Topic B: Finding a Number when a Percent of it is Given
In problems when a certain percentage of a number is given, the missing term is the whole. You will be told the % and the part (is), and asked to find the whole (of), which is 100%.
Example A
20% of what number is 14?
- The part = 14
- The whole (“what number”) = unknown (call it N)
- The percent = 20%
- Write the proportion:
[latex]\dfrac{14}{\textit{N}} = \dfrac{20}{100}[/latex] - Solve the proportion:
- Simplify:
[latex]\dfrac{14}{\textit{N}} = \dfrac{\cancel{20}1}{\cancel{100}5}[/latex] - [latex]\dfrac{14}{\textit{N}} = \dfrac{1}{5}[/latex]
- Cross-multiply to solve:
[latex]\begin{equation}\begin{split} 14\times5 &= \textit{N}\times1 \\ 70 &= \textit{N} \end{split}\end{equation}[/latex][latex]20\%\text{ of }70 = 14[/latex]
- Simplify:
Check by finding 20% of 70. The answer should be 14.
[latex]\begin{equation}\begin{split} \dfrac{20}{100} &= \dfrac{\textit{N}}{70} \\ 140 &= 100\textit{N} \\ 14 &= \textit{N} \end{split}\end{equation}[/latex]
Example B
33⅓% of is 60.
- Part = 60
- Whole = N
- Percent = 33⅓% = ⅓
Set up proportion:
[latex]\begin{equation}\begin{split} \dfrac{60}{\textit{N}} &= \dfrac{1}{3} \\ 3\times60 &= \textit{N}\times1 \\ 180 &= \textit{N} \end{split}\end{equation}[/latex]
33⅓% of 180 is 60.
To check the answer, find 33⅓% of 180. The answer should be 60.
Exercise 1
Set up the proportion. Do not solve the question.
- 18 is 50% of what number?
- 24 is 15% of what number?
- 200% of what number is 86?
- 66⅔% of what number= 500?
Answers to Exercise 1
- [latex]\dfrac{18}{\textit{P}}=\dfrac{50}{100}[/latex]
- [latex]\dfrac{24}{\textit{N}}=\dfrac{15}{100}[/latex]
- [latex]\dfrac{86}{\textit{P}}=\dfrac{200}{100}[/latex]
- [latex]\dfrac{500}{\textit{P}}=\dfrac{66\tfrac{2}{3}}{100}[/latex] or [latex]\dfrac{2}{3}[/latex]
Exercise 2
Solve the following. Check your answers to see if you set up the proportion correctly.
- 60 is 75% of what number?
- 950 is 95% of:
- 125 = 33⅓% of what number?
- 120% of is 6.
- 270 is 100% of what number?
Answers to Exercise 2
- [latex]\dfrac{60}{\textit{X}}=\dfrac{75}{100},\text{ } 80[/latex]
- [latex]\dfrac{950}{\textit{P}}=\dfrac{95}{100},\text{ } 1,000[/latex]
- [latex]\dfrac{125}{\textit{N}}=\dfrac{33\tfrac{1}{3}}{100},\text{ } 375[/latex]
- [latex]\dfrac{120}{100}=\dfrac{6}{\textit{J}}, \text{ }5[/latex]
- [latex]\dfrac{100}{100}=\dfrac{270}{\textit{N}}, \text{ }270[/latex]
Exercise 3
Solve the questions.
- 480 is 66⅔% of .
- 40% of is 50.
- 33⅓% of what number is 99?
- 3 is 150% of what number?
- 122 is 80% of .
Answers to Exercise 3
- [latex]720[/latex]
- [latex]125[/latex]
- [latex]297[/latex]
- [latex]2[/latex]
- [latex]152.5[/latex]
Solving Problems when the Percent of a Number is Given
Read the problems carefully. More than one step may be needed. Look at the wording so you will recognize problems missing the whole and be able to tell them from problems missing the part.
Exercise 4
Solve the problems. Round money to the nearest cent.
- When the business was declared bankrupt, all the creditors (people owed money by the business) were paid 55% of the money owed them.
- If a creditor received $8,000 from the bankrupt business, how much money had he really been owed?
- How much money did the creditor lose on this business?
- In telethons and other fund-raising events, records show that about 80% of the money pledged is actually collected. The local telethon organizers need to raise $12,000. To actually raise $12,000, their goal for pledges should be what amount?
- The shoe store sold all merchandise at 25% off in a huge clearance sale. They took in $3,500 in the first day of the sale. If the same shoes had been sold at the regular price, how much money would they have taken in? (Note—this problem has 2 steps. The merchandise was 25% off, so it sold for 100% − 25% = 75% of the original price.)
- A common rate of commission earned by real estate agents is 3½%. If an agent had a gross income of $63,000 from commissions in one year, what was the value of the houses sold?
Answers to Exercise 4
-
- $14,545.45
- $6,545.45
- $15,000.00
- $4,666.67
- $1,800,000
Topic B: Self-Test
Mark /8 Aim 6/8
- Solve these questions.
(4 marks)- 2.5% of what number is 160?
- 5 is 4.5% of .
- 180 is 90% of what number?
- 28 is 35% of what number?
- Solve these problems.
(4 marks)- If a bank insists that new house buyers have a cash down payment of 12%, what house price can a couple afford if they have saved a $15,000 down payment?
- Jim has really cut down on his smoking. He now smokes 7 cigarettes a day, which he says is only 20% of what he used to smoke. How many cigarettes a day did Jim smoke before he started cutting down?
Answers to Topic B Self-Test
-
- [latex]6,400[/latex]
- [latex]111.11[/latex]
- [latex]200[/latex]
- [latex]80[/latex]
-
- $125,000
- 35 cigarettes per day