Topic A: Finding What Percent One Number is of Another
In problems where you must find what percent one number is of another, the missing term is the percent. You will be told the part (is) and the whole (of), you know the 100, and you solve for the missing percent.
Example A
4 is what percent of 5?
- 4 is the part (is)
- 5 is the whole (of)
- % is unknown (call it P)
- Write the proportion:
[latex]\dfrac{4}{5} = \dfrac{\textit{P}}{100}[/latex] - Cross-multiply to solve:
[latex]\begin{equation}\begin{split} 4\times100 &= 5\times\textit{P} \\ 400 &= 5\textit{P} \\ 400\div5 &= \textit{P} \\ 80\% &= \textit{P} \end{split}\end{equation}[/latex]
Be sure to write the percent sign – %.
In percent problems, the number after “of” usually is the whole. The number close to “is” usually is the part. You may find it helpful to think “is over of”.
Like this: [latex]\dfrac{\text{is}}{\text{of}}[/latex].
An equal to sign (=) can substitute for “is”.
12 is what percent of 15?
[latex]\dfrac{\text{is}}{\text{of}}[/latex] will help to find [latex]\dfrac{\text{part}}{\text{whole}}[/latex].
[latex]\dfrac{12\text{ (is)}}{15\text{ (of)}}=\dfrac{\textit{N}}{100}[/latex]
Example B
What percent of 85 is 60?
- 60 = the part (close to “is”)
- 85 = the whole (after “of”)
- % = the unknown
- Set up the proportion: [latex]\dfrac{\text{is}}{\text{of}}[/latex]
[latex]\dfrac{60}{85} = \dfrac{\textit{P}}{100}[/latex]
- Simplify:
[latex]\dfrac{\cancel{60}12}{\cancel{85}17} = \dfrac{\textit{P}}{100}[/latex] - [latex]\dfrac{12}{17} = \dfrac{\textit{P}}{100}[/latex]
- Cross-multiply to solve:
[latex]12\times100 = 1,200[/latex], and [latex]17\times\textit{P}=17\textit{P}[/latex][latex]\begin{equation}\begin{split} 1,200 &= 17\textit{P} \\ 1,200\div17 &= \textit{P} \\ 70.588\% &= \textit{P} \end{split}\end{equation}[/latex]
Round to 70.6%.
Exercise 1
The following examples ask you to find what percent one number is of another. The missing term is the percent. Look carefully at the wording and decide which number is the part (close to “is”) and which number is the whole thing (after “of”).Write the proportion but do not solve the problem.
- 3 is what % of 6?
- 12 is % of 5?
- What % of 27 is 9?
- What % of ½ is ¼?
- % of 50 is 25?
- % of 64 = 48
Answers to Exercise 1
- [latex]\dfrac{3}{6}=\dfrac{\textit{N}}{100}[/latex]
- [latex]\dfrac{\textit{F}}{100}=\dfrac{12}{5}[/latex]
- [latex]\dfrac{9}{27}=\dfrac{\textit{X}}{100}[/latex]
- [latex]\dfrac{\tfrac{1}{4}}{\tfrac{1}{2}}=\dfrac{\textit{P}}{100}[/latex]
- [latex]\dfrac{\textit{X}}{100}=\dfrac{25}{50}[/latex]
- [latex]\dfrac{48}{64}=\dfrac{\textit{N}}{100}[/latex]
Exercise 2
Solve each question by first setting up the proportion [latex]\dfrac{\text{is (part)}}{\text{of (whole)}}=\dfrac{\%}{100}[/latex]. Review p. 70 as necessary.
- 3 is what percent of 60?
- 3 is what percent of 4?
- 1 is what percent of 3?
- What % of 50 is 35?
- What percent of 350 is 42?
- 15 is % of 12.
- 14 is % of 700.
- What percent of 96 is 12?
- 2 is % of 125.
- 46 is % of 40.
Answers to Exercise 2
- [latex]5\%[/latex]
- [latex]75\%[/latex]
- [latex]33.\overline{3}\%[/latex]
- [latex]70\%[/latex]
- [latex]12\%[/latex]
- [latex]125\%[/latex]
- [latex]2\%[/latex]
- [latex]12.5\%[/latex]
- [latex]1.6\%[/latex]
- [latex]115\%[/latex]
Exercise 3
Solve the following by setting up the proportion.
- 16 is % of 64.
- 17 is % of 85.
- What % of 52 is 13?
- What percent of 125 is 75?
- 1 is % of 200.
- 36 = % of 12.
- % of 72 = 27.
- % of 48 = 18.
- 125 = % of 75.
- What % of 18 is 24?
Answers to Exercise 3
- [latex]25\%[/latex]
- [latex]20\%[/latex]
- [latex]25\%[/latex]
- [latex]60\%[/latex]
- [latex]0.5\%[/latex]
- [latex]300\%[/latex]
- [latex]37.5\%[/latex]
- [latex]37.5\%[/latex]
- [latex]166.\overline{6}[/latex] or [latex]166\tfrac{2}{3}\%[/latex]
- [latex]133.\overline{3}[/latex] or [latex]133\tfrac{1}{3}\%[/latex]
Finding the Percent of an Increase or Decrease
You learned in to find the amount of an increase (gain) or decrease (loss) when given the percent of the increase or decrease.
Now you are going to find the percent of the increase or decrease when you are given the amounts. This is called the rate of the increase or decrease.
Problems which ask you to find the percent of increase or decrease often involve two steps:
- Find the amount of change (either increase or decrease) by finding the difference between the two amounts given. Subtract to find the difference.
- Find the percentage of increase/decrease. Always compare the change (amount of increase or decrease) to the amount before the change (the original amount) using this proportion.
Example A
The rent went from $375 a month to $427.50 a month. What is the percent of the increase?
- Find the change (the amount of increase) by finding the difference between the amounts.[latex]$427.50 - 375 = $52.50[/latex]The amount of increase is $52.50.
- Find the % of increase.
The amount of increase is $52.50.
The original amount (the amount before the increase) is $375.What % of $375 is 52.50?
[latex]\begin{equation}\begin{split} \dfrac{\textit{X}}{100} &= \dfrac{52.50}{375} \\
375\textit{X} &= 5250 \\ \textit{X} &= \dfrac{5250}{375} \\ \textit{X} &= 14\% \end{split}\end{equation}[/latex]
The rent increase is 14%.
Example B
The hours of operation at the college were reduced from 35 hours a week to 30 hours a week. What is the percent of this cut in operations?
- Find the amount of change (a decrease) by finding the difference between the amounts.[latex]35\text{ hours }- 30\text{ hours }= 5\text{ hours}[/latex]The amount of decrease is 5 hours.
- Find the % of increase.
Decrease is 5 hours.
Original amount is 35 hours.What percent is 5 of 35?
[latex]\begin{equation}\begin{split} \dfrac{5}{35} &= \dfrac{\textit{P}}{100} \\ 500 &= 35\textit{P} \\ \dfrac{500}{35} &= \textit{P} \\ 14\tfrac{2}{7}\% &= \textit{P} \end{split}\end{equation}[/latex]
The hours of operation at the college were cut 14²⁄₇%.
Exercise 4
Solve the following problems.
- Ms. Lister’s bi-weekly unemployment cheque increased from $405 to $435. What percent increase is this?
- Joan weighed 72 kg before she went on a programme of strict exercise and careful eating. She now weighs 60 kg. What is the percent of her weight loss?
- The car dealership gives a special deal if the customer does not have a trade-in and pays cash. The dealers will only charge $10,650 for a car listed at $12,000. What is the percent savings in this deal?
- A regular toilet uses 20 litres of water per flush. By purchasing a new low flow toilet, the water use is 6 litres per flush. What is the percent savings of water per flush if the new tank is used?
Answers to Exercise 4
- 7.4% increase
- 16.6% decrease
- 11.25% savings
- 70% savings
Other Problems
Many situations compare one number to another.
- 24 out of 25 on the test
- 6 out of 10 people are overweight
- The government spends 27¢ of every federal tax dollar on the national debt
These numbers are often more easily thought about if written as a percent.
The following problems ask you to find what percent one number is of another. Often several steps are involved to calculate the part or to calculate the whole (as in question e) You may be asked to use the % after you find it. Remember the whole thing = 100%.
Exercise 5
Solve the following problems.
- The Doal family net income is $2,300 per month. Their mortgage payment is $750 each month. What percent is the mortgage payment of their monthly income?
- Jean played on the college volleyball team and missed a lot of classes when she travelled to tournaments. She missed nine of the 42 English classes last semester.
- What percent of her English classes did she miss?
- What percent of her English classes did she attend?
- Four women and six men serve on the Village Council. What percent of the council members are women?
- Gail bought a $500 G.I.C. (Guaranteed Investment Certificate) one year ago. She was delighted to receive her annual interest cheque of $52.50 today. What percent interest did Gail’s G.I.C. pay for that year?
- If a math book had 320 pages and you still had 110 pages left to do, what percent of the book had you finished? (2 steps)
Answers to Exercise 5
A. 32.6% monthly income
B.
1. 21.4% of English classes
i2. 78.6% of English classes
C. 40% of members are women.
D. 10.5% interest
E. 65.625% or 65⅝% of the book
School Grades
When looking at test results, the mark shows how you did on the test.
If you get [latex]\tfrac{7}{10}[/latex] on a test, you know you got 7 answers right, and 3 answers wrong.
Sometimes it is also helpful to see your mark as a percentage.
Example A
By solving for N, the percentage can be found:
[latex]\begin{equation}\begin{split} \dfrac{7}{10}&= \dfrac{\textit{N}}{100} \\ 7 \times 100 &= \textit{N} \times 10 \\ \dfrac{\cancel{700}70}{\cancel{10}}&= \dfrac{\cancel{10}\textit{N}}{\cancel{10}} \\ \dfrac{70}{1} &= \dfrac{\textit{N}}{1} \\ 70&=\textit{N} \end{split}\end{equation}[/latex]
So, [latex]\tfrac{7}{10} = 70\%[/latex].
Now, you can see that the test mark of [latex]\tfrac{7}{10}[/latex] equals [latex]70\%[/latex].
Example B
The test result was [latex]\tfrac{15}{43}[/latex], what was the percent on the test?
[latex]\begin{equation}\begin{split} \dfrac{15}{43} &= \dfrac{\textit{N}}{100} \\ 15\times100 &= \textit{N}\times43 \\ \dfrac{1500}{43} &= \dfrac{\cancel{43}\textit{N}}{\cancel{43}} \\ \dfrac{1500}{43} &= \dfrac{\textit{N}}{1} \\ 34.88\% &= \textit{N} \end{split}\end{equation}[/latex]
If you round, [latex]\textit{N} = 35\%[/latex].
Not such a great mark!
Example C
Find the percent of the following grade: [latex]\tfrac{89}{97}[/latex].
[latex]\begin{equation}\begin{split} \dfrac{89}{97} &= \dfrac{\textit{P}}{100} \\ 89\times100 &= \textit{P}\times97 \\ \dfrac{8900}{97} &= \dfrac{\cancel{97}\textit{P}}{\cancel{97}} \\ \dfrac{8900}{97} &= \dfrac{\textit{P}}{1} \\ 91.75\% &= \textit{P} \end{split}\end{equation}[/latex]
[latex]\textit{P} = 91.75\%[/latex] or [latex]92\%[/latex].
Exercise 6
Find the percents for the following test grades. Round your answer to the nearest percent.
- [latex]\tfrac{33}{42}[/latex]
- [latex]\tfrac{24}{40}[/latex]
- [latex]\tfrac{90}{120}[/latex]
- [latex]\tfrac{100}{110}[/latex]
- [latex]\tfrac{10}{20}[/latex]
Answers to Exercise 6
- [latex]79\%[/latex]
- [latex]60\%[/latex]
- [latex]75\%[/latex]
- [latex]91\%[/latex]
- [latex]50\%[/latex]
Topic A: Self-Test
Mark /12 Aim 10/12
- Solve to find the missing percents.
(4 marks)- 12 is % of 60.
- 15 is % of 75.
- 8.2 = % of 32.8.
- What percent of 64 is 48?
- Problems.
(4 marks)- The $140 jacket was on sale for $126. What percent is the savings?
- The rent on the apartment went from $320 a month to $400 dollars a month. What percent is this rent increase?
- Find the percents for the following grades on tests. Round your answer to the nearest whole percent.
(4 marks)- [latex]\tfrac{12}{15}[/latex]
- [latex]\tfrac{10}{19}[/latex]
- [latex]\tfrac{71}{92}[/latex]
- [latex]\tfrac{132}{140}[/latex]
Answers to Topic A Self-Test
A.
1. 20%
2. 20%
3. 25%
4. 75%
B.
1. 10% savings
2. 25% increase
C.
1. 80%
2. 53%
3. 77%
4. 94%