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

Worked Examples

Worked Example 1

A class has students wearing hats out of 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 red apples out of apples.

a) Solve the proportion of red apples by division.

b) Express the proportion as a fraction, decimal and percentage.

Worked Example 3

A test score is correct out of 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)

b)

c)

State which proportion is greatest.

Worked Example 5

Compare the proportions of shaded parts:

a)

b)

c)

Write them in ascending order.

Worked Example 6

Two classes are compared.

Class A has absent out of .

Class B has absent out of .

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 students bringing lunch from home out of 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 green counters out of 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 correct out of 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)

b)

c)

State which proportion is greatest.

Problem 5

Compare the proportions:

a)

b)

c)

Write them in ascending order.

Problem 6

Two teams are compared.

Team A wins games out of .

Team B wins games out of .

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

Exercise 1.

Write each proportion as a fraction, decimal and percentage:

a) out of

b) out of

c) out of

Exercise 2.

Write each proportion as a fraction, decimal and percentage:

a) out of

b) out of

c) out of

Exercise 3.

Express each as a decimal and percentage:

a)

b)

c)

Exercise 4.

Express each as a fraction and percentage:

a)

b)

c)

Exercise 5.

Express each as a fraction and decimal:

a)

b)

c)

Exercise 6.

Compare the proportions and state the larger:

a) and

b) and

c) and

Exercise 7.

Write the proportions in ascending order:

a)

b)

c)

Exercise 8.

Write the proportions in descending order:

a)

b)

c)

Reasoning

Exercise 9.

Explain why finding a proportion usually involves division.

Exercise 10.

A student says that out of is . Explain the mistake.

Exercise 11.

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

Exercise 12.

A student says , and are different amounts because they look different. Explain why this is incorrect.

Problem-solving

Exercise 13.

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

Exercise 14.

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

Exercise 15.

In one bag, of marbles are blue. In another bag, of marbles are blue. Which bag has the greater proportion of blue marbles?

Exercise 16.

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

Exercise 17.

Class A has students wearing sport shoes out of . Class B has students wearing sport shoes out of . Compare the proportions.

Exercise 18.

A survey shows that out of people prefer apples, while out of 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