Middle Years

Lesson

When buying products, two things that are important to understand are mark ups and discounts.

Mark up

A mark up is a percentage increase in the price of a product.

Discount

A discount is a percentage decrease in the price of a product.

A mark up indicates that the price of the product has increased from its original price and the percentage of the mark up indicates by how much. For example: if the price of a jumper has been marked up by $20%$20%, then its price has been increased by $20%$20%.

Similarly, the percentage of a discount indicates how much the price of a product has decreased from its original price.

Since mark ups and discounts are simply percentage increases and decreases, we can calculate them using the same methods.

An encyclopedia priced at $\$110$$110 is marked up by $40%$40%. What is the marked up price of the encyclopedia?

**Think:** We are given that the original price of the encyclopedia is $\$110$$110. Since it has been marked up, we are calculating a percentage increase. The amount that we are increasing by is $40%$40%.

**Do:** To increase the original price by $40%$40%, we multiply it by $140%$140%. This gives us:

Marked up price = $140%\times110$140%×110

Evaluating the expression tells us that the marked up price is $\$154$$154.

A pair of boots priced at $\$80$$80 is discounted by $35%$35%. What is the discounted price of the pair of boots?

Suppose a video game was marked up by $20%$20%, later discounted by $25%$25% and finally discounted again by $40%$40%. If the original price of the video game was $\$50$$50, what is its current price?

If we calculate the changing price of the video game through each mark up and discount we find that:

- The video game starts at the price of $\$50$$50.
- After a mark up of $20%$20%, its price is $\$50\times120%$$50×120% which is equal to $\$60$$60.
- After a discount of $25%$25%, its price is $\$60\times75%$$60×75% which is equal to $\$45$$45.
- After a final discount of $40%$40%, its price is $\$45\times60%$$45×60% which is equal to $\$27$$27.

As such, we find that the current price of the game is $\$27$$27.

However, this is a lot of steps. Notice that the only operation we applied to the price was multiplication. Since multiplication does not need to be applied one at a time, we can reach the same answer by calculating:

Current price = $\$50\times120%\times75%\times60%$$50×120%×75%×60%

The advantages of this approach are that we can cancel out factors if we convert the percentages into fractions, or simply use a calculator to evaluate the expression in one step. Since the order in which we multiply doesn't matter either, we can also choose which percentage changes we want to apply first to make calculations easier.

Caution

Mark ups and discounts of the same percentage do not cancel each other out.

For example: if an item is marked up by $10%$10% and then discounted by $10%$10%, its final price will be:

Final Price = Original Price $\times$× $110%\times90%$110%×90%

Evaluating the left hand side, we get:

Final Price = Original Price $\times$× $99%$99%

As we can see, the final price and the original price are not the same.

A hula hoop priced at $\$10$$10 is marked up by $40%$40% and then later discounted by $40%$40%.

Which of these expressions is equal to the new price?

$10\times140%\div140%$10×140%÷140%

A$10\times40%\div40%$10×40%÷40%

B$10\times140%\times60%$10×140%×60%

C$10\times140%\times140%$10×140%×140%

DCalculate the new price of the hula hoop.

Give your answer in dollars, to the nearest cent.

Which of the following is true?

If you apply a mark up of $40%$40% and then a discount of $40%$40%, the final price will always be more than the original.

AIf you apply a mark up of $40%$40% and then a discount of $40%$40%, the final price will always be less than the original.

BIf you apply a mark up of $40%$40% and then a discount of $40%$40%, the final price will always be the same as the original.

C

A bladesmith began her career selling her swords for $\$1500$$1500. After a few months she marked up the price by $16%$16%.

A few weeks after that a video of her making the swords went viral, and she marked up the price by $11%$11%.

In a few days she is due to appear on a popular podcast, and in anticipation of the increased demand she marks up the price by $13%$13%.

What is the price of the bladesmith's swords after all the mark ups?

Give your answer in dollars, to the nearest cent.