one hundred and twenty per cent.

2:34Always distinguish the final amount from the increase itself. One includes the original whole, while the other describes only the extra part.

Percentage comparisons, profit and loss

2:42To express one quantity as a percentage of another, first divide by the quantity representing the whole. Suppose eighteen students out of sixty travel by bus. 18 ÷ 60 gives zero point three. Multiplying by one hundred expresses that fraction as thirty per cent. The order of division is important. We want the bus travellers as a fraction of all sixty students, so sixty belongs in the denominator. Reversing the division would answer a different question.

3:11If the quantities have different units, convert them into matching units before comparing them. Percentage increase compares the increase with the original amount. A price rises from eighty pounds to ninety-two pounds. Subtracting eighty from ninety-two gives an increase of twelve pounds. Dividing twelve by eighty and multiplying by one hundred gives a percentage increase of fifteen per cent. Now reverse the change, so the price falls from ninety-two pounds to eighty pounds.

3:36The decrease is still twelve pounds, but the original amount is now ninety-two pounds. 12 ÷ 92, multiplied by one hundred, gives thirteen point zero four three and further digits. To three significant figures, the percentage decrease is thirteen point zero per cent. The amounts changed by the same twelve pounds, but the percentages differ. Twelve represents a smaller fraction of ninety-two than of eighty. For percentage change, divide the change by the starting amount, not the final amount.

4:06This is one of the most important errors to avoid. Profit is the amount by which selling price exceeds cost price. An item bought for eighty pounds and sold for one hundred pounds makes twenty pounds profit. Dividing twenty by eighty and multiplying by one hundred gives twenty-five per cent profit. Unless the question specifies otherwise, percentage profit is based on cost price. A loss occurs when the selling price is below the cost price.

4:32Selling that same item for sixty-eight pounds gives a loss of twelve pounds. Calculating 12 ÷ 80 × 100 gives fifteen per cent loss. Dividing by the selling price would use the wrong whole.

Multipliers and repeated changes

4:45When the percentage change is given, a multiplier lets us calculate the new amount directly. Increasing something by fifteen per cent leaves us with one hundred and fifteen per cent of the original. Dividing that percentage by one hundred gives a multiplier of one point one five. Multiplying by it includes both the original amount and the increase. A discount reduces the price, so we calculate the percentage remaining. A fifteen per cent discount leaves eighty-five per cent of the original price.

5:12The multiplier is therefore zero point eight five. Multiplying two hundred and forty pounds by zero point eight five gives a discounted price of two hundred and four pounds. For an increase, add the percentage to one hundred before converting to a multiplier. For a decrease, subtract it from one hundred instead. Multiplying by zero point one five finds only fifteen per cent of the original. It does not give the final amount after a fifteen per cent increase or decrease.

5:40We can now explain the changes from the opening example. Increasing one hundred pounds by twenty per cent gives one hundred and twenty pounds. Reducing this new amount by twenty per cent gives ninety-six pounds. The overall decrease is four pounds, which is four per cent of the original hundred pounds. The second percentage acts on one hundred and twenty pounds, not on the starting hundred pounds. Its reduction is therefore larger than the earlier increase.

6:04For repeated percentage changes, apply each multiplier to the current amount. Do not simply add or subtract the percentage rates.

Simple and compound interest

6:12Interest is money earned on savings or charged on borrowing. The principal is the original amount invested or borrowed. With simple interest, every period uses that same principal. With compound interest, each period uses the current balance, including interest already added. Suppose five hundred pounds earns simple interest at four per cent per year for three years. Multiplying five hundred by zero point zero four gives annual interest of twenty pounds.

6:40Three years therefore produce sixty pounds interest. Adding this to the principal gives a final amount of five hundred and sixty pounds. This explains the simple interest formula rather than just asking you to remember it. Interest equals principal multiplied by the percentage rate divided by one hundred, multiplied by the number of periods. On screen, P represents principal, r is the percentage rate, and t is time. An annual rate requires the time to be measured in years.

7:07For compound interest, assume the same investment earns four per cent annually, with interest added at each year end. The multiplier is one point zero four. After one year, 500 × 1.04 gives five hundred and twenty pounds. That entire balance becomes the starting amount for year two. Multiplying five hundred and twenty by one point zero four gives five hundred and forty pounds and eighty pence. Applying the multiplier again gives five hundred and sixty-two point four three two pounds.

7:37To the nearest penny, the final balance is five hundred and sixty-two pounds and forty-three pence. Subtracting the original five hundred pounds gives interest of sixty-two pounds and forty-three pence, to the nearest penny. A power provides a shorter way to write repeated multiplication by the same factor. The final amount equals 500 × 1.04³. In general, multiply the principal by the multiplier raised to the power of the number of periods.

8:03That multiplier is one plus the percentage rate over one hundred. This assumes a constant rate and no additional deposits or withdrawals. Simple interest adds the same amount each period because its reference amount stays fixed. Compound interest also earns interest on earlier interest, so the additions grow. The compound formula gives the final balance, not the interest alone. To find the interest, subtract the principal.

Reverse percentages and exam practice

8:27Reverse percentages recover the original amount from a changed amount. A sale price of sixty-eight pounds follows a fifteen per cent discount. The sale price therefore represents eighty-five per cent of the original. Dividing sixty-eight by zero point eight five gives an original price of eighty pounds. We divide because this undoes the multiplication used for the discount. Adding fifteen per cent to sixty-eight pounds would use the sale price as the whole.

8:52That is not the quantity on which the discount was calculated. Checking 80 × 0.85 confirms the result of sixty-eight pounds. In examinations, read carefully whether you must find a percentage of a quantity, a percentage change, or an original amount. Paper Two is non-calculator, so practise convenient percentage parts, fraction methods and written division. Paper Four allows a calculator, but you still need to choose the correct whole and multiplier.

9:20Simple and compound interest formulae are not given. Consider a final exam-style question combining two different percentage relationships. A shop sells an item for one hundred and sixty-two pounds after a ten per cent discount. The shop makes thirty-five per cent profit on its cost price. Calculate the marked price, the cost price and the profit. The selling price represents ninety per cent of the marked price. Dividing one hundred and sixty-two by zero point nine gives a marked price of one hundred and eighty pounds.

9:49Separately, the selling price represents one hundred and thirty-five per cent of the cost price. Dividing one hundred and sixty-two by one point three five gives a cost price of one hundred and twenty pounds. Subtracting one hundred and twenty from one hundred and sixty-two gives a profit of forty-two pounds. As a check, 42 ÷ 120 × 100 gives thirty-five per cent. Notice that the discount refers to marked price, while the profit refers to cost price.

10:17Labelling these quantities prevents the two relationships from becoming confused. Keep full calculator values during a calculation rather than rounding intermediate results. Unless instructed otherwise, give answers that are not exact to three significant figures. For money, follow the requested accuracy, usually the nearest penny when an amount needs rounding. Exact answers do not need artificial rounding. The central question is always which amount represents one hundred per cent. Finding a percentage means taking a fraction of that whole.

10:49Percentage change compares the change with the original amount. Forward changes multiply by a multiplier, while reverse percentages divide by it. Repeated changes act on successive amounts, which explains why equal increases and decreases do not cancel. Simple interest uses a fixed principal, while compound interest uses an updated balance. Keep the final amount separate from the change, profit or interest. With these relationships understood, the calculations become a connected system rather than isolated rules.