Finding Percents
Finding a percent based on a portion of a number means figuring out what percent a specific portion represents. When thinking about percents, remember that percent means "per 100" (think, per (each) and cent (100 as in century)). Remember that portions do not have to be written as percents. They can also be written as decimals or fractions (which are fairly easily converted to percents). So 15% can also be written as .15 or $\dfrac{15}{100}$.
Students often do this with test scores. Say you took a test with 20 questions. You got 17 questions correct. You want to know what grade you got, so you find "What percent of the questions did I get correct?"
There are several ways to find the 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).
Our example: What percent is 17 out of 20?
We like to use proportions that follow this format: $\dfrac{part}{whole}=\dfrac{\text{%}}{100}$
The "part" is the portion you care about (in this case, the number of questions you got right). "Whole" is the total number of questions. "%" is what you are trying to figure out. "100" always represents "whole" in percents.
$$\eqalign{\text{part}&=17\\\text{whole}&=20\\\text{%}&=x\\\text{100}&=100}$$
Some people remember this proportion set us as: $\dfrac{is}{of}=\dfrac{\text{%}}{100}$
If you rephrase our question to: What % of 20 is 17?, then 17 is your "is" number (numerator, on top); 20 is your "of" number (the denominator or bottom number). Use a variable to fill in for the percent you are trying to find.
$$\eqalign{\text{is}&=17\\\text{of}&=20\\\text{%}&=x\\\text{100}&=100}$$
With either set up, our proportion for this problem would be: $\dfrac{17}{20}=\dfrac{x}{100}$
We put an $x$ in for the "%" because that's what we're trying to solve for. Now, just cross multiply to solve the proportion for $x$.
$$\eqalign{\dfrac{17}{20}&=\dfrac{x}{100} && \text{Set up proportion}\\ 17\times100 &=20\times x &&\text{Cross multiply}\\20x&=1700 &&\text{Solve for variable}\\x&=85 &&\text{You got an 85%}}$$
The process of thinking, "What part do I care about?" does not have to refer to something you actually care about (like how many questions you got right), but how many you're thinking about (so that you can solve the math problem).
Example: What portion of the following shape is shaded?
Again, we start with a proportion:
The "part" is the portion you care about (in this case, the number of shaded rectangles). "Whole" is the total number of rectangles. "%" is what you are trying to figure out. "100" always represents "whole" in percents.
$$\eqalign{\text{part}&=1\\\text{whole}&=6\\\text{%}&=x\\\text{100}&=100}$$
$$\eqalign{\dfrac{1}{6}&=\dfrac{x}{100} && \text{Set up proportion}\\ 1\times100 &=6\times x &&\text{Cross multiply}\\6x&=100 &&\text{Solve for variable}\\x&=16.67 &&\text{You got an 16.67%}}$$
Overall, for us, the easiest way to find a percent, is to create a fraction and then use a proportion to turn that fraction into a percent.
Many people prefer a simpler method of finding percent. They follow these steps:
- Write the number as a fraction: $\dfrac{\text{number you care about}}{\text{total}}$
- Divide the top by the bottom.
- Turn the decimal answer into a percent.
This method is simple!
$$\eqalign{\dfrac{17}{20}&=.85 && \text{Write fraction}\\ 17\div 20&=.85 &&\text{Divide}\\.85 \times 100 &= 85 &&\text{Multiply by 100 for the percent answer}}$$
However, as you'll see in practice problems 7-12 below, sometimes turning a decimal into a percent confuses students. We prefer the EdBoost method because it eliminates decimal point confusion, and and it provides one simple method for answering a range of percent questions. Try the mixed percent practice page (download link below) to see how working lots of percent problems at the same time can be confusing.
Practice Problems:
Finding Percents (based on portions)
- What percent of 20 is 5?
- What percent of 80 is 9?
- What percent of 35 is 7?
- What percent of 68 is 17?
- What percent of 250 is 30?
- What percent of 500 is 60?
- What percent of 70 is 80.5?
- What percent of 50 is .5?
- What percent of 30 is 75?
- What percent of 80 is 8?
- What percent of 80 is 80?
- What percent of 80 is 800?
Answer Key:
Test Prep Practice
Pre Algebra: Finding Percents
1.
In the figure above, what percent of the rectangle is shaded?
(A) 42.9%
(B) 33.3%
(C) 66.7%
(D) 6.6%
(E) 70%
The figure is drawn to scale.
2. In the figure above, about what percent of the circle is not shaded?
(A) 80%
(B) 75%
(C) 40%
(D) 30%
(E) 20%
3. In the figure above, what percent of the triangles are shaded?
(A) 40%
(B) 66.7%
(C) 33.3%
(D) 62.5%
(E) 37.5%
4. In the figure above, what percent of the rectangle is not shaded?
Answer Key: