2.03c - Percentage change and original values

Increasing and decreasing amounts

A coat costs $£70$ after a $12.5\%$ reduction. Would adding $12.5\%$ recover its original price? Keep this question in mind: the answer depends on which price represents $100\%$.

Start with a known original price. A $15\%$ increase means add $15\%$ of that original amount; a $15\%$ decrease means subtract the same share. Calculate the change before combining it with the starting amount.

The original amount, not the final one, is the reference for the stated change.

Change only the direction of the change. The percentage amount is still $£12$, but now it is taken away.

An increase can exceed $100\%$: a $120\%$ increase adds more than the original amount, leaving $220\%$ of it. This is different from “becomes $120\%$ of the original”, which means a $20\%$ increase.

Keep the percentage and direction, but change the starting amount.

Percentage multipliers

A multiplier combines finding the change and adding or subtracting it into one multiplication.

For a $15\%$ decrease, $100\%-15\%=85\%$ remains. So multiplying by $0.85$ gives the final amount directly. A $15\%$ increase leaves $115\%$, giving multiplier $1.15$. This works because the original amount is one whole copy of itself.

Use a fraction instead of a decimal when it makes the arithmetic easier.

Change only the direction. Subtracting the same percentage makes the multiplier less than one.

You can also read the change from a multiplier. $1.08$ means an $8\%$ increase; $0.92$ means an $8\%$ decrease. For two known positive amounts, divide final by original to obtain the multiplier. For example, $70\div80=0.875$, so the decrease is $(1-0.875)\times100\%=12.5\%$. Equivalently, divide the size of the change by the original and multiply by $100\%$.

Use the same fraction multiplier with a new original amount.

Finding the original amount

To reverse a percentage change, divide the final amount by the multiplier that produced it.

Return to the coat costing $£70$ after a $12.5\%$ reduction. The $£70$ represents $87.5\%$, or seven eighths, of the old price. Dividing it into seven equal parts gives one eighth of the old price; taking eight such parts recovers the whole.

[DIAGRAM: asset_name: Recovering the original price; asset_slug: reverse-percentage-bar; caption: The sale price is seven eighths of the original, so each eighth is ten pounds.; description: An eight-cell horizontal bar. Seven equal pale blue cells represent the final 87.5 percent price of 70 pounds and the remaining pale peach cell represents the removed 12.5 percent. Each cell is labelled 10 pounds. A bracket across all eight cells identifies the original 100 percent price of 80 pounds.]

For a positive final amount and a multiplier between zero and one, dividing by the multiplier makes the amount larger, as an original sale price should be.

Keep the final price at $£63$ and change only the direction of the original change. We now reverse an increase, so the original amount must be smaller than the final amount.

The opening proposal fails because the percentage would use a different whole. Adding $12.5\%$ of $£70$ gives $£78.75$, not $£80$. The reduction was $12.5\%$ of the old price, which was larger. The inverse of multiplying by $0.875$ is dividing by $0.875$, not multiplying by $1.125$.

Reverse an increase with a new final price.

Quick Recap

• Calculate a percentage change from the original amount.
• An increase uses multiplier $1+\frac p{100}$; a decrease uses $1-\frac p{100}$.
• Reverse a nonzero multiplier by division to recover the original whole.
• Equal percentage rises and falls are measured from their own starting amounts, so they do not generally undo each other.

The coat's original price was $£80$: divide $£70$ by $0.875$, rather than adding $12.5\%$ of the sale price.