5 Ratios, rates, and money

5.02 Equivalent and simplified ratios

5.03 Find part or whole measures with ratios

5.04 Rates

5.05 Best buys

Level 7

5.05 Best buys

Lesson

Worksheet

Practice

Lesson

Introduction

Every business is trying to sell something, and the way they display their prices can be confusing. Sometimes they can be deliberately confusing. By using some mathematical strategies we can compare two or more different offers and determine the best buy.

Cost per unit

If the two options we are choosing between involve different quantities, we can use what we know about rates to find out how much a single unit costs.

Examples

Example 1

Isabelle is buying juice for her nephew's birthday party. A 3.2 \text{ L} bottle of apple juice costs \$13.76. A 2.1 \text{ L} bottle of orange juice costs \$6.30.

How much does apple juice cost per litre?

Worked Solution

Create a strategy

Divide the cost of the bottle by the number of litres.

Apply the idea

\displaystyle \text{Apple juice price}	\displaystyle =	\displaystyle 13.76 \div 3.2	Divide 13.76 by 3.2
	\displaystyle =	\displaystyle \$ 4.30	Perform the division

How much does orange juice cost per litre?

Worked Solution

Create a strategy

Divide the cost of the bottle by the number of litres.

Apply the idea

\displaystyle \text{Orange juice price}	\displaystyle =	\displaystyle 6.30 \div 2.1	Divide 6.30 by 2.1
	\displaystyle =	\displaystyle \$ 3.00	Perform the division

Which juice is the best buy?

Orange juice

Apple juice

Worked Solution

Create a strategy

Compare the answer in parts (a) and (b) and choose which juice is cheaper per litre.

Apply the idea

The price of apple juice per litre is \$4.30 while the price of orange juice per litre is \$3.00. So the best buy is the orange juice.

So the correct answer is option A.

Idea summary

In choosing between two options, we can find the cost per unit to determine which is the better buy.

Quantity for unit cost

We can also compare prices by determining how much a single dollar buys. The idea is very similar to the previous one.

Examples

Example 2

A 1.35 \text{ kg} packet of mince costs \$15. How many grams of mince does each dollar buy?

Worked Solution

Create a strategy

To get the weight per dollar, divide the total weight of the mince by the number of dollars it costs.

Apply the idea

\displaystyle \$1\text{ buys}	\displaystyle =	\displaystyle 1.35\text{ kg} \div \$15	Divide by 15
	\displaystyle =	\displaystyle 0.09 \text{ kg}	Perform the division
	\displaystyle =	\displaystyle 0.09 \times 1000 \text{ g}	Convert to grams
	\displaystyle =	\displaystyle 90 \text{ g}	Evaluate

Idea summary

We can also compare prices by determining how much a single dollar buys.

Notice that when using this method, the rate with the larger unit value represents the better value, as we get more for the same price.

Discounts

Stores frequently offer a percentage discount on items which converts the original price into a sale price. This discount is often presented as "30\% off" or "reduced by 25\%", and it is important to know how to calculate the new price.

When a sale price has a percentage discount applied, we find that percentage of the original price and subtract it to find the sale price: \text{Sale price}=\text{Original price}-(\text{Original price}\times\text{Discount percent})

We can take a shortcut by thinking about what happens after the discount is applied. If an item is reduced by 30\%, then the sale price is 70\% of the original price. Similarly, a 25\% discount means the sale price is 75\% of the original price. The remaining percent is 100 minus the discount.

Examples

Example 3

A pair of Bevi jeans normally costs \$120, and the store is offering a 20\% discount. A similar pair of Genie jeans normally costs \$150, and the store is offering a 30\% discount. Which pair of jeans is the best buy?

Worked Solution

Create a strategy

Find the remaining percentages and multiply them by the original prices.

Apply the idea

A 20\% discount is the same as paying 80\% of the original price, so we should find 80\% of \$120. Similarly, a 30\% discount is the same as paying 70\% of the original price, so we should find 70\% of \$150.

For Bevi jeans:

\displaystyle \text{Sale price}	\displaystyle =	\displaystyle 80\% \times 120	Find the remaining percent
	\displaystyle =	\displaystyle \dfrac{80}{100} \times 120	Convert the percentage to a fraction
	\displaystyle =	\displaystyle \$96	Evaluate

For Genie jeans:

\displaystyle \text{Sale price}	\displaystyle =	\displaystyle 70\% \times 150	Find the remaining percent
\displaystyle \text{Sale price}	\displaystyle =	\displaystyle \dfrac{70}{100} \times 150	Convert the percentage to a fraction
	\displaystyle =	\displaystyle \$105

\$96 \lt \$105 so the best buy is the pair of Bevi jeans.

Idea summary

We can also use a shortcut by multiplying the original price by the remaining percent which is 100\% minus the discount percentage.

5.05 Best buys

Introduction

Ideas

Cost per unit

Examples

Example 1

Quantity for unit cost

Examples

Example 2

Discounts

Examples

Example 3

Outcomes

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