Finding a Whole Number (from a percent)
Sometimes you are given a percent and a number, and asked to find the whole number that that percent is based on.
There are several ways to find a whole number based on a percent. EdBoost's preferred way to to use a proportion (not necessarily because it's the easiest way to do this operation, but because there are several related operations, and using a proportion makes it clear how to solve all kinds of percent problems).
Example: 20 is 80% of what number?
We like to use proportions that follow this format: $\dfrac{part}{whole}=\dfrac{\text{%}}{100}$
The "part" is the percentage portion of the number you are given. "Whole" is the whole number you are trying to find. "%" is the percent you are given. "100" always represents "whole" in percents.
$$\eqalign{\text{part}&=20\\\text{whole}&=x\\\text{%}&=80\\\text{100}&=100}$$
Some people remember this proportion set us as: $\dfrac{is}{of}=\dfrac{\text{%}}{100}$
If you rephrase our question to: 20 is 80% of what number?, then your "is" number is 20, so 20 goes in the numerator (top) spot, your percent is 80, and your "of" number is unknown, so put an $x$ in the denominator.
$$\eqalign{\text{is}&=20\\\text{of}&=x\\\text{%}&=80\\\text{100}&=100}$$
With either set up, our proportion for this problem would be: $\dfrac{20}{x}=\dfrac{80}{100}$
We put an $x$ in for the "whole" because that's what we're trying to solve for. Now, just cross multiply to solve the proportion for $x$.
$$\eqalign{\dfrac{20}{x}&=\dfrac{80}{100} && \text{Set up proportion}\\ 20\times100&=x\times80 &&\text{Cross multiply}\\2000&=80x &&\text{Solve for variable}\\x&=25}$$
This is the type of percent problem that most people aren't sure, off the tops of their heads, how to set up. But, you can:
- Turn the percent into a decimal.
- Divide the whole number by the decimal.
$$\eqalign{80\text{%}&=.8 && \text{Turn percent into decimal}\\ 20\div .8&=25 &&\text{Divide to find the answer}}$$
This math is easy to execute. But the set up is not always easy to remember. We like to use proportions because they are a solid way of solving all "find the percent" type problems (see Percent Mixed Review downloads below).
Practice Problems:
Percents: Finding the Whole
1. 10% of what number is 8?
2. 5 is 4% of what number?
3. 50% of what number is 46?
4. 34% of what number is 51?
5. 14 is 20% of what number?
6. 5% of what number is 3.5?
7. 1 is 25% of what number?
8. 15 is 60% of what number?
9. 50% of what number is 80?
10. 150% of what number is 30?
11. 68% of what number is 130?
12. 43% of what number is 130?
Answer Key: