030. Finding Percentages of Quantities

Learning Intentions

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

Pre-requisite Summary

• Understand that a percentage means “out of ”
• Be able to convert a percentage to a fraction such as
• 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 as a fraction.

b) Find of .

c) Explain why multiplying by a fraction works.

Worked Example 2

a) Write as a fraction.

b) Find of .

Worked Example 3

a) Find of .

b) Find of .

c) Find of .

Worked Example 4

a) A shirt costs $60. Find of $60.

b) Interpret what this percentage represents in the context of the problem.

Worked Example 5

a) A class has students. are present. How many students are present?

b) State the multiplication needed.

Worked Example 6

a) A tank holds L of water and is full. How much water is in the tank?

b) Explain the steps used.

Problems

Problem 1

a) Write as a fraction.

b) Find of .

c) Explain why multiplying by a fraction works.

Problem 2

a) Write as a fraction.

b) Find of .

Problem 3

a) Find of .

b) Find of .

c) Find of .

Problem 4

a) A book costs $80. Find of $80.

b) Interpret what this percentage represents in the context of the problem.

Problem 5

a) A class has students. are present. How many students are present?

b) State the multiplication needed.

Problem 6

a) A tank holds L of water and is full. How much water is in the tank?

b) Explain the steps used.

Exercises

Understanding and Fluency

Exercise 1.

Convert each percentage to a fraction in simplest form, then find the percentage of the number:

a) of

b) of

c) of

Exercise 2.

Convert each percentage to a fraction in simplest form, then find the percentage of the number:

a) of

b) of

c) of

Exercise 3.

Find each percentage of the number:

a) of

b) of

c) of

Exercise 4.

Find each percentage of the number:

a) of

b) of

c) of

Exercise 5.

Find each percentage of the number:

a) of

b) of

c) of

Exercise 6.

Find each percentage of the number:

a) of

b) of

c) of

Reasoning

Exercise 7.

Explain why finding of a number is the same as finding of the number.

Exercise 8.

A student says that of is . Explain the mistake.

Exercise 9.

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

Exercise 10.

A student finds of by doing . Explain why this is incorrect.

Problem-solving

Exercise 11.

A jacket costs $120. Find of the cost.

Exercise 12.

A school has students. of them walk to school. How many students walk to school?

Exercise 13.

A container holds mL of juice. of the juice is poured out. How much juice is poured out?

Exercise 14.

A test has marks. A student earns of the marks. How many marks is that?

Exercise 15.

A farm has trees. are orange trees. How many orange trees are there?

Exercise 16.

A phone battery is at of its full charge of mAh. How much charge remains?

Exercise 17.

A bag contains marbles. are red. How many red marbles are there?

Exercise 18.

A shop discounts an item priced at $90 by . 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 ”
• 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