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Maths Quest 7 VC 3e

LESSON 5.4 The unitary method and best buys

LEARNING INTENTION

At the end of this lesson you should be able to:

use the unitary method to determine the best value option
apply the unitary method to find amounts from a partial amount.

eles-3891

5.4.1 Unit price

The unit price is the price per unit. Examples include the price per $100 mL$ , the price per $kg$ or the price of a single item.
The unitary method can be used to work out the best buy for an item by comparing the unit price.
Supermarkets in Australia are now obliged to show the unit price on most products.
To determine the ‘best buy’ when comparing similar items:
- decide what unit you will use in your comparison
- divide the price of each item by its number of units
- identify the item with the smallest unit price. This is the best value item.

Price per unit

$Price per unit = \frac{price}{number of units}$

Remember to represent the same units when you are making comparisons.

WORKED EXAMPLE 9 Determining the best buy using the unitary method

Three shampoos are sold in the following quantities.

Brand A: $200 mL$ for $$ 5 . 38$
Brand B: $300 mL$ for $$ 5 . 98$
Brand C: $400 mL$ for $$ 8 . 04$

Determine which shampoo is the best buy.

THINK		WRITE
1.	Determine the number of $100 - mL$ units for each shampoo.	$\begin{array}{rcl} Brand A & = & \frac{200}{100} = 2 units \\ Brand B & = & \frac{300}{100} = 3 units \\ Brand C & = & \frac{400}{100} = 4 units \end{array}$
2.	Determine the price per unit for each shampoo. $Price per unit = \frac{price}{100 -mL units}$	$\begin{matrix} Brand A & = & \frac{5.38}{2} = $ 2.69 per 100 mL \\ Brand B & = & \frac{5.98}{3} = $1 .99 per 100 mL \\ Brand C & = & \frac{8.04}{4} = $2 .01 per 100 mL \end{matrix}$
3.	Write the answer.	Brand B: $300 mL$ of shampoo for $$ 5.98$ is the best buy.

eles-3892

5.4.2 Unitary method in percentages

The unitary method can be used to determine the whole amount when given a partial amount and the percentage of the total that it corresponds to.

Determining the whole amount from a partial amount

Step 1: Find $1 %$ of the total amount.
Step 2: Multiply the value of the $1 %$ by $100$ to get the whole amount $(100)$ .

WORKED EXAMPLE 10 Determining total price from a partial amount

Calculate the whole amount if $5 %$ is represented by $$ 25$ .

THINK		WRITE
1.	Five per cent of a number equals $25$ . Calculate $1 %$ of the $$ 25$ by dividing the number by $5$ .	$\begin{array}{l} 5 % & = & $ 25 \\ 1 % & = & $ 25 \div 5 \\ = & $ 5 \end{array}$
2.	Calculate $100 %$ by multiplying the amount by $100$ .	$\begin{array}{l} 100 % & = & $ 5 \times 100 \\ = & $ 500 \end{array}$
3.	Write the answer.	The whole amount is $$ 500$ .

The same method can also be applied when given a percentage that is greater than $100 %$ .

WORKED EXAMPLE 11 Determining values from an increased amount

Marcie receives a pay rise of $12 %$ , taking her salary to $$ 72 000$ a year. Calculate her annual salary before the pay rise, to the nearest dollar.

THINK		WRITE
1.	A pay rise of $12 %$ means that Marcie’s new salary is $112 %$ of her old salary.	$112 % of previous salary = $ 72 000$
2.	Calculate what $1 %$ of her previous salary was by dividing by $112$ .	$\begin{matrix} 1% of previous salary & = & \frac{$ 72 000}{112} \\ = & $ 642.8571 \end{matrix}$
3.	Calculate $100 %$ of her previous salary by multiplying by $100$ .	$\begin{matrix} 100% of previous salary & = & $ 642.8571 \times 100 \\ = & $ 64 285.71 \end{matrix}$
4.	Write the answer, rounding to the nearest dollar.	Marcie’s previous salary was $$ 64 286$ .

ACTIVITY: Price comparison websites

Use the internet to research different price comparison websites. Investigate what goods and services price comparison websites display and present your research to your class.

Resources

eWorkbook: Topic 5 Workbook (worksheets, code puzzle and project) (ewbk-1906)

LESSON 5.4 The unitary method and best buys

Students may complete the Quick quiz for this lesson to ensure they have practised and understood the skills covered.

LEARNING INTENTIONS

At the end of this lesson students should be able to:

use the unitary method to determine the best value option
apply the unitary method to find amounts from a partial amount.

Level	Sequenced activities	Questions
Practise	Teacher Video: Unit price (eles-3891) Teacher Video: Unitary method in percentages (eles-3892) Interactivity: Unitary method (int-4154) Interactivity: Unitary method in percentages (int-4155) Individual Pathway Interactivity: Practise (int-4396)	1, 4, 8, 9, 11, 17, 21
Consolidate	Teacher Video: Unit price (eles-3891) Teacher Video: Unitary method in percentages (eles-3892) Interactivity: Unitary method (int-4154) Interactivity: Unitary method in percentages (int-4155) Individual Pathway Interactivity: Consolidate (int-4396)	2, 5, 6, 12, 13, 15, 18, 22
Master	Teacher Video: Unit price (eles-3891) Teacher Video: Unitary method in percentages (eles-3892) Interactivity: Unitary method (int-4154) Interactivity: Unitary method in percentages (int-4155) Individual Pathway Interactivity: Master (int-4396)	3, 7, 10, 14, 16, 19, 20, 23

5.4 Exercise

Select your pathway

Questions:
1, 4, 8, 9, 11, 17, 21

Questions:
2, 5, 6, 12, 13, 15, 18, 22

Questions:
3, 7, 10, 14, 16, 19, 20, 23

Fluency

Q1

Which of the following is the best buy?

Chocolate weight	Cost
150 g	$3.25
250 g	$4.75
325 g	$5.50

1 mark(s)

Q2

Which of the following is the best buy?

No. of pages	Cost
80	$1.98
160	$3.38
200	$3.98

1 mark(s)

Q3

Compare the cost of 400 g of biscuits for $2.48 and 500 g for $3.10. Which is the better buy?

1 mark(s)

Q4

Answer each of the following questions.

a. 10% is $18. The original amount is $

Correct answers:

1 mark(s)

b. 20% is $6. The original amount is $

Correct answers:

1 mark(s)

c. 25% is $60. The original amount is $

Correct answers:

1 mark(s)

d. 40% is $900. The original amount is $

Correct answers:

1 mark(s)

Q5

Answer each of the following questions.

a. 90% is $380. The original amount is $

(State your answer rounded to the nearest cent)

Correct answers:

1 mark(s)

b. 200% is $800. The original amount is $

Correct answers:

1 mark(s)

c. 120% is $420. The original amount is $

Correct answers:

1 mark(s)

d. 9% is $54. The original amount is $

Correct answers:

1 mark(s)

Q6

Answer each of the following questions.

a. 40% is $120, therefore 60% = $

Correct answers:

1 mark(s)

b. 20% is $6, therefore 38% = $

Correct answers:

1 mark(s)

c. 25% is $60, therefore 50% = $

Correct answers:

1 mark(s)

Q7

Answer each of the following questions.

a. 50% is $900, therefore 150% = $

Correct answers:

1 mark(s)

b. 75% is $250, therefore 12% = $

Correct answers:

1 mark(s)

c. 14% is $4200, therefore 86% = $

Correct answers:

1 mark(s)

Q8

Given that 2 kg of apples cost $3.80, the cost of 4.5 kg = $

Correct answers:

1 mark(s)

Understanding

Q9

You spend 40% of your monthly allowance on a pair of used inline skates.

What is your monthly allowance if you spend $20 on the skates?

Correct answers:

1 mark(s)

Q10

Hannah saves 32% of her wages each week. If she saves $220, what is her weekly wage?

Correct answers:

1 mark(s)

Q11

Alex receives a 15% pay rise. He now receives $97 290 a year.

How much was his annual income before the pay rise?

Correct answers:

1 mark(s)

Q12

Which is the best buy?

500 g of cheese for $12.80, 200 g cheese for $5.62 or 90 g cheese for $2.10?

1 mark(s)

Q13

After a discount of 15%, a pair of wireless headphones is worth $183.

What was its value before the discount? (Round your answer to the nearest cent)

Correct answers:

1 mark(s)

Q14

During a sale, a retailer allows a discount of 15% off the marked price. His sale price of $60 still gives him a profit of 10%.

Answer the following questions, rounding your answers to the nearest cent.

a. What did the article cost him?

Correct answers:

1 mark(s)

b. What was the marked price?

Correct answers:

1 mark(s)

Q15

A discount of 15% reduced the price of a pen by $3.20.

Answer the following questions, rounding your answers to the nearest cent.

a. What was the original price of the pen?

Correct answers:

1 mark(s)

b. What was its selling price?

Correct answers:

1 mark(s)

Q16

Supermarkets are obliged by law to display the unit price when selling the same items in different quantities.

Calculate the price per 100 g for the following amounts of chocolate. Give your answers correct to the nearest cent.

a. 250 g for $5.95 = $ per 100 g

Correct answers:

1 mark(s)

b. 400 g for $8.50 = $ per 100 g

Correct answers:

1 mark(s)

c. 60 g for $2.00 = $ per 100 g

Correct answers:

1 mark(s)

Reasoning

Q17

In some situations it is not always practical to go for the best buy. Why might this be? List all of the possible reasons that you can think of.

1 mark(s)

Q18

Explain some of the different situations in which you would use the unitary method.

1 mark(s)

Q19

While shopping in a sale, Carly spots a sign that says ‘Buy 2 items, get the third half price (the least expensive item will be counted as the third item)’.

If she finds a shirt for $25.99, a skirt for $87.99 and a belt for $15.97 and pays cash for the items, how much will Carly end up paying?

Total cost = $

Correct answers:

1 mark(s)

Q20

Penny loves to make bracelets.

To make them, she needs to buy some beads. Brand A costs $7.50 for a box that contains 1000 beads; Brand B costs $17.50 for a box that contains 2500 beads.

Which brand is the better buy? Explain your reasoning.

2 mark(s)

Problem Solving

Q21

Ice-cream sticks are sold in the following multi-packs.

Cost($)	Number of ice-cream sticks
$$ 3.25$	$6$
$$ 4.99$	$10$
$$ 7.50$	$15$

Which is the best buy?

1 mark(s)

Q22

Some players from a soccer club went to a café to celebrate their win. Each had a burger and a drink. The bill came to $80. When it came time to pay the bill, 3 people left without paying, so the remaining members had to pay an extra $2 each to cover the bill.

If there were originally $n$ people in the group, and the burger and drink deal cost $$ b$ , write an equation using these pronumerals to show how the bill was settled. Do not attempt to solve your equation.

80 =

Correct answers:

1 mark(s)

Q23

The price of entry into a theme park has increased by 10% every year since the theme park opened.

If the latest price rise increased the tickets to $8.80, what was the price of a ticket 2 years ago? (Round your answer to the nearest cent)

Correct answers:

1 mark(s)

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5.4 Textbook Questions

Exercise 5.4 The unitary method and best buys