118e. The Unitary Method with Percentages

Learning Intentions

• To understand that the unitary method involves finding the value of ‘one unit’ as an intermediate step
• Use the unitary method to Solve a quantity when only a percentage is known
• use the unitary method to find a new percentage when a different percentage is known
• Apply the unitary method to find the original price when a price has been increased or decreased by a percentage

Pre-requisite Summary

• Know that a percentage means “out of ”
• Be able to convert between percentages, fractions and decimals
• Understand that finding is often the same as dividing by
• Be able to multiply and divide whole numbers and decimals accurately
• Understand that an increase makes a quantity larger and a decrease makes a quantity smaller
• Know that the original price is the price before the increase or decrease
• Understand that the unitary method finds the value of one unit first

Worked Examples

Worked Example 1

Use the unitary method to find the value of the whole quantity:

a) of a number is

b) of a quantity is $

Worked Example 2

Use the unitary method to find a new percentage:

a) If of a quantity is , find

b) If of an amount is $ , find

Worked Example 3

Use the unitary method to find a quantity:

a) of a length is cm. Find the whole length

b) of a mass is kg. Find the whole mass

Worked Example 4

Use the unitary method to find the original price after an increase:

a) After a increase, a price is $

b) After a increase, a price is $

Worked Example 5

Use the unitary method to find the original price after a decrease:

a) After a discount, a price is $

b) After a discount, a price is $

Worked Example 6

Use the unitary method in mixed contexts:

a) of a class is students. Find the total number of students

b) If of a price is $ , find of the price

Problems

Problem 1

Use the unitary method to find the value of the whole quantity:

a) of a number is

b) of a quantity is $

Problem 2

Use the unitary method to find a new percentage:

a) If of a quantity is , find

b) If of an amount is $ , find

Problem 3

Use the unitary method to find a quantity:

a) of a length is cm. Find the whole length

b) of a mass is kg. Find the whole mass

Problem 4

Use the unitary method to find the original price after an increase:

a) After a increase, a price is $

b) After a increase, a price is $

Problem 5

Use the unitary method to find the original price after a decrease:

a) After a discount, a price is $

b) After a discount, a price is $

Problem 6

Use the unitary method in mixed contexts:

a) of a class is students. Find the total number of students

b) If of a price is $ , find of the price

Exercises

Understanding and Fluency

Exercise 1.

Complete each statement about the unitary method:

a) The unitary method first finds the value of ______ unit

b) When working with percentages, finding often means dividing by ______

c) To find the whole amount from a known percentage, we often find ______ first

Exercise 2.

Use the unitary method to find the whole quantity:

a) of a number is

b) of a quantity is

c) of an amount is $

Exercise 3.

Use the unitary method to find the whole quantity:

a) of a length is cm

b) of a mass is kg

c) of a class is students

Exercise 4.

Use the unitary method to find a new percentage:

a) If of a quantity is , find

b) If of a quantity is , find

c) If of an amount is $ , find

Exercise 5.

Use the unitary method to find a new percentage:

a) If of a quantity is , find

b) If of a price is $ , find

c) If of a mass is kg, find

Exercise 6.

Use the unitary method to find the original price after an increase:

a) After a increase, the new price is $

b) After a increase, the new price is $

c) After a increase, the new price is $

Exercise 7.

Use the unitary method to find the original price after a decrease:

a) After a discount, the sale price is $

b) After a discount, the sale price is $

c) After a discount, the sale price is $

Exercise 8.

Solve each using the unitary method:

a) of a number is . Find the number

b) A price after a increase is $ . Find the original price

c) A jumper is sold for $ after a discount. Find the original price

d) If of a quantity is , find

Reasoning

Exercise 9.

Explain why the unitary method is called the “unitary” method.

Exercise 10.

A student says that if of a quantity is , then the whole quantity is . Explain the mistake.

Exercise 11.

Noah says that if a price is increased by , then the new price is of the original price. Is he correct? Explain.

Exercise 12.

Explain why a sale price after a discount represents of the original price.

Exercise 13.

A student says that if a new price is $ after a increase, then the original price is found by subtracting . Describe the error.

Problem-solving

Exercise 14.

A school says that of its students travel by bus, and this is students. How many students are at the school altogether?

Exercise 15.

A phone is sold for $ after a increase. What was the original price?

Exercise 16.

A bike is on sale for $ after a discount. What was the original price?

Exercise 17.

In a survey, of the people asked chose option A, and this was people. How many people were surveyed?

Exercise 18.

A bag of rice weighs kg when full. If of the rice has been used, how many kilograms remain?

Exercise 19.

A shop knows that of the original price of a jacket is $ .

a) Find of the original price

b) Find the original price

c) Find the price after a discount

Potential Misunderstandings

• Students may think the unitary method means multiplying first, rather than finding the value of one unit first
• Students may forget that with percentages, the “one unit” is often
• Students may divide by the wrong percentage when trying to find
• Students may confuse the known percentage with the whole amount
• Students may think finding the whole quantity means multiplying by every time, even when the known percentage is not
• Students may not recognise that after an increase, the new price represents more than of the original price
• Students may not recognise that after a decrease, the sale price represents less than of the original price
• Students may try to find the original price after a discount by subtracting the discount percentage from the new price
• Students may confuse the original price with the changed price
• Students may calculate correctly but then multiply by the wrong percentage to find the required value