048. Comparing Proportions

Learning Intentions

• To understand that finding one number as a proportion of a total generally involves division
• express a proportion as a fraction, decimal or percentage
• use fractions, decimals and percentages to compare proportions.

Pre-requisite Summary

• Understand that a proportion compares a part to a whole
• Know that a fraction can represent a part of a total
• Be able to divide one whole number by another
• Be able to convert between fractions, decimals and percentages
• Understand that equivalent fractions, decimals and percentages can represent the same proportion
• Be able to compare numbers written in the same form
• Know that percentages are proportions out of $100$

Worked Examples

Worked Example 1

A class has $12$ students wearing hats out of $30$ students.
a) Write the proportion wearing hats as a fraction.
b) Write the proportion as a decimal.
c) Write the proportion as a percentage.

Worked Example 2

A basket contains $8$ red apples out of $20$ apples.
a) Find the proportion of red apples by division.
b) Express the proportion as a fraction, decimal and percentage.

Worked Example 3

A test score is $18$ correct out of $24$ questions.
a) Write the score as a fraction.
b) Convert the fraction to a decimal.
c) Convert the proportion to a percentage.

Worked Example 4

Compare the proportions:
a) ${\displaystyle \frac{3}{5}}$
b) $0.58$
c) $62\mathrm{\%}$
State which proportion is greatest.

Worked Example 5

Compare the proportions of shaded parts:
a) ${\displaystyle \frac{7}{10}}$
b) $0.68$
c) $72\mathrm{\%}$
Write them in ascending order.

Worked Example 6

Two classes are compared.
Class A has $9$ absent out of $30$.
Class B has $7$ absent out of $20$.
a) Find each absence proportion.
b) Express each as a fraction, decimal or percentage.
c) Decide which class has the greater absence proportion.

Problems

Problem 1

A class has $15$ students bringing lunch from home out of $25$ students.
a) Write the proportion as a fraction.
b) Write the proportion as a decimal.
c) Write the proportion as a percentage.

Problem 2

A jar contains $6$ green counters out of $15$ counters.
a) Find the proportion of green counters by division.
b) Express the proportion as a fraction, decimal and percentage.

Problem 3

A quiz score is $16$ correct out of $20$ questions.
a) Write the score as a fraction.
b) Convert the fraction to a decimal.
c) Convert the proportion to a percentage.

Problem 4

Compare the proportions:
a) ${\displaystyle \frac{2}{3}}$
b) $0.61$
c) $65\mathrm{\%}$
State which proportion is greatest.

Problem 5

Compare the proportions:
a) ${\displaystyle \frac{3}{4}}$
b) $0.72$
c) $78\mathrm{\%}$
Write them in ascending order.

Problem 6

Two teams are compared.
Team A wins $8$ games out of $20$.
Team B wins $12$ games out of $25$.
a) Find each winning proportion.
b) Express each as a fraction, decimal or percentage.
c) Decide which team has the greater winning proportion.

Exercises

Understanding and Fluency

1. Write each proportion as a fraction, decimal and percentage:
   a) $4$ out of $10$
   b) $7$ out of $20$
   c) $18$ out of $25$

2. Write each proportion as a fraction, decimal and percentage:
   a) $9$ out of $12$
   b) $5$ out of $8$
   c) $14$ out of $20$

3. Express each as a decimal and percentage:
   a) ${\displaystyle \frac{1}{2}}$
   b) ${\displaystyle \frac{3}{4}}$
   c) ${\displaystyle \frac{7}{10}}$

4. Express each as a fraction and percentage:
   a) $0.6$
   b) $0.35$
   c) $0.82$

5. Express each as a fraction and decimal:
   a) $40\mathrm{\%}$
   b) $75\mathrm{\%}$
   c) $12\mathrm{\%}$

6. Compare the proportions and state the larger:
   a) ${\displaystyle \frac{2}{5}}$ and $45\mathrm{\%}$
   b) $0.7$ and ${\displaystyle \frac{3}{4}}$
   c) $62\mathrm{\%}$ and $0.58$

7. Write the proportions in ascending order:
   a) ${\displaystyle \frac{1}{2}},55\mathrm{\%},0.48$
   b) $0.9,{\displaystyle \frac{4}{5}},85\mathrm{\%}$
   c) $72\mathrm{\%},{\displaystyle \frac{3}{4}},0.71$

8. Write the proportions in descending order:
   a) $0.25,20\mathrm{\%},{\displaystyle \frac{1}{5}}$
   b) ${\displaystyle \frac{7}{8}},0.82,84\mathrm{\%}$
   c) $66\mathrm{\%},{\displaystyle \frac{2}{3}},0.61$

Reasoning

9. Explain why finding a proportion usually involves division.

10. A student says that $8$ out of $20$ is ${\displaystyle \frac{20}{8}}$. Explain the mistake.

11. Explain why converting proportions into the same form helps when comparing them.

12. A student says $0.4$, ${\displaystyle \frac{4}{10}}$ and $40\mathrm{\%}$ are different amounts because they look different. Explain why this is incorrect.

Problem-solving

13. A school has $18$ left-handed students out of $60$ students. Express this proportion as a fraction, decimal and percentage.

14. A basketball player scores $21$ shots out of $30$ attempts. What proportion of shots were scored? Give your answer as a fraction, decimal and percentage.

15. In one bag, $12$ of $40$ marbles are blue. In another bag, $9$ of $25$ marbles are blue. Which bag has the greater proportion of blue marbles?

16. A shop sold $45$ hats out of $60$ items sold in one hour. Express the proportion of hats sold as a fraction, decimal and percentage.

17. Class A has $14$ students wearing sport shoes out of $28$. Class B has $18$ students wearing sport shoes out of $30$. Compare the proportions.

18. A survey shows that $32$ out of $50$ people prefer apples, while $21$ out of $30$ people prefer oranges in a different group. Which preference proportion is greater?

Potential Misunderstandings

• Students may reverse the part and the total when writing a proportion
• Students may think a proportion is found by subtraction instead of division
• Students may confuse the total with the remainder rather than the whole group
• Students may not simplify a fraction correctly when expressing a proportion
• Students may convert between fractions, decimals and percentages incorrectly
• Students may compare proportions in different forms without converting them to a common form
• Students may think the larger numerator always means the larger proportion without considering the total
• Students may not recognise that a fraction, decimal and percentage can represent the same proportion