Introduction
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.
Ideas
- Flat changes in money
- Finding cost and sale prices
- Percentage profit and loss
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.
The net change in money is the directed number indicating how much money was earned after buying and selling an object. \text{Net change}=\text{Sale price}-\text{Cost price}
A positive net change indicates money gained and a negative net changes indicates money lost.
Examples
Example 1
After buying and then selling a bicycle, the net change in Laura's money was -\$44.
Which of the following describes her change in money?
A
Profit of \$44
B
Loss of \$44
C
Loss of -\$44
D
Breaking Even
Worked Solution
Create a strategy
To describe the change look at the sign of the value.
Apply the idea
If the net change in money is a negative value, it is a loss.
We can describe her change in money as a loss of \$44.
The correct option is B.
Idea summary
The net change in money is the directed number indicating how much money was earned after buying and selling an object.\text{Net change}=\text{Sale price}-\text{Cost price}
A positive net change indicates money gained and a negative net changes indicates money lost.
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:\text{Profit} = \text{Sale price} - \text{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:\text{Loss} = \text{Cost price} - \text{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.
Exploration
The following applet shows a visual representation of the relationship between profit and loss.
Drag the blue points on the top bar left and right.
Loading interactive...
We can see that when the selling price is less than the cost, we get a loss. When the selling price is greater than the cost, we get a profit.
Examples
Example 2
Sandy bought a stove for \$238 and made a profit of \$39.
What was the sale price of the item?
Worked Solution
Create a strategy
We can calculate sale price using the formula:\text{Profit} = \text{Sale price} - \text{Cost price}
Apply the idea
| \displaystyle 39 | \displaystyle = | \displaystyle \text{Sale price} - 238 | Substitute the values |
| \displaystyle \text{Sale price} | \displaystyle = | \displaystyle 39 + 238 | Add 238 to both sides |
| \displaystyle = | \displaystyle \$277 | Evaluate |
Idea summary
\text{Profit} = \text{Sale price} - \text{Cost price}
\text{Loss} = \text{Cost price} - \text{Sale price}
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.
Examples
Example 3
Charlie bought a cake for \$220 and sold it for \$209.
Which of the following describe Charlie's change in money after buying and selling the cake?
Select all that apply.
A
Loss of 11\%
B
Loss of \$11
C
Profit of 5\%
D
Profit of \$11
E
Loss of 5\%
F
Loss of \$5
Worked Solution
Create a strategy
Compare the size of the cost and sale price.
Apply the idea
Since the cost price was greater than the sale price, the change in money will be a loss.
| \displaystyle \text{Loss} | \displaystyle = | \displaystyle \text{Cost price} - \text{Sale price} | Use the formula |
| \displaystyle = | \displaystyle 220 - 209 | Substitute the values | |
| \displaystyle = | \displaystyle \$11 | Evaluate the subtraction |
| \displaystyle \text{Percentage loss} | \displaystyle = | \displaystyle \frac{\text{loss}}{\text{cost price}}\times 100\% | Find the percentage loss |
| \displaystyle = | \displaystyle \frac{11}{220}\times 100\% | Substitute the values | |
| \displaystyle = | \displaystyle \frac{110}{22}\% | Evaluate | |
| \displaystyle = | \displaystyle 5\% | Simplify |
So he lost \$11 which is equal to 5\%. Options B and E are correct.
Idea summary
To find the percentage loss or profit, we generally divide the loss or profit by the cost price and multiply by 100\%:
\text{Percentage profit}=\dfrac{\text{profit}}{\text{cost price}}\times 100\%
\text{Percentage loss}=\dfrac{\text{loss}}{\text{cost price}}\times 100\%
If we are specifically told to find the loss or profit as a percentage of the sale price, then we replace the cost price in the denominator with the sale price.
