POP

E1.13 Percentages, Including Reverse Percentages

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01 — Explanation

How do I find a percentage increase or decrease using a multiplier?

Convert the percentage change into a single multiplier and apply it directly:

Change	Multiplier
Increase of $r\%$	$1+\dfrac{r}{100}$
Decrease of $r\%$	$1-\dfrac{r}{100}$

\text{Original} \times \text{Multiplier} = \text{Final}

02 — Explanation

What is a reverse percentage question, and how is it different?

A standard percentage question gives you the original amount and asks for the result after a change. A reverse percentage question gives you the final amount and asks you to work backwards to the original ( $100\%$ ) value — which means dividing by the multiplier, not applying the percentage again.

Examiner Tip: The most common error here is finding the percentage of the final value and subtracting or adding it, as if the change had been applied to that number. The percentage was always applied to the unknown original value — you must divide the final amount by the multiplier to undo it.

03 — Explanation

How do I set up and solve a reverse percentage problem?

\text{Original} = \frac{\text{Final}}{\text{Multiplier}}

Worked Example

A shop increases all prices by $12\%$ . After the increase, a chair costs $\$ 140$. Find the price before the increase.

A $12\%$ increase means the final price is $112\%$ of the original: multiplier $=1.12$ .
Set up the equation: $\text{Original}\times1.12 = 140$ .
Solve: $\text{Original} = \dfrac{140}{1.12} = \$ 125$.

Examiner Tip: Double-check the direction of your multiplier against the wording — "increase" always gives a multiplier greater than $1$ ; "decrease" always gives one less than $1$ . Using $0.88$ for what's actually an increase is an easy slip under time pressure.

04 — Explanation

What's the difference between simple and compound interest?

Simple interest pays the same fixed amount every year, always calculated on the original principal. Compound interest instead pays interest on the running total, so each year's interest is calculated on a slightly larger amount than the last — the multiplier method from above applies directly, using $t$ as the number of years:

\text{Simple: } I = P \times \frac{r}{100} \times t \qquad \text{Compound: } A = P\left(1+\frac{r}{100}\right)^t

where $P$ is the principal, $r\%$ is the annual rate, and $A$ is the total amount after $t$ years.

Worked Example

$\$ 2000 $is invested for$ 3 $years. Compare the total value under (a) simple interest at$ 5\% $per year, and (b) compound interest at$ 5\%$ per year.

Simple interest: $I = 2000 \times \dfrac{5}{100}\times3 = \$ 300 $, so the total is$ 2000+300=\ $2300$ .
Compound interest: $A = 2000\times(1.05)^3 = 2000\times1.157625 = \$ 2315.25$.
Compound interest earns $\$ 15.25$ more, because the second and third years' interest are calculated on an already-grown amount.

Examiner Tip: Never apply the compound-interest multiplier as $1+\dfrac{r}{100}\times t$ — the rate must be raised to the power $t$ , not multiplied by it. That confusion is the most common way marks are lost on this topic.

05 — Explanation

How do I calculate profit or loss as a percentage of the original cost?

\text{Percentage profit/loss} = \frac{\text{Profit or loss}}{\text{Original cost price}} \times 100

Always divide by the cost price (what was originally paid), never by the selling price — the percentage is measuring the change relative to the starting point of the transaction.