Percent Increase and Decrease

Why we measure change against the original

Suppose a price moves from one amount to another. The plain amount of change is just the difference:

If the new value is bigger, the change is positive and the quantity went up. If the new value is smaller, the change is negative and the quantity went down. That difference alone does not tell you how big a deal the change is, because the same difference can be huge or tiny depending on the size of the thing it happened to. A raise of $1{,}000$ dollars is life-changing on a $2{,}000$ dollar salary and barely noticeable on a $200{,}000$ dollar salary.

So compare the change with a baseline: the original value, the amount you had before the change happened. Percent change is the change compared with that original, written as a percent, and it asks the same kind of question as before, what percent one number is of another, now applied to the change and the original:

A positive result is a percent increase; a negative result is a percent decrease.

The original goes on the bottom because it is the amount the change is being measured against, the whole the change is a part of. Putting the new value on the bottom instead answers a different question. Doing that is the single most common error in this whole topic.

(The original must not be $0$: dividing by $0$ is undefined, so a quantity that starts at $0$ has no percent change.)

Finding the new value with a multiplier

Often you know the original amount and the percent change, and you want the new value. There are two ways to get it, and they always agree.

The two-step way matches how you would explain it in words. To increase $200$ by $15\%$, first find $15\%$ of the original, then add it on:

For a decrease you subtract instead. To decrease $200$ by $15\%$, find the same $30$ and take it away: $200 - 30 = 170$.

The one-step way does it with a single multiplication.

So to grow an amount by $p\%$, multiply by $1 + \frac{p}{100}$; to shrink it by $p\%$, multiply by $1 - \frac{p}{100}$. This number is called the multiplier. A $15\%$ increase leaves you with $115\%$ of the original, which is why its multiplier is $1.15$. A $15\%$ decrease leaves you with $85\%$, so its multiplier is $0.85$.

Discounts, tax, tip, and markup

These multipliers are exactly the arithmetic behind everyday money problems. A discount (a sale) is a percent decrease: ”$25\%$ off” means you pay $100\% - 25\% = 75\%$ of the original price, a multiplier of $0.75$. Sales tax, a tip, and a markup are percent increases added on top of a price: a $7\%$ tax means you pay $107\%$ of the listed price, a multiplier of $1.07$, and a $40\%$ markup on cost means the selling price is $1.40$ times the cost. Decide whether the percent is added or removed, build the matching multiplier, and multiply.

The grid below is the original price, split into $100$ squares. Shade only the squares a shopper still pays for. Shade $65$ and the readout gives $0.65$, the multiplier from the backpack above; try $70$ for a $30\%$ discount and $90$ for a $10\%$ discount. The number named in a sale is the part that leaves; the number you multiply by is the part that stays, so a $30\%$ discount multiplies by $0.70$, never by $0.30$.

Two changes in a row

Worked Example 5 could add the tax and tip together only because both were figured on the same menu price. Often a second percent is figured on the result of the first change instead, and then the percents cannot simply be added.

Whenever a second percent applies to a moving amount rather than a fixed one, apply each multiplier to the value that came before it, not to the original both times. Applying multiplier $m_1$ and then $m_2$ is the same as multiplying once by their product $m_1 \times m_2$, and multiplication can be done in either order, so it makes no difference which of the two changes you apply first, only which amount each one is based on.

Finding the original from the new value

Say a $20\%$ discount leaves an item priced at $40$ dollars. The discount left $100\% - 20\% = 80\%$ of the original, so the $40$ you paid is $80\%$ of the original price:

You know a part ($80\%$ of the original) and want the whole, so divide the part by that decimal to recover it:

The original price was $50$ dollars. This is the general pattern whenever you know a new value and the percent that produced it: read off what percent of the original the new value is, then undo the multiplication with division, instead of multiplying to find a part. The reason you divide instead of adding the percent back is the trap in Worked Example 8 below: $20\%$ of the sale price is a smaller amount than $20\%$ of the original, so adding $20\%$ of the sale price back on cannot recover it. The same logic handles an increase: a value that is the result of a $25\%$ increase is $125\%$ of the original, so you divide by $1.25$ to get back to it.

(A $100\%$ decrease has a multiplier of $0$, and dividing by $0$ is undefined, so no division can recover an original from a value that was reduced all the way to $0$.)

Common mistakes