Percentages, Changes and Compounding

Percentages are another way to write the fractions from the previous lesson. Twenty-five per cent and one quarter express the same proportion. The hundred is a standard comparison scale, not necessarily the actual number of objects involved. A quarter of twelve items is still twenty-five per cent, even though there are only twelve items. Keep asking, “A percentage of what?” Twenty-five per cent of a small budget and twenty-five per cent of a large budget describe the same share but different amounts of money. In this lesson, all numerical examples are illustrative rather than observations of real prices or returns.

We can make the arithmetic visible by splitting twelve per cent into ten per cent plus two per cent. Ten per cent of two hundred and fifty pounds is twenty-five pounds. One per cent is two pounds fifty, so two per cent is five pounds. Adding twenty-five and five gives thirty pounds, matching the decimal method. This is a useful mental check, but the decimal method generalises to awkward percentages as well. The percentage and its base work together. Never assume that a percentage reported beside a number must use that number as its base; read the description carefully.

The number one in each formula represents the entire original amount. With an increase, we keep that original amount and add the extra share. With a decrease, we keep the original amount minus the removed share. This explains why a twenty-per-cent discount leaves eighty per cent of the price rather than twenty per cent. For a positive starting amount, an increase multiplier is above one and a decrease multiplier lies below one. A complete one-hundred-per-cent decrease gives zero. Larger percentage decreases need careful interpretation and are generally unsuitable for ordinary non-negative quantities such as stock remaining.

Calculate the absolute change first: subtract the original value from the new value. Then compare that change with the original value, not the new one. A rise of ten may be large or small depending on where we started. Ten added to forty is a quarter of the starting amount, hence twenty-five per cent. If the new amount is lower, subtraction gives a negative number and the formula records that direction. For a starting value of zero, division by zero prevents a finite percentage calculation. Report the absolute change instead. Changes involving negative bases also require extra contextual interpretation.

The two answers are not competing calculations. They answer different questions. Percentage points give the direct distance between two percentage levels, just as degrees give a distance between temperatures. A percentage change measures that distance relative to the original level. Our increase of six points is one fifth of the original thirty-per-cent level, so its relative size is twenty per cent. Always include the words “percentage points” when you mean a subtraction between percentage levels. Saying only that a rate “rose six per cent” would usually suggest a different relative calculation and could mislead your reader.

Slow down at the change of base. The first ten per cent is ten per cent of one hundred, which is ten. The second is ten per cent of one hundred and ten, which is eleven. Removing eleven after adding ten leaves us one pound below the start. This is why adding the signed percentages, plus ten and minus ten, gives the wrong answer for successive changes. Multipliers retain the changing base automatically. Equal increases and decreases are not reverse operations. To undo a known multiplier, divide by that multiplier rather than applying the same percentage with the opposite sign.

An exponent is a compact way to record repeated multiplication: a multiplier squared means the multiplier times itself. Three periods require three copies. The percentage changes may describe output, costs or another quantity; compounding is a mathematical structure, not a promise about real-world performance. It differs from repeatedly adding the same original-base amount. Before using the formula, identify the length of each period and make sure the rate refers to that same period. An annual percentage cannot simply be relabelled as a monthly percentage. Fees, withdrawals, capacity limits or changing rates would need to be included separately in the model.

This question specifies the order and the base of the charge, so we do not need to guess how the seller calculates it. Begin with the discount. Fifteen divided by one hundred is zero point one five, and multiplying one hundred and twenty by that decimal gives eighteen pounds. Subtract eighteen from one hundred and twenty to get one hundred and two pounds. The multiplier route reaches the same intermediate value because eighty-five per cent remains. Write that intermediate value down clearly. The five-per-cent charge is attached to this discounted amount, rather than to the original list price.

Five per cent of one hundred and two pounds is five pounds ten, so adding the charge produces one hundred and seven pounds ten. We then compare that final total with the original list price. The reduction is twelve pounds ninety, which is ten point seven five per cent of one hundred and twenty pounds. The combined multiplier provides a second check: zero point eight nine two five means the final amount is eighty-nine point two five per cent of the original. Its distance below one hundred per cent is ten point seven five percentage points on that original-price index.

This is a deliberately simple model with a constant growth rate and no rounding to whole units. The units could describe a divisible measure of output, so fractional values are allowed. First turn four per cent into the decimal zero point zero four, then add one to include the original output. Multiplying five hundred by one point zero four gives five hundred and twenty. You can check that by finding four per cent of five hundred, which is twenty, and adding it to the start. The crucial next step is to use five hundred and twenty as the new base.

The second increase is twenty point eight units, slightly larger than the first increase of twenty. That additional zero point eight is four per cent of the first year's extra twenty units. Adding the second increase to five hundred and twenty gives five hundred and forty point eight. To find the overall rate, return to the original five hundred-unit base. The gain of forty point eight divided by five hundred is zero point zero eight one six, or eight point one six per cent. Adding two lots of twenty would produce five hundred and forty, which corresponds to a different, non-compounded rule.

Working backwards means undoing the operation that produced the observed value. A fifteen-per-cent increase multiplies the original by one point one five, so division by one point one five reverses it. Sixty pounds is the result, and multiplying sixty by one point one five checks the answer. Do not remove fifteen per cent of sixty-nine: that would reverse a different operation based on sixty-nine itself. For a discount, the same logic applies with a multiplier below one. If a full one-hundred-per-cent discount leaves zero, however, the original cannot be recovered from that final zero alone.

Suppose a tiny activity doubles while a much larger activity grows only slightly. The tiny activity has the larger percentage increase, but it may still add fewer units. This is why absolute changes and relative changes often belong together. Time periods matter too: a growth figure for one month cannot be compared fairly with a whole-year figure without additional analysis. When rounding a repeated calculation, keep the full intermediate values so that early rounding does not accumulate. Finally, a percentage describes a relationship between numbers. It does not show whether a policy, a price change or some other event caused the movement.

Before attempting the exercises, practise choosing the operation in words. Are you finding part of a whole, moving a quantity forwards, comparing endpoints, or working backwards? Those are different tasks even when the same percentage appears in the question. Write the base next to each calculation and label percentage-point answers explicitly. For successive changes, a small timeline can help: starting amount, first new amount, final amount. Check the final answer against a combined multiplier. Enter a percentage such as seven point five as 7.5 when the field already displays the percent unit, rather than entering its decimal multiplier.