115r. Review Percentages and Percentage Calculations

Learning Intentions

• To understand that a percentage is a number out of $100$
• convert percentages to fractions and decimals
• convert fractions and decimals to percentages
• express one quantity as a percentage of another quantity
• find a certain percentage of a quantity

Pre-requisite Summary

• Know that a fraction represents part of a whole
• Know that a decimal represents parts of a whole using place value
• Understand that $\mathrm{\%}$ means “per hundred” or “out of $100$”
• Be able to simplify fractions
• Be able to convert between some simple fractions and decimals
• Be able to multiply and divide whole numbers and decimals
• Understand that “of” in percentage problems usually means multiply
• Be able to compare two quantities using division

Worked Examples

Worked Example 1

Write each percentage as a fraction and a decimal:
a) $35\mathrm{\%}$
b) $8\mathrm{\%}$
c) $125\mathrm{\%}$

Worked Example 2

Write each fraction or decimal as a percentage:
a) ${\displaystyle \frac{3}{4}}$
b) $0.6$
c) ${\displaystyle \frac{7}{20}}$

Worked Example 3

Express one quantity as a percentage of another quantity:
a) $12$ as a percentage of $20$
b) $18$ as a percentage of $24$

Worked Example 4

Find each percentage of a quantity:
a) $25\mathrm{\%}$ of $60$
b) $10\mathrm{\%}$ of $350$
c) $50\mathrm{\%}$ of $84$

Worked Example 5

Find each percentage of a quantity:
a) $15\mathrm{\%}$ of $80$
b) $40\mathrm{\%}$ of $65$
c) $5\mathrm{\%}$ of $240$

Worked Example 6

Express one quantity as a percentage of another quantity:
a) $45$ out of $60$
b) $9$ out of $15$
c) $27$ out of $90$

Problems

Problem 1

Write each percentage as a fraction and a decimal:
a) $45\mathrm{\%}$
b) $12\mathrm{\%}$
c) $150\mathrm{\%}$

Problem 2

Write each fraction or decimal as a percentage:
a) ${\displaystyle \frac{1}{2}}$
b) $0.35$
c) ${\displaystyle \frac{9}{25}}$

Problem 3

Express one quantity as a percentage of another quantity:
a) $14$ as a percentage of $28$
b) $21$ as a percentage of $30$

Problem 4

Find each percentage of a quantity:
a) $20\mathrm{\%}$ of $90$
b) $25\mathrm{\%}$ of $48$
c) $10\mathrm{\%}$ of $630$

Problem 5

Find each percentage of a quantity:
a) $30\mathrm{\%}$ of $70$
b) $75\mathrm{\%}$ of $40$
c) $5\mathrm{\%}$ of $180$

Problem 6

Express one quantity as a percentage of another quantity:
a) $32$ out of $40$
b) $7$ out of $20$
c) $54$ out of $72$

Exercises

Understanding and Fluency

1. Write each percentage as a fraction out of $100$, then simplify where possible, and write it as a decimal:
   a) $20\mathrm{\%}$
   b) $65\mathrm{\%}$
   c) $4\mathrm{\%}$
   d) $140\mathrm{\%}$

2. Convert each fraction or decimal to a percentage:
   a) ${\displaystyle \frac{1}{4}}$
   b) ${\displaystyle \frac{3}{5}}$
   c) $0.08$
   d) $0.92$

3. Convert each fraction to a percentage:
   a) ${\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{3}{10}}$
   c) ${\displaystyle \frac{7}{25}}$
   d) ${\displaystyle \frac{9}{20}}$

4. Convert each decimal to a percentage:
   a) $0.4$
   b) $1.2$
   c) $0.075$
   d) $0.125$

5. Express one quantity as a percentage of another quantity:
   a) $6$ out of $12$
   b) $15$ out of $25$
   c) $18$ out of $40$
   d) $27$ out of $30$

6. Express one quantity as a percentage of another quantity:
   a) $8$ as a percentage of $20$
   b) $24$ as a percentage of $60$
   c) $35$ as a percentage of $50$
   d) $16$ as a percentage of $64$

7. Find each percentage of a quantity:
   a) $10\mathrm{\%}$ of $90$
   b) $50\mathrm{\%}$ of $36$
   c) $25\mathrm{\%}$ of $120$
   d) $5\mathrm{\%}$ of $200$

8. Find each percentage of a quantity:
   a) $15\mathrm{\%}$ of $60$
   b) $30\mathrm{\%}$ of $90$
   c) $40\mathrm{\%}$ of $75$
   d) $75\mathrm{\%}$ of $28$

Reasoning

9. Explain why $35\mathrm{\%}={\displaystyle \frac{35}{100}}=0.35$.

10. A student says that $8\mathrm{\%}=8$ because the number is $8$. Explain the mistake.

11. Noah says that to convert a decimal to a percentage, you divide by $100$. Is he correct? Explain.

12. Explain why finding $25\mathrm{\%}$ of a quantity is the same as finding one quarter of it.

13. A student says that $15$ is $75\mathrm{\%}$ of $20$ because $15-20=5$. Describe the error.

Problem-solving

14. A test has $40$ questions. A student gets $32$ correct. Express this score as a percentage.

15. A shirt costs $\mathrm{\$}80$. It is discounted by $25\mathrm{\%}$. Find the discount amount.

16. A water bottle is $60\mathrm{\%}$ full and holds $900$ mL when full. How much water is in the bottle?

17. In a class of $30$ students, $18$ students walk to school. What percentage of the class walks to school?

18. A packet contains $240$ g of cereal. One serving is $15\mathrm{\%}$ of the packet. How many grams is one serving?

19. A basketball player scored $27$ successful shots out of $45$ attempts. Express this as a percentage.

Potential Misunderstandings

• Students may think a percentage is just a whole number with a percent sign, rather than a number out of $100$
• Students may forget that $\mathrm{\%}$ means “per hundred”
• Students may confuse converting to a decimal with converting to a percentage
• Students may move the decimal point in the wrong direction when converting between decimals and percentages
• Students may forget to simplify a fraction after writing a percentage as a fraction
• Students may think percentages cannot be greater than $100\mathrm{\%}$
• Students may express one quantity as a percentage of another using subtraction instead of division
• Students may reverse the two quantities when finding a percentage
• Students may forget that “of” means multiply in percentage-of-a-quantity questions
• Students may calculate the percentage amount correctly but forget to include the unit