117. Percentage Change, Profit, Loss and Error

Learning Intentions

• To know the meaning of the terms percentage change, percentage profit, percentage loss and percentage error
• calculate the percentage change (increase or decrease) when prices are increased or decreased

Pre-requisite Summary

• Know that a percentage is a number out of $100$
• Be able to find the increase or decrease between two values
• Understand that percentage change compares the change to the original amount
• Know the meaning of increase and decrease
• Be able to subtract decimal numbers accurately
• Be able to divide one quantity by another
• Know the meaning of profit and loss from buying and selling situations
• Understand that an error is the difference between an estimated or measured value and the actual value

Worked Examples

Worked Example 1

State the meaning of each term:
a) percentage change
b) percentage profit
c) percentage loss
d) percentage error

Worked Example 2

Find the percentage increase:
a) a price increases from $\mathrm{\$}40$ to $\mathrm{\$}50$
b) a price increases from $\mathrm{\$}80$ to $\mathrm{\$}92$

Worked Example 3

Find the percentage decrease:
a) a price decreases from $\mathrm{\$}60$ to $\mathrm{\$}45$
b) a price decreases from $\mathrm{\$}120$ to $\mathrm{\$}90$

Worked Example 4

Find the percentage profit or percentage loss:
a) cost price $\mathrm{\$}30$, selling price $\mathrm{\$}36$
b) cost price $\mathrm{\$}50$, selling price $\mathrm{\$}40$

Worked Example 5

Find the percentage error:
a) actual value $20$, measured value $19$
b) actual value $80$, measured value $84$

Worked Example 6

A store changes prices. Find the percentage change and state whether it is an increase or decrease:
a) from $\mathrm{\$}25$ to $\mathrm{\$}30$
b) from $\mathrm{\$}200$ to $\mathrm{\$}170$
c) from $\mathrm{\$}48$ to $\mathrm{\$}54$

Problems

Problem 1

State the meaning of each term:
a) percentage change
b) percentage profit
c) percentage loss
d) percentage error

Problem 2

Find the percentage increase:
a) a price increases from $\mathrm{\$}50$ to $\mathrm{\$}65$
b) a price increases from $\mathrm{\$}90$ to $\mathrm{\$}108$

Problem 3

Find the percentage decrease:
a) a price decreases from $\mathrm{\$}70$ to $\mathrm{\$}56$
b) a price decreases from $\mathrm{\$}150$ to $\mathrm{\$}120$

Problem 4

Find the percentage profit or percentage loss:
a) cost price $\mathrm{\$}40$, selling price $\mathrm{\$}48$
b) cost price $\mathrm{\$}60$, selling price $\mathrm{\$}51$

Problem 5

Find the percentage error:
a) actual value $25$, measured value $27$
b) actual value $100$, measured value $96$

Problem 6

A store changes prices. Find the percentage change and state whether it is an increase or decrease:
a) from $\mathrm{\$}32$ to $\mathrm{\$}40$
b) from $\mathrm{\$}90$ to $\mathrm{\$}72$
c) from $\mathrm{\$}75$ to $\mathrm{\$}81$

Exercises

Understanding and Fluency

1. State the meaning of each term:
   a) percentage change
   b) percentage profit
   c) percentage loss
   d) percentage error

2. Find the increase or decrease in each situation first, then find the percentage change:
   a) from $\mathrm{\$}20$ to $\mathrm{\$}25$
   b) from $\mathrm{\$}45$ to $\mathrm{\$}36$
   c) from $\mathrm{\$}80$ to $\mathrm{\$}100$

3. Find the percentage increase:
   a) from $\mathrm{\$}30$ to $\mathrm{\$}36$
   b) from $\mathrm{\$}120$ to $\mathrm{\$}150$
   c) from $\mathrm{\$}200$ to $\mathrm{\$}230$

4. Find the percentage decrease:
   a) from $\mathrm{\$}50$ to $\mathrm{\$}40$
   b) from $\mathrm{\$}90$ to $\mathrm{\$}72$
   c) from $\mathrm{\$}160$ to $\mathrm{\$}120$

5. Find the percentage profit or percentage loss:
   a) cost price $\mathrm{\$}25$, selling price $\mathrm{\$}30$
   b) cost price $\mathrm{\$}80$, selling price $\mathrm{\$}68$
   c) cost price $\mathrm{\$}150$, selling price $\mathrm{\$}180$

6. Find the percentage profit or percentage loss:
   a) cost price $\mathrm{\$}40$, selling price $\mathrm{\$}50$
   b) cost price $\mathrm{\$}70$, selling price $\mathrm{\$}56$
   c) cost price $\mathrm{\$}90$, selling price $\mathrm{\$}99$

7. Find the percentage error:
   a) actual $40$, measured $38$
   b) actual $75$, measured $78$
   c) actual $120$, measured $114$

8. A price changes. State whether it is an increase or decrease, then find the percentage change:
   a) from $\mathrm{\$}24$ to $\mathrm{\$}30$
   b) from $\mathrm{\$}200$ to $\mathrm{\$}150$
   c) from $\mathrm{\$}48$ to $\mathrm{\$}60$
   d) from $\mathrm{\$}84$ to $\mathrm{\$}63$

Reasoning

9. Explain why percentage change is calculated using the original value, not the new value.

10. A student says that when a price changes from $\mathrm{\$}40$ to $\mathrm{\$}50$, the percentage change is ${\displaystyle \frac{10}{50}}\times 100\mathrm{\%}=20\mathrm{\%}$. Explain the mistake.

11. Noah says that percentage profit is found by dividing the profit by the selling price. Is he correct? Explain.

12. Explain why a decrease from $\mathrm{\$}100$ to $\mathrm{\$}80$ is a $20\mathrm{\%}$ decrease.

13. A student says that if the measured value is greater than the actual value, then the percentage error must be negative. Describe the error.

Problem-solving

14. A shirt was marked up from $\mathrm{\$}40$ to $\mathrm{\$}52$. Find the percentage increase.

15. A bicycle was on sale, reduced from $\mathrm{\$}360$ to $\mathrm{\$}288$. Find the percentage decrease.

16. A shop bought a lamp for $\mathrm{\$}25$ and sold it for $\mathrm{\$}30$. Find the profit and the percentage profit.

17. A shop bought a game for $\mathrm{\$}80$ and sold it for $\mathrm{\$}68$. Find the loss and the percentage loss.

18. A piece of wood has an actual length of $2.5$ m, but it is measured as $2.4$ m. Find the percentage error.

19. The price of a phone increased from $\mathrm{\$}600$ to $\mathrm{\$}690$, then later decreased to $\mathrm{\$}621$.
    a) Find the first percentage change
    b) Find the second percentage change

Potential Misunderstandings

• Students may think percentage change compares the change to the new value instead of the original value
• Students may confuse the amount of change with the percentage change
• Students may forget to decide first whether the change is an increase or a decrease
• Students may calculate the difference correctly but divide by the wrong amount
• Students may confuse percentage profit with the actual profit amount
• Students may confuse percentage loss with subtracting a percent sign from the loss
• Students may use the selling price instead of the cost price when finding percentage profit or percentage loss
• Students may think percentage error can be found without first finding the error amount
• Students may forget that percentage error compares the error to the actual value
• Students may assume the same dollar change always gives the same percentage change, even when the original amounts are different