029. Fractions and Percentages

Learning Intentions

• To understand that a percentage can be thought of as a fraction with a denominator of $100$
• convert a percentage to a fraction in simplest form
• convert a fraction to a percentage

Pre-requisite Summary

• Understand that a fraction represents part of a whole
• Know the meaning of numerator and denominator
• Be able to write equivalent fractions
• Be able to simplify fractions using common factors
• Understand that “percent” means “out of $100$”
• Be able to multiply or divide by powers of ten
• Recognise common benchmark fractions such as ${\displaystyle \frac{1}{2}},{\displaystyle \frac{1}{4}},{\displaystyle \frac{3}{4}}$

Worked Examples

Worked Example 1

a) Explain why $35\mathrm{\%}$ can be written as ${\displaystyle \frac{35}{100}}$.
b) Explain what the denominator $100$ represents.
c) Write $8\mathrm{\%}$ as a fraction with denominator $100$.

Worked Example 2

Convert each percentage to a fraction in simplest form:
a) $20\mathrm{\%}$
b) $45\mathrm{\%}$
c) $75\mathrm{\%}$

Worked Example 3

Convert each percentage to a fraction in simplest form:
a) $12\mathrm{\%}$
b) $60\mathrm{\%}$
c) $90\mathrm{\%}$

Worked Example 4

Convert each fraction to a percentage:
a) ${\displaystyle \frac{1}{2}}$
b) ${\displaystyle \frac{1}{4}}$
c) ${\displaystyle \frac{3}{5}}$

Worked Example 5

Convert each fraction to a percentage:
a) ${\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{2}{25}}$
c) ${\displaystyle \frac{7}{10}}$

Worked Example 6

a) Decide whether ${\displaystyle \frac{18}{100}}$ and $18\mathrm{\%}$ represent the same quantity.
b) Convert ${\displaystyle \frac{9}{20}}$ to a percentage.
c) Convert $40\mathrm{\%}$ to a fraction in simplest form.

Problems

Problem 1

a) Explain why $42\mathrm{\%}$ can be written as ${\displaystyle \frac{42}{100}}$.
b) Explain what the denominator $100$ represents.
c) Write $6\mathrm{\%}$ as a fraction with denominator $100$.

Problem 2

Convert each percentage to a fraction in simplest form:
a) $25\mathrm{\%}$
b) $40\mathrm{\%}$
c) $70\mathrm{\%}$

Problem 3

Convert each percentage to a fraction in simplest form:
a) $15\mathrm{\%}$
b) $50\mathrm{\%}$
c) $80\mathrm{\%}$

Problem 4

Convert each fraction to a percentage:
a) ${\displaystyle \frac{1}{5}}$
b) ${\displaystyle \frac{1}{2}}$
c) ${\displaystyle \frac{3}{10}}$

Problem 5

Convert each fraction to a percentage:
a) ${\displaystyle \frac{3}{4}}$
b) ${\displaystyle \frac{1}{20}}$
c) ${\displaystyle \frac{9}{25}}$

Problem 6

a) Decide whether ${\displaystyle \frac{32}{100}}$ and $32\mathrm{\%}$ represent the same quantity.
b) Convert ${\displaystyle \frac{7}{20}}$ to a percentage.
c) Convert $65\mathrm{\%}$ to a fraction in simplest form.

Exercises

Understanding and Fluency

1. Write each percentage as a fraction with denominator $100$:
   a) $7\mathrm{\%}$
   b) $26\mathrm{\%}$
   c) $94\mathrm{\%}$

2. Convert each percentage to a fraction in simplest form:
   a) $10\mathrm{\%}$
   b) $30\mathrm{\%}$
   c) $50\mathrm{\%}$

3. Convert each percentage to a fraction in simplest form:
   a) $24\mathrm{\%}$
   b) $36\mathrm{\%}$
   c) $84\mathrm{\%}$

4. Convert each fraction to a percentage:
   a) ${\displaystyle \frac{1}{10}}$
   b) ${\displaystyle \frac{1}{2}}$
   c) ${\displaystyle \frac{3}{4}}$

5. Convert each fraction to a percentage:
   a) ${\displaystyle \frac{2}{5}}$
   b) ${\displaystyle \frac{3}{20}}$
   c) ${\displaystyle \frac{9}{10}}$

6. Convert each fraction to a percentage:
   a) ${\displaystyle \frac{7}{25}}$
   b) ${\displaystyle \frac{11}{20}}$
   c) ${\displaystyle \frac{17}{100}}$

7. Match each fraction and percentage:
   a) ${\displaystyle \frac{1}{4}}$
   b) ${\displaystyle \frac{3}{5}}$
   c) ${\displaystyle \frac{9}{20}}$

8. Convert in either direction as needed:
   a) $55\mathrm{\%}$
   b) ${\displaystyle \frac{4}{25}}$
   c) $125\mathrm{\%}$

Reasoning

9. Explain why $65\mathrm{\%}={\displaystyle \frac{65}{100}}$.

10. A student says $40\mathrm{\%}={\displaystyle \frac{40}{10}}$. Explain the mistake.

11. Explain why converting ${\displaystyle \frac{3}{4}}$ to a percentage involves making an equivalent fraction with denominator $100$.

12. A student says ${\displaystyle \frac{2}{5}}=25\mathrm{\%}$. Explain why this is incorrect.

Problem-solving

13. A test score is $18$ out of $20$. Write this fraction as a percentage.

14. A shirt is labelled as being made from $60\mathrm{\%}$ cotton. Write this percentage as a fraction in simplest form.

15. A class completed ${\displaystyle \frac{3}{5}}$ of a project. What percentage of the project was completed?

16. A water tank is ${\displaystyle \frac{7}{10}}$ full. Write this as a percentage.

17. A student answered $9$ out of $12$ questions correctly. Write the fraction correct as a percentage.

18. A battery is at $25\mathrm{\%}$ charge. Write this as a fraction in simplest form.

Potential Misunderstandings

• Students may think percent means “out of $10$” instead of “out of $100$”
• Students may write a percentage as a fraction with the wrong denominator
• Students may forget to simplify the fraction after writing it over $100$
• Students may confuse the numerator and denominator when converting
• Students may not recognise that converting a fraction to a percentage often requires an equivalent fraction with denominator $100$
• Students may incorrectly scale only the numerator or only the denominator
• Students may assume that the percent sign can be ignored without changing the meaning
• Students may think all fractions convert to percentages without using equivalent fractions or multiplication

Next: 030. Finding Percentages of Quantities