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 find 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 $100$”
• Be able to convert between percentages, fractions and decimals
• Understand that finding $1\mathrm{\%}$ is often the same as dividing by $100$
• 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) $20\mathrm{\%}$ of a number is $14$
b) $5\mathrm{\%}$ of a quantity is $\mathrm{\$}8$

Worked Example 2

Use the unitary method to find a new percentage:
a) If $10\mathrm{\%}$ of a quantity is $9$, find $35\mathrm{\%}$
b) If $25\mathrm{\%}$ of an amount is $\mathrm{\$}12$, find $75\mathrm{\%}$

Worked Example 3

Use the unitary method to find a quantity:
a) $15\mathrm{\%}$ of a length is $18$ cm. Find the whole length
b) $8\mathrm{\%}$ of a mass is $6$ kg. Find the whole mass

Worked Example 4

Use the unitary method to find the original price after an increase:
a) After a $20\mathrm{\%}$ increase, a price is $\mathrm{\$}72$
b) After a $15\mathrm{\%}$ increase, a price is $\mathrm{\$}92$

Worked Example 5

Use the unitary method to find the original price after a decrease:
a) After a $25\mathrm{\%}$ discount, a price is $\mathrm{\$}45$
b) After a $10\mathrm{\%}$ discount, a price is $\mathrm{\$}81$

Worked Example 6

Use the unitary method in mixed contexts:
a) $40\mathrm{\%}$ of a class is $12$ students. Find the total number of students
b) If $5\mathrm{\%}$ of a price is $\mathrm{\$}3.50$, find $120\mathrm{\%}$ of the price

Problems

Problem 1

Use the unitary method to find the value of the whole quantity:
a) $25\mathrm{\%}$ of a number is $18$
b) $4\mathrm{\%}$ of a quantity is $\mathrm{\$}6$

Problem 2

Use the unitary method to find a new percentage:
a) If $20\mathrm{\%}$ of a quantity is $16$, find $45\mathrm{\%}$
b) If $50\mathrm{\%}$ of an amount is $\mathrm{\$}30$, find $15\mathrm{\%}$

Problem 3

Use the unitary method to find a quantity:
a) $12\mathrm{\%}$ of a length is $24$ cm. Find the whole length
b) $6\mathrm{\%}$ of a mass is $9$ kg. Find the whole mass

Problem 4

Use the unitary method to find the original price after an increase:
a) After a $25\mathrm{\%}$ increase, a price is $\mathrm{\$}100$
b) After a $10\mathrm{\%}$ increase, a price is $\mathrm{\$}66$

Problem 5

Use the unitary method to find the original price after a decrease:
a) After a $20\mathrm{\%}$ discount, a price is $\mathrm{\$}56$
b) After a $15\mathrm{\%}$ discount, a price is $\mathrm{\$}68$

Problem 6

Use the unitary method in mixed contexts:
a) $30\mathrm{\%}$ of a class is $9$ students. Find the total number of students
b) If $4\mathrm{\%}$ of a price is $\mathrm{\$}2.40$, find $125\mathrm{\%}$ of the price

Exercises

Understanding and Fluency

1. Complete each statement about the unitary method:
   a) The unitary method first finds the value of ______ unit
   b) When working with percentages, finding $1\mathrm{\%}$ often means dividing by ______
   c) To find the whole amount from a known percentage, we often find ______ first

2. Use the unitary method to find the whole quantity:
   a) $10\mathrm{\%}$ of a number is $7$
   b) $20\mathrm{\%}$ of a quantity is $18$
   c) $4\mathrm{\%}$ of an amount is $\mathrm{\$}5$

3. Use the unitary method to find the whole quantity:
   a) $25\mathrm{\%}$ of a length is $15$ cm
   b) $8\mathrm{\%}$ of a mass is $12$ kg
   c) $40\mathrm{\%}$ of a class is $14$ students

4. Use the unitary method to find a new percentage:
   a) If $5\mathrm{\%}$ of a quantity is $4$, find $15\mathrm{\%}$
   b) If $10\mathrm{\%}$ of a quantity is $11$, find $70\mathrm{\%}$
   c) If $25\mathrm{\%}$ of an amount is $\mathrm{\$}20$, find $50\mathrm{\%}$

5. Use the unitary method to find a new percentage:
   a) If $20\mathrm{\%}$ of a quantity is $16$, find $5\mathrm{\%}$
   b) If $50\mathrm{\%}$ of a price is $\mathrm{\$}36$, find $125\mathrm{\%}$
   c) If $4\mathrm{\%}$ of a mass is $3$ kg, find $12\mathrm{\%}$

6. Use the unitary method to find the original price after an increase:
   a) After a $20\mathrm{\%}$ increase, the new price is $\mathrm{\$}84$
   b) After a $10\mathrm{\%}$ increase, the new price is $\mathrm{\$}55$
   c) After a $25\mathrm{\%}$ increase, the new price is $\mathrm{\$}150$

7. Use the unitary method to find the original price after a decrease:
   a) After a $25\mathrm{\%}$ discount, the sale price is $\mathrm{\$}60$
   b) After a $10\mathrm{\%}$ discount, the sale price is $\mathrm{\$}72$
   c) After a $20\mathrm{\%}$ discount, the sale price is $\mathrm{\$}96$

8. Solve each using the unitary method:
   a) $35\mathrm{\%}$ of a number is $21$. Find the number
   b) A price after a $15\mathrm{\%}$ increase is $\mathrm{\$}92$. Find the original price
   c) A jumper is sold for $\mathrm{\$}63$ after a $30\mathrm{\%}$ discount. Find the original price
   d) If $2\mathrm{\%}$ of a quantity is $1.6$, find $100\mathrm{\%}$

Reasoning

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

10. A student says that if $20\mathrm{\%}$ of a quantity is $14$, then the whole quantity is $14\times 20=280$. Explain the mistake.

11. Noah says that if a price is increased by $20\mathrm{\%}$, then the new price is $20\mathrm{\%}$ of the original price. Is he correct? Explain.

12. Explain why a sale price after a $25\mathrm{\%}$ discount represents $75\mathrm{\%}$ of the original price.

13. A student says that if a new price is $\mathrm{\$}88$ after a $10\mathrm{\%}$ increase, then the original price is found by subtracting $10$. Describe the error.

Problem-solving

14. A school says that $15\mathrm{\%}$ of its students travel by bus, and this is $27$ students. How many students are at the school altogether?

15. A phone is sold for $\mathrm{\$}138$ after a $15\mathrm{\%}$ increase. What was the original price?

16. A bike is on sale for $\mathrm{\$}96$ after a $20\mathrm{\%}$ discount. What was the original price?

17. In a survey, $5\mathrm{\%}$ of the people asked chose option A, and this was $12$ people. How many people were surveyed?

18. A bag of rice weighs $40$ kg when full. If $15\mathrm{\%}$ of the rice has been used, how many kilograms remain?

19. A shop knows that $8\mathrm{\%}$ of the original price of a jacket is $\mathrm{\$}6.40$.
    a) Find $1\mathrm{\%}$ of the original price
    b) Find the original price
    c) Find the price after a $25\mathrm{\%}$ 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 $1\mathrm{\%}$
• Students may divide by the wrong percentage when trying to find $1\mathrm{\%}$
• Students may confuse the known percentage with the whole amount
• Students may think finding the whole quantity means multiplying by $100$ every time, even when the known percentage is not $1\mathrm{\%}$
• Students may not recognise that after an increase, the new price represents more than $100\mathrm{\%}$ of the original price
• Students may not recognise that after a decrease, the sale price represents less than $100\mathrm{\%}$ 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 $1\mathrm{\%}$ correctly but then multiply by the wrong percentage to find the required value