030. Finding Percentages of Quantities

Learning Intentions

• To understand that finding percentages of a number can be found by multiplying by a fraction
• find a percentage of a number
• apply percentages to worded problems

Pre-requisite Summary

• Understand that a percentage means “out of $100$”
• Be able to convert a percentage to a fraction such as $25\mathrm{\%}={\displaystyle \frac{25}{100}}={\displaystyle \frac{1}{4}}$
• Be able to simplify fractions to make calculations easier
• Be able to multiply a whole number by a fraction
• Understand that “of” in percentage questions means multiply
• Be able to interpret simple worded problems involving parts of a whole

Worked Examples

Worked Example 1

a) Write $20\mathrm{\%}$ as a fraction.
b) Find $20\mathrm{\%}$ of $45$.
c) Explain why multiplying by a fraction works.

Worked Example 2

a) Write $35\mathrm{\%}$ as a fraction.
b) Find $35\mathrm{\%}$ of $80$.

Worked Example 3

a) Find $50\mathrm{\%}$ of $126$.
b) Find $25\mathrm{\%}$ of $84$.
c) Find $10\mathrm{\%}$ of $230$.

Worked Example 4

a) A shirt costs $60. Find $15\mathrm{\%}$ of $60.
b) Interpret what this percentage represents in the context of the problem.

Worked Example 5

a) A class has $28$ students. $75\mathrm{\%}$ are present. How many students are present?
b) State the multiplication needed.

Worked Example 6

a) A tank holds $240$ L of water and is $35\mathrm{\%}$ full. How much water is in the tank?
b) Explain the steps used.

Problems

Problem 1

a) Write $30\mathrm{\%}$ as a fraction.
b) Find $30\mathrm{\%}$ of $50$.
c) Explain why multiplying by a fraction works.

Problem 2

a) Write $40\mathrm{\%}$ as a fraction.
b) Find $40\mathrm{\%}$ of $70$.

Problem 3

a) Find $50\mathrm{\%}$ of $148$.
b) Find $25\mathrm{\%}$ of $96$.
c) Find $10\mathrm{\%}$ of $340$.

Problem 4

a) A book costs $80. Find $15\mathrm{\%}$ of $80.
b) Interpret what this percentage represents in the context of the problem.

Problem 5

a) A class has $32$ students. $75\mathrm{\%}$ are present. How many students are present?
b) State the multiplication needed.

Problem 6

a) A tank holds $300$ L of water and is $35\mathrm{\%}$ full. How much water is in the tank?
b) Explain the steps used.

Exercises

Understanding and Fluency

1. Convert each percentage to a fraction in simplest form, then find the percentage of the number:
   a) $20\mathrm{\%}$ of $35$
   b) $25\mathrm{\%}$ of $48$
   c) $10\mathrm{\%}$ of $260$

2. Convert each percentage to a fraction in simplest form, then find the percentage of the number:
   a) $50\mathrm{\%}$ of $92$
   b) $75\mathrm{\%}$ of $44$
   c) $5\mathrm{\%}$ of $180$

3. Find each percentage of the number:
   a) $15\mathrm{\%}$ of $60$
   b) $40\mathrm{\%}$ of $90$
   c) $35\mathrm{\%}$ of $120$

4. Find each percentage of the number:
   a) $60\mathrm{\%}$ of $150$
   b) $12\mathrm{\%}$ of $200$
   c) $30\mathrm{\%}$ of $70$

5. Find each percentage of the number:
   a) $80\mathrm{\%}$ of $45$
   b) $45\mathrm{\%}$ of $200$
   c) $65\mathrm{\%}$ of $40$

6. Find each percentage of the number:
   a) $5\mathrm{\%}$ of $320$
   b) $90\mathrm{\%}$ of $70$
   c) $55\mathrm{\%}$ of $60$

Reasoning

7. Explain why finding $25\mathrm{\%}$ of a number is the same as finding ${\displaystyle \frac{1}{4}}$ of the number.

8. A student says that $30\mathrm{\%}$ of $50$ is $30\times 50=1500$. Explain the mistake.

9. Explain why “of” means multiply in a percentage question.

10. A student finds $20\mathrm{\%}$ of $80$ by doing $80÷20=4$. Explain why this is incorrect.

Problem-solving

11. A jacket costs $120. Find $25\mathrm{\%}$ of the cost.

12. A school has $240$ students. $60\mathrm{\%}$ of them walk to school. How many students walk to school?

13. A container holds $500$ mL of juice. $40\mathrm{\%}$ of the juice is poured out. How much juice is poured out?

14. A test has $80$ marks. A student earns $75\mathrm{\%}$ of the marks. How many marks is that?

15. A farm has $360$ trees. $15\mathrm{\%}$ are orange trees. How many orange trees are there?

16. A phone battery is at $35\mathrm{\%}$ of its full charge of $4000$ mAh. How much charge remains?

17. A bag contains $200$ marbles. $12\mathrm{\%}$ are red. How many red marbles are there?

18. A shop discounts an item priced at $90 by $10\mathrm{\%}$. How much is the discount?

Potential Misunderstandings

• Students may think a percentage can be applied by adding the percent number instead of multiplying by a fraction
• Students may forget that a percentage means “out of $100$”
• Students may convert a percentage to a fraction incorrectly
• Students may not simplify the fraction before multiplying, even when simplification would help
• Students may confuse finding a percentage of a quantity with increasing a quantity by that percentage
• Students may treat “of” as addition rather than multiplication
• Students may calculate the percentage amount correctly but misinterpret what it means in a worded problem
• Students may place the number and the percentage in the wrong order of operations

Next: 031. Introducing Ratios and Simplifying