When referring to increases and decreases in money, we will often use the terms profit and loss. Profit is used when referring to an increase in money while a loss refers to a decrease in money. These two terms can be used when talking about flat changes in money or percentage changes in money.

Flat changes in money

When calculating flat changes in money, we will often refer to the cost price and the sale price. The cost price is how much money we paid for an object and the sale price is how much money we earned by selling the object.

We can find the flat change in money as a directed number by subtracting the cost price from the sale price. This will tell us the net change in money.

Net change in money

The net change in money is the directed number indicating how much money was earned after buying and selling an object.

$Net change = Sale Price - Cost Price$

A positive net change indicates money gained and a negative net changes indicates money lost.

Worked Example

Stephen bought a cinnamon roll for $\$3$$3 and sold it for $\$7$$7.

What was his net change in money as a directed number? Did Stephen gain or lose money?

Think: We are given that the cost price of the roll was $\$3$$3 and the sale price was $\$7$$7. We can calculate the net change by subtracting the cost price from the sale price.

Do: Subtracting the cost price from the sale price, we get:

Net change	$=$=		$\$7-\$3$$7−$3
Net change	$=$=		$\$4$$4

Since the net change in money was positive, we also know that Stephen gained money.

Reflect: When calculating net change, we can also use a number line.

On this number line we can see how Stephen starts at zero, loses $\$3$$3 buying the roll and then gains $\$7$$7 by selling it, resulting in a net change of $\$4$$4.

Another way to refer to Stephen's net change in money is 'a profit of $\$4$$4'.

Instead of having both positive and negative net change, we can instead refer to the change as either profit or loss. By referring to money gained as profit and money lost as loss, we can discuss money without needing negative numbers.

Using these terms, we can visualise Stephen's change in money like this:

As we can see, spending money moves Stephen's position in the direction of loss while gaining money moves him in he direction of profit.

If Stephen neither gained nor lost money then he would make $\$0$$0 which is referred to as breaking even.

Practice question

Question 1

After buying and then selling $a bicycle$ , the net change in $Laura$ 's money was $-\$44$−$44.

Which of the following describes $her$ change in money?

Profit of $\$44$$44
A
Loss of $\$44$$44
B
Loss of $-\$44$−$44
C
Breaking even
D

Finding cost and sale prices

Since profit is a positive net change in money, we can only have a profit when the sale price is greater than the cost price. Since profit is always a positive value, we can find it using the equation:

Profit

$Profit = Sale Price - Cost Price$

Similarly, since a loss only occurs when the cost price is greater than the sale price, we can find it using the equation:

Loss

$Loss = Cost Price - Sale Price$

Using either of these equations, as long as we are given two out of the three values, we can always find the third.

Worked Example

Julia sold a watermelon for $\$8$$8 and made a loss of $\$3.50$$3.50. How much did she buy the watermelon for?

Think: Since Julia made a loss, we want to use the equation $Loss = Cost Price - Sale Price$ to find the amount she bought the watermelon for.

Do: By substituting the known values into the equation, we get:

$3.50$3.50 $= Cost Price -$ $8$8

By adding $8$8 to both sides of the equation, we get:

$3.50+8$3.50+8 $= Cost Price$

Evaluating the addition tells us that Julia bought the watermelon for $\$11.50$$11.50.

Practice question

Question 2

$Sandy$ $bought$ a $stove$ for $\$238$$238 and made a $profit$ of $\$39$$39.

What was the $sale$ price of the item?

Percentage profit and loss

The percentage profit or loss made when selling an object is the flat profit or loss as a percentage of the cost price.

In other words, the percentage profit or loss is equal to the profit or loss as a percentage of the cost price.

Worked Example

Alfonso bought a suit of armour for $\$320$$320 and sold it for $\$400$$400.

What was the percentage profit or loss that Alfonso made?

Think: We know that the cost price was $\$320$$320 and the sale price was $\$400$$400. Since the sale price is greater than the cost price, we know Alfonso made a profit, and we can find that profit using the equation $Profit = Sale Price - Cost Price$ .

Do: By subtracting the cost price from the sale price, we find that Alfonso's flat profit was $\$80$$80.

We can find the profit as a percentage of the cost price using the equation:

$Percentage profit =$ $\frac{80}{320}\times100%$80320×100%

Evaluating the expression tells us that Alfonso's percentage profit was $25%$25%.

Reflect: Notice that $\$400$$400 is equal to $125%$125% of $\$320$$320. We can see that the sale price is $25%$25% greater than the cost price, telling us that the percentage profit was $25%$25% as expected.

Practice question

Question 3

$Charlie$ bought a $cake$ for $\$220$$220 and sold it for $\$209$$209.

Which of the following describe $Charlie$ 's change in money after buying and selling the $cake$ ?

Select all that apply.

Loss of $11%$11%
A
Loss of $\$11$$11
B
Profit of $5%$5%
C
Profit of $\$11$$11
D
Loss of $5%$5%
E
Loss of $\$5$$5
F

4.07 Profit and loss

Flat changes in money

Worked Example

Practice question

Question 1

Finding cost and sale prices

Worked Example

Practice question

Question 2

Percentage profit and loss

Worked Example

Practice question

Question 3

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