14.01 Unit costs

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 or unit cost.

 

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.

Worked example

At the neighbourhood weekend market two stalls, Chokblox and Candy Kingdom both sell chocolate fudge in bulk. Chokblox sells an enormous $2$2 kg slab for $\$30$$30, while Candy Kingdom sells a smaller $500$500 g piece for $\$7$$7. Which is the best buy?

Think: Since they each sell different amounts, it is hard to compare the offers. Instead we will try to figure out how much the fudge costs per kg from each stall.

Do: The cost at Chokblox can be represented by the rate:

$\$30$$30/$2$2 kg

Dividing both sides by $2$2 gives us the cost per kg:

$\$15$$15/$1$1 kg

Similarly, the cost at Candy Kingdom can be represented by the rate:

$\$7$$7/$500$500 g

Since there are $1000$1000 g in a kg, multiplying both sides by $2$2 gives us the cost per kg:

$\$14$$14/$1$1 kg

This second rate represents a cheaper price per kg, so the best buy is the fudge from Candy Kingdom.

 

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.

Worked example

The two market stalls Chokblox and Candy Kingdom also sell pick and mix chocolate. Chokblox charges $\$3$$3 for every $570$570 g,  and Candy Kingdom charges $\$4$$4 for every $820$820 g. Which is the best buy?

Think: If we tried to find the price per kg (or price per g), we would have to deal with fractions. Instead we will try to answer a different question - which stall gives us the most chocolate for $\$1$$1?

Let's switch the order of the rate around, so we can see how much we get per dollar.

Do: The cost at Chokblox can be represented by the rate:

$570$570 g/$\$3$$3 

Dividing both sides by $3$3 gives us the quantity we can buy for $\$1$$1:

$190$190 g/$\$1$$1

The cost at Candy Kingdom can be represented by the rate:

$820$820 g/$\$4$$4

Dividing both sides by $4$4 gives us the quantity we can buy for $\$1$$1:

$205$205 g/$\$1$$1

Since we can buy more chocolate for a single dollar from Candy Kingdom, it is once again the best buy.

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%$30% off!" or "reduced by $25%$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:

$\text{Sale price}=\text{Original price}-\left(\text{Original price}\times\text{Discount percent}\right)$Sale price=Original price−(Original price×Discount percent)

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

Worked example

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

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

Do: To find $80%$80% of $\$120$$120 we multiply the price by $80$80 and divide by $100$100:

$\text{Sale price}=120\times\frac{80}{100}=96$Sale price=120×80100​=96

This means the sale price of the Bevi jeans is $\$96$$96.

To find $70%$70% of $\$150$$150 we multiply the price by $70$70 and divide by $100$100:

$\text{Sale price}=150\times\frac{70}{100}=105$Sale price=150×70100​=105

This means the sale price of the Genie jeans is $\$105$$105. The best value is the pair of Bevi jeans.

Practice questions

Question 1

Question 2

Question 3