3.06 Proportional reasoning with percents
Recall that a percent is a ratio where the denominator is $100$100. Because of this definition, we can use proportional reasoning strategies to solve problems with percents.
Proportions can be used to represent percent problems as follows:
| $\frac{\text{percent}}{100}$percent100 | $=$= | $\frac{\text{part}}{\text{whole}}$partwhole |
Worked example
Question 1
Evaluate: Use a proportion to answer the question, "What percent of 20 is 3?"
Think: We can translate the statement to a proportion. Then use proportional reasoning to solve for the unknown.
The percent is the unknown. So we can use the variable $x$x to represent it.
The number $3$3 is the part and $20$20 is the whole.
Do:
| $\frac{\text{percent}}{100}$percent100 | $=$= | $\frac{\text{part}}{whole}$partwhole |
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| $\frac{x}{100}$x100 | $=$= | $\frac{3}{20}$320 |
$x$x is the unknown percent. $20$20 is the whole. |
| $\frac{x}{100}$x100 | $=$= | $\frac{3\times5}{20\times5}$3×520×5 |
Multiplying the fraction by $\frac{5}{5}$55 gives us a common denominator of $100$100. |
| $\frac{x}{100}$x100 | $=$= | $\frac{15}{100}$15100 |
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| $x$x | $=$= | $15$15 |
If the denominators in a proportion are the same, the numerators must also be the same. |
So the number $3$3 is $15%$15% of $20$20.
Reflect: Is there another method that we might use to check our solution?
Proportional reasoning with benchmark percents
Suppose we want to check our solution to the first worked example using a different method. Let's see how we can apply proportional reasoning to percents in a different way.
Worked example
QUESTION 2
Evaluate: Find $15%$15% of $20$20.
Think: It might be easiest to find $10%$10% of $20$20.
We can then use half of that amount to find $5%$5% of $20$20. If we add the two amounts, that will give us $15%$15% of $20$20.
Do: First, find $10%$10% of $20$20.
| $10%$10% of $20$20 | $=$= | $0.10\times20$0.10×20 |
Since $10%=0.10$10%=0.10 |
| $=$= | $2$2 |
Evaluate |
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| $5%$5% of $20$20 | $=$= | $\frac{1}{2}\times2$12×2 |
Since $5%$5% is half of $10%$10% |
| $=$= | $1$1 | ||
| $15%$15% | $=$= | $10%+5%$10%+5% | |
| $=$= | $2+1$2+1 | ||
| $=$= | $3$3 |
So $15%$15% of $20$20 is $3$3.
Reflect: What other percents can we calculate using the benchmark of $10%$10%?
Practice questions
Question 3
Translate the following percentage problem to a proportion. Do not solve or simplify the proportion.
'What percent of $92$92 is $23$23?'
Let the unknown number be $x$x.
Question 4
Translate the following percentage problem to a proportion. Do not solve or simplify the proportion.
'$60%$60% of what number is $144$144?'
Let the unknown number be $x$x.
Question 5
We want to find $45%$45% of $5$5 hours.
How many minutes are there in $5$5 hours?
What is $10%$10% of $300$300 minutes?
What is $5%$5% of $300$300 minutes?
Hence find $45%$45% of $300$300 minutes.