Percentage Change, Profit and Loss

A shop raises the price of a $\$200$ jacket by 10% in autumn, then drops it by 10% in the winter sale. A customer says "it is back to $\$200$, so the sale is pointless". Are they right? Write down your instinct before you calculate anything, then keep it in mind: this single question catches out more students in the exam than any other percentage idea.

Almost every percentage question in this module is faster and safer if you stop calculating the change separately and use a single multiplier instead.

Start from 100%, which is the whole original amount, then adjust:

• Increase by $r$%: multiplier $= 1 + \dfrac{r}{100}$. A 15% rise gives $1.15$.
• Decrease by $r$%: multiplier $= 1 - \dfrac{r}{100}$. A 15% fall gives $0.85$.

Then new price = original price × multiplier. A $\$80$ jacket marked up 15% costs $80 \times 1.15 = \$92$. The same jacket discounted 15% costs $80 \times 0.85 = \$68$.

Doing it in one step matters because it makes the reverse calculation obvious: if multiplying by 1.15 goes forwards, then dividing by 1.15 goes backwards. You will use that in Card 3.

Increase by r%: multiplier = 1 + r/100. Decrease by r%: multiplier = 1 − r/100. New price = original × multiplier. 15% up on 80 is 80 x 1.15 = 92; 15% down is 80 x 0.85 = 68 (dollars).

Pause, copy the two multiplier rules into your book (increase: 1 + r/100, decrease: 1 − r/100), with one worked number beside each so you can see the pattern: 80 × 1.15 = 92 and 80 × 0.85 = 68.

The dollar answer comes first, and it is a subtraction:

Profit or loss = selling price − cost price

A positive answer is a profit. A negative answer is a loss, and it is normal to report the loss as a positive number with the word "loss" attached rather than as a negative profit.

To compare deals fairly, convert to a percentage of what the seller had to put in, which is the cost price:

Percentage profit $= \dfrac{\text{profit}}{\text{cost price}} \times 100$

A bike bought for $\$250$ and sold for $\$310$ makes $\$60$ profit, which is $\dfrac{60}{250} \times 100 = 24\%$. A pen bought for $\$2$ and sold for $\$3$ makes only $\$1$, but that is a 50% profit, so the pen is the better deal per dollar invested even though the bike returns more cash.

Profit or loss = selling price − cost price. Positive is profit, negative is a loss. Percentage profit = profit ÷ cost price × 100. Bike: 310 - 250 = 60 dollars profit, and 60/250 = 24%.

Pause, copy both formulas into your book (profit or loss = selling price − cost price; percentage profit = profit ÷ cost price × 100) and underline "cost price" in the second one, because dividing by the selling price is the most common lost mark here.

Because a percentage change is a multiplication, the way back is a division by the same multiplier:

Original price = new price ÷ multiplier

A jacket sells for $\$92$ after a 15% increase. The multiplier was $1.15$, so the original price is $92 \div 1.15 = \$80$. A coat costs $\$308$ after a 12% discount, so the multiplier was $0.88$ and the original price is $308 \div 0.88 = \$350$.

This is the single most-tested idea in the whole percentage strand, because the wrong method looks so reasonable. Taking 15% off the $\$92$ gives $92 \times 0.85 = \$78.20$, which is not $\$80$. The reason is that the 15% was calculated on the original $\$80$, not on the new $\$92$, so you cannot take the percentage off the new figure.

A quick way to check any reverse answer: run it forwards again. $80 \times 1.15 = 92$ ✓, and $350 \times 0.88 = 308$ ✓. If it does not come back, the division was wrong.

Original price = new price ÷ multiplier. After a 15% rise: 92 / 1.15 = 80 dollars. After a 12% discount: 308 / 0.88 = 350 dollars. Never take the percentage off the new price. Check by multiplying forwards.

Pause, copy the reverse rule into your book (original = new ÷ multiplier) with both worked numbers, 92 / 1.15 = 80 and 308 / 0.88 = 350, and write beside it the warning that 92 x 0.85 = 78.20 is the wrong method.