Parts, Wholes, and Percent Equations

Finding a percent of a number. Here you know the percent and the whole, and you want the part.

For the tank, you took $25$ of $100$ equal shares of $60$: you divided $60$ by $100$ and then multiplied by $25$. Doing both is the same as multiplying $60$ by $\frac{25}{100}$. So in a phrase like "$25\%$ of $60$," where $60$ is the stated whole, the word of tells you to multiply:

$25\%$ of $60$ is $\frac{25}{100} \times 60 = 0.25 \times 60 = 15$.

The part comes out in the same unit as the whole, which is liters here.

You can check a result with easy percents of the same whole. $10\%$ of an amount is the amount divided by $10$, because $\frac{10}{100} = \frac{1}{10}$. Here $10\%$ of $60$ is $6$, and $25\%$ is two and a half times as much: $2.5 \times 6 = 15$.

Finding what percent one number is of another. Now you know the part and the whole, and the percent is missing. The question "What percent of $60$ is $15$?" asks: if the whole, $60$, counts as $100$ shares, how many shares is the part, $15$?

Start from the percent equation, $a = \frac{p}{100} \times w$. Divide both sides by $w$. The two sides stay equal because you divide both by the same number, and you may divide by $w$ because the stated whole is positive, so it is not zero. That gives $\frac{a}{w} = \frac{p}{100}$. Now multiply both sides by $100$, which also keeps them equal, to get $p$ alone:

In words: divide the part by the whole, then multiply by $100$. For the tank, $\frac{15}{60} = 0.25$ and $0.25 \times 100 = 25$, so $15$ is $25\%$ of $60$.

Check two things before you divide. The number on the bottom must be the stated whole: the amount after the word of in a question like "What percent of $60$ is $15$?" And the part and the whole must be in the same unit, so that the units cancel and leave a plain number.

Solving percent equations. Put numbers in place of two of the values in the percent equation, and the third value is an unknown. To solve the equation means to find the number for the unknown that makes the equation true.

You have solved two kinds already: part missing and percent missing. The third kind has the whole missing, as in "$15$ is $25\%$ of what number?" Think in shares first. If $25$ of the $100$ shares are $15$, then one share is $15 \div 25 = 0.6$, and all $100$ shares are $100 \times 0.6 = 60$.

An equation gets you there in fewer steps. A sentence shaped like "$a$ is $p\%$ of $w$," with $w$ as the stated whole and $a$ in the same unit, becomes an equation word by word: is becomes $=$ and of becomes $\times$. With $w$ for the unknown whole, a positive number, you get $15 = 0.25 \times w$.

Here $w$ is multiplied by $0.25$. Divide both sides by $0.25$ to undo that. Dividing both sides by the same number keeps them equal, as long as that number is not zero. So $w = 15 \div 0.25 = 60$. Check in the sentence: $25\%$ of $60$ is $0.25 \times 60 = 15$.

Takeaway. One equation, $a = \frac{p}{100} \times w$, links the part $a$, the percent $p\%$ and the whole $w$.

• To find the part, multiply the whole by the percent written as a fraction or a decimal.
• To find the percent, divide the part by the whole, then multiply by $100$.
• To find the whole, divide the part by the percent written as a decimal.

These rules work under the same conditions: the whole is the stated, positive amount that the percent is taken of, the part and the whole are in the same unit, and the percent is not zero when you divide by it. Check every answer in the original sentence, and say whether it is exact or rounded.

Next, you will use the percent equation to describe how much an amount goes up or down. That is called percent of change.