116. Percentage Change

Learning Intentions

• To know the meaning of the terms discount, mark-up, profit, loss, selling price, retail price and wholesale price
• find the new value if an amount is increased or decreased by a percentage
• To understand that percentage mark-ups and discounts correspond to increasing and decreasing a price by a percentage

Pre-requisite Summary

• Know that a percentage is a number out of $100$
• Be able to convert a percentage to a decimal or fraction
• Be able to find a percentage of a quantity
• Understand that increasing means making a value larger
• Understand that decreasing means making a value smaller
• Know that money amounts are usually written in dollars and cents
• Be able to add and subtract decimal numbers accurately

Worked Examples

Worked Example 1

State the meaning of each term:
a) discount
b) mark-up
c) profit

Worked Example 2

State the meaning of each term:
a) loss
b) selling price
c) wholesale price
d) retail price

Worked Example 3

Find the new value after an increase:
a) Increase $\mathrm{\$}80$ by $15\mathrm{\%}$
b) Increase $\mathrm{\$}240$ by $10\mathrm{\%}$

Worked Example 4

Find the new value after a decrease:
a) Decrease $\mathrm{\$}60$ by $20\mathrm{\%}$
b) Decrease $\mathrm{\$}150$ by $5\mathrm{\%}$

Worked Example 5

A shop buys an item for $\mathrm{\$}40$ wholesale and adds a $25\mathrm{\%}$ mark-up.
a) Find the mark-up amount
b) Find the selling price

Worked Example 6

A jacket has a retail price of $\mathrm{\$}120$ and is sold at a discount of $30\mathrm{\%}$.
a) Find the discount amount
b) Find the new selling price

Problems

Problem 1

State the meaning of each term:
a) discount
b) mark-up
c) profit

Problem 2

State the meaning of each term:
a) loss
b) selling price
c) wholesale price
d) retail price

Problem 3

Find the new value after an increase:
a) Increase $\mathrm{\$}90$ by $20\mathrm{\%}$
b) Increase $\mathrm{\$}300$ by $5\mathrm{\%}$

Problem 4

Find the new value after a decrease:
a) Decrease $\mathrm{\$}70$ by $10\mathrm{\%}$
b) Decrease $\mathrm{\$}200$ by $15\mathrm{\%}$

Problem 5

A shop buys an item for $\mathrm{\$}50$ wholesale and adds a $30\mathrm{\%}$ mark-up.
a) Find the mark-up amount
b) Find the selling price

Problem 6

A pair of shoes has a retail price of $\mathrm{\$}140$ and is sold at a discount of $25\mathrm{\%}$.
a) Find the discount amount
b) Find the new selling price

Exercises

Understanding and Fluency

1. Match each term to its meaning:
   a) discount
   b) mark-up
   c) profit
   d) loss

2. State whether each situation describes a discount, mark-up, profit or loss:
   a) a store reduces the price of a bike by $20\mathrm{\%}$
   b) a shop increases the price of a toy by $15\mathrm{\%}$
   c) an item is sold for more than it cost
   d) an item is sold for less than it cost

3. Find the new value after each increase:
   a) $\mathrm{\$}50$ increased by $10\mathrm{\%}$
   b) $\mathrm{\$}120$ increased by $25\mathrm{\%}$
   c) $\mathrm{\$}400$ increased by $5\mathrm{\%}$

4. Find the new value after each decrease:
   a) $\mathrm{\$}90$ decreased by $20\mathrm{\%}$
   b) $\mathrm{\$}250$ decreased by $12\mathrm{\%}$
   c) $\mathrm{\$}80$ decreased by $15\mathrm{\%}$

5. Find the mark-up amount and the new price:
   a) $\mathrm{\$}40$ marked up by $10\mathrm{\%}$
   b) $\mathrm{\$}75$ marked up by $20\mathrm{\%}$
   c) $\mathrm{\$}160$ marked up by $5\mathrm{\%}$

6. Find the discount amount and the new price:
   a) $\mathrm{\$}100$ discounted by $30\mathrm{\%}$
   b) $\mathrm{\$}240$ discounted by $25\mathrm{\%}$
   c) $\mathrm{\$}360$ discounted by $10\mathrm{\%}$

7. Decide whether the price has increased or decreased, then find the new value:
   a) a book costing $\mathrm{\$}24$ has a $15\mathrm{\%}$ discount
   b) a chair costing $\mathrm{\$}80$ has a $12\mathrm{\%}$ mark-up
   c) a game costing $\mathrm{\$}60$ has a $5\mathrm{\%}$ discount

8. Complete each statement:
   a) A mark-up means the price is ______ by a percentage
   b) A discount means the price is ______ by a percentage
   c) Wholesale price is usually the price paid by the ______
   d) Retail price is usually the price paid by the ______

Reasoning

9. Explain why a $20\mathrm{\%}$ discount and a $20\mathrm{\%}$ mark-up do not have the same effect on a price.

10. A student says that a $15\mathrm{\%}$ discount on $\mathrm{\$}200$ means subtracting $\mathrm{\$}15$. Explain the mistake.

11. Noah says that profit means the same as selling price. Is he correct? Explain.

12. Explain why a mark-up corresponds to increasing the original price by a percentage.

13. A student says that if the wholesale price is $\mathrm{\$}50$ and the selling price is $\mathrm{\$}60$, then the profit is $60\mathrm{\%}$. Describe the error.

Problem-solving

14. A store buys a bag for $\mathrm{\$}40$ wholesale and sells it for $\mathrm{\$}52$.
    a) Find the profit
    b) State whether this is a mark-up or a discount situation

15. A bicycle has a retail price of $\mathrm{\$}480$ and is on sale for $25\mathrm{\%}$ off. What is the sale price?

16. A toy store buys a board game for $\mathrm{\$}24$ and adds a mark-up of $50\mathrm{\%}$. What is the new selling price?

17. A jacket costs $\mathrm{\$}150$ and the store offers a $12\mathrm{\%}$ discount. How much is the discount, and what is the new price?

18. A café buys mugs for $\mathrm{\$}8$ each wholesale and sells them for $\mathrm{\$}10$ each. Find the profit on one mug.

19. A computer monitor has a wholesale price of $\mathrm{\$}220$. The shop uses a $30\mathrm{\%}$ mark-up. What is the retail price?

Potential Misunderstandings

• Students may think discount means the final price rather than the amount taken off
• Students may think mark-up means subtracting a percentage instead of adding a percentage
• Students may confuse profit with selling price
• Students may confuse wholesale price with retail price
• Students may think a percentage increase means adding the percentage number directly as dollars
• Students may think a percentage decrease means subtracting the percentage number directly as dollars
• Students may calculate the percentage amount correctly but then forget to add or subtract it from the original amount
• Students may confuse the original price with the new price when finding profit or loss
• Students may think all discounted items are sold at a loss
• Students may not recognise that mark-ups and discounts describe percentage increases and decreases in price