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 (the quantity went up). If the new value is smaller, the change is negative (the quantity went down). That difference alone is not enough, 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 a rounding error on a $200{,}000$ dollar salary.

So we need a baseline to compare the change against. The only sensible baseline is the original value, the amount you had before the change happened. The change grew out of that starting amount, so the fair question is “how big is this change compared to where we started.” Asking what fraction the change is of the original, and then writing that fraction as a percent, is the percent change:

The original value must not be zero. Dividing by zero is undefined, so a quantity that starts at zero has no percent change at all.

This is the second percent task from the last lesson, “what percent is one number of another,” applied to the change and the original. You take the ratio $\frac{\text{change}}{\text{original}}$ and rescale it to a denominator of $100$, which is what multiplying by $100\%$ does. A positive result is a percent increase; a negative result is a percent decrease.

The original goes on the bottom for a concrete reason: it is the whole that the change is being measured as a part of. Putting the new value on the bottom would answer a different and less natural question. Doing that is the single most common error in this whole topic.

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 $1 - \frac{p}{100}$. The number you multiply by is called the multiplier. Returning to the $15\%$ increase on $200$:

the same $230$ as before, in one step. The decrease is $\left(1 - \frac{15}{100}\right) \times 200 = 0.85 \times 200 = 170$. A useful way to read the multiplier: a $15\%$ increase leaves you with $115\%$ of the original. A $15\%$ decrease leaves you with $85\%$ of the original, and those two readings are why the multipliers are $1.15$ and $0.85$.

Discounts, tax, tip, and markup

These multipliers are exactly the arithmetic behind everyday money problems. Every one of them is either a percent decrease or a percent increase on a starting price.

A discount (a sale) is a percent decrease. ”$25\%$ off” means you pay $100\% - 25\% = 75\%$ of the original, so the sale price is $0.75$ times the original. Sales tax and a tip are percent increases added to a bill. A $7\%$ tax means you pay $107\%$ of the listed price, a multiplier of $1.07$. A markup is a percent increase a store adds to its cost to set the selling price. A $40\%$ markup on cost means the price is $1.40$ times the cost. In every case, decide whether the percent is added or removed, build the matching multiplier, and multiply.

Read the whole grid below, all $100$ squares of it, as the original price, and shade only the squares a shopper still pays for.

Shade $65$ of them and the readout hands you $0.65$, the multiplier from the backpack above. The $35$ unshaded squares are the discount and the shaded ones are the sale price. Shade $70$ instead and you have the multiplier for $30\%$ off, and $90$ gives the one for $10\%$ off. The pattern to catch is that the number named in the sale is the part that leaves. The number you multiply by in the sale is the part that stays, so a $30\%$ discount multiplies by $0.70$ and never by $0.30$.

Finding the original from the new value

Sometimes you know the new value and the percent change, and you want the original amount that came before it. You just have to read off what percent of the original the new value is, and then undo the multiplication with division.

That division needs a multiplier that is not zero. A $100\%$ decrease has a multiplier of $0$, so the new value is $0$ whatever the original was. No division can recover the original from that.

After $20\%$ off, the sale price is what is left of the original, which is $100\% - 20\% = 80\%$ of it. So the sale price is $80\%$ of the original, which means

This is now the third percent task from the last lesson. You know a part ($80\%$ of the original is the sale price) and you want the whole (the original). To recover a whole from a known part, you divide. If the sale price is $48$ dollars, then $48$ is $80\%$ of the original, so

The original price was $60$ dollars. The check runs forward: $80\%$ of $60$ is $0.80 \times 60 = 48$, the sale price you were given. The same logic handles an increase. If a number is the result of a $25\%$ increase, it is $125\%$ of the original. So you divide the new value by $1.25$ to get back to the original.

The reason you divide rather than just subtracting the percent is the trap of the next section. $20\%$ of the sale price is not the same as $20\%$ of the original, so adding $20\%$ of $48$ back on cannot recover it. You must divide by the multiplier that produced the new value.

Common mistakes