Percent of Change and New Amounts

Every problem in this lesson has three quantities, all measured in the same unit:

• The original amount is the amount before the change. In this lesson it is always a positive number.
• The new amount is the amount after the change.
• The amount of change is the difference between them: how much was gained or lost. To find it, subtract the smaller amount from the larger one.

If the new amount is greater than the original amount, the change is an increase. If the new amount is less than the original amount, the change is a decrease.

You already know that a percent is a rate per $100$ of a stated positive whole, and that $\text{part} = \text{percent} \times \text{whole}$. For a percent of change, the whole is the original amount and the part is the amount of change.

Try it with the first playlist. The original amount is $20$ songs, which is positive, and the new amount is $25$ songs. The amount of change is $25 - 20 = 5$ songs. Write the change as a fraction of the original amount, then scale it to a rate per $100$:

$\frac{5}{20} = \frac{5 \times 5}{20 \times 5} = \frac{25}{100} = 25\%$

Multiplying the numerator and the denominator by $5$ keeps the ratio the same, so the first playlist grew by $25\%$. That is its percent of increase.

The second playlist also changed by $5$ songs, but its original amount is $200$ songs: $\frac{5}{200} = \frac{2.5}{100} = 2.5\%$. Dividing the numerator and the denominator by $2$ keeps the ratio the same, so its percent of increase is only $2.5\%$.

Now reverse the question. You know the positive original amount and the percent of change, and you want the new amount.

The percent of change tells you what part of the original amount is added or taken away. So first find the amount of change with $\text{part} = \text{percent} \times \text{whole}$, using the original amount as the whole. Then:

• after an increase, add the amount of change to the original amount;
• after a decrease, subtract the amount of change from the original amount.

Suppose a game score of $80$ points increases by $15\%$. Since $15\%$ means $15$ per $100$, write it as $0.15$. The amount of change is $0.15 \times 80 = 12$ points, so the new amount is $80 + 12 = 92$ points.

You can also get there in one step. The original amount is $100\%$ of itself, so after a $15\%$ increase you have $100\% + 15\% = 115\%$ of it:

Here is why that works. The original amount is $1 \times 80$ and the amount of change is $0.15 \times 80$. Each is a number times $80$, so you can add the two numbers first, $1 + 0.15 = 1.15$, and then multiply by $80$ once. That is the distributive property at work. The number $1.15$ is the multiplier: the number you multiply the original amount by to get the new amount. It is $115\%$ written as a decimal.

In general, let $r$ stand for the percent of change written as a decimal, such as $r = 0.15$ for $15\%$. For a positive original amount:

• after an increase, $\text{new amount} = \text{original amount} \times (1 + r)$;
• after a decrease, $\text{new amount} = \text{original amount} \times (1 - r)$.

An amount that cannot go below zero can lose at most $100\%$ of itself, so for a decrease of such an amount, $r$ is at most $1$.

The takeaway: a percent of change always measures the change against the positive original amount, never the new amount. To find it, divide the amount of change by the original amount and write the result as a percent. To go the other way, find that percent of the original amount, then add it for an increase or subtract it for a decrease. In one step, multiply the original amount by $1 + r$ for an increase or by $1 - r$ for a decrease, where $r$ is the percent of change written as a decimal.

Next, you will work backward to find the original amount before a percent change, and you will solve percent of change word problems.