Markup and Markdown

A store buys an item at its **cost price** and sells it at a higher **selling price**. The extra amount is the **markup**. Later the store may lower the price: that reduction is a **markdown** (also called a **discount**), and the new price is the **sale price**. Both use a **multiplier**: $$\text{Selling price} = \text{Cost} \times (1 + p)$$ $$\text{Sale price} = \text{Original price} \times (1 - d)$$ Here $p$ is the markup rate and $d$ is the discount rate, written as decimals (25% = $0.25$). **Which price is the base?** - A markup percent is always a percent **of the cost price**. - A markdown percent is always a percent **of the original price** (the price before the sale). Two changes in a row multiply: 20% off and then 10% off gives $0.80 \times 0.90 = 0.72$, so you pay 72% and save 28%, not 30%.

Selling Price from a Markup

A sports shop buys a backpack for 40 dollars and adds a 35% markup. What is the selling price?

Find the markup amount (a percent of the cost): $40 \times 0.35 = 14$ dollars = Markup = 14 dollars

Add it to the cost price: $40 + 14 = 54$ dollars = 54 dollars

Check with the multiplier: $40 \times 1.35 = 54$ = Same answer

Answer: The backpack sells for **54 dollars**.

Sale Price after a Markdown

A jacket costs 85 dollars. During a sale it is marked down 40%. What is the sale price?

Find the discount (a percent of the original price): $85 \times 0.40 = 34$ dollars = Discount = 34 dollars

Subtract it from the original price: $85 - 34 = 51$ dollars = 51 dollars

Check with the multiplier: you pay $100\% - 40\% = 60\%$: $85 \times 0.60 = 51$ = Same answer

Answer: The sale price is **51 dollars**.

Finding the Percent of Markup

A bike shop buys a bike for 250 dollars and sells it for 340 dollars. What is the percent of markup?

Find the markup amount: $340 - 250 = 90$ dollars = Markup = 90 dollars

Divide by the cost price (the base of a markup): $\frac{90}{250} = 0.36$ = 0.36

Write it as a percent: $0.36 \times 100\% = 36\%$ = 36%

Answer: The markup is **36%** of the cost price.

Finding the Original Price

A pair of jeans is 30% off and the sale price is 63 dollars. What was the original price?

What percent of the original price do you pay?: $100\% - 30\% = 70\%$, so the multiplier is $0.70$ = Multiplier = 0.70

Write the equation: $\text{Original} \times 0.70 = 63$ = Original × 0.70 = 63

Divide by the multiplier: $\text{Original} = \frac{63}{0.70} = 90$ dollars = 90 dollars

Check: $90 \times 0.70 = 63$ = Correct

Answer: The original price was **90 dollars**. (Adding 30% of 63 to 63 gives 81.90 dollars, which is wrong, because the 30% was taken from 90, not from 63.)

Two Discounts in a Row

A coat costs 200 dollars. It is marked down 20%, and at the checkout you get an extra 10% off the reduced price. What do you pay, and what is the total discount in percent?

Apply the first discount: $200 \times 0.80 = 160$ dollars = 160 dollars

Apply the second discount to the NEW price: $160 \times 0.90 = 144$ dollars = 144 dollars

Find the total saving: $200 - 144 = 56$ dollars = 56 dollars saved

Compare it with the original price: $\frac{56}{200} = 0.28 = 28\%$ (shortcut: $0.80 \times 0.90 = 0.72$) = 28%, not 30%

Answer: You pay **144 dollars**. The total discount is **28%**, because the 10% was taken from 160, not from 200.

In the lesson Percent Increase and Decrease you learned the multipliers $1 + p$ and $1 - d$. Markup and markdown are where those multipliers meet real money: - **Shops** add a markup to cover rent, staff and profit before an item reaches the shelf. - **Sales** like 40% off are markdowns from the original price. - **Stacked coupons** (an extra 15% off at the checkout) are successive discounts, and they always save less than the two percents added together. - **Price tags** often show only the sale price, so finding the original price is a useful check. - **Small businesses** need to know whether a sale price is still above the cost price. With these skills you can check any price tag and see which deal really is better.

Mistake: Adding two discounts: 20% off and then 10% off = 30% off

Why: The second discount is taken from the already reduced price, which is smaller than the original price, so it saves less.

Correct: Multiply the multipliers: $0.80 \times 0.90 = 0.72$. You pay 72%, so the total discount is 28%.

Mistake: Finding the original price by adding the percent to the sale price ($63 + 30\%$ of $63 = 81.90$)

Why: The 30% was a percent of the original price, not of the sale price. Using the sale price as the base gives a number that is too small.

Correct: Divide by the multiplier: $63 \div 0.70 = 90$. Check: $90 \times 0.70 = 63$.

Mistake: Dividing the markup by the selling price ($\frac{90}{340} \approx 26\%$)

Why: A markup percent is measured against the cost price, the price the store paid.

Correct: Markup percent $= \frac{\text{markup}}{\text{cost}} \times 100\%$, here $\frac{90}{250} = 36\%$.

Mistake: Thinking a 25% markdown undoes a 25% markup

Why: The markdown is taken from the larger selling price, so it removes more than the markup added: $80 \times 1.25 = 100$, then $100 \times 0.75 = 75$.

Correct: The price ends 5 below the cost of 80. To get back from 100 to 80 you need a 20% markdown: $100 \times 0.80 = 80$.

Clothing Store Pricing

Fashion shops often mark up new items a lot and mark them down at the end of the season. The question is whether the sale price still covers the cost.

A shirt bought for 20 dollars with a 100% markup sells for $20 \times 2 = 40$ dollars. Half price later is 20 dollars: no profit at all.

Stacked Coupons on Sale Days

On big sale days shops often offer a discount plus an extra coupon at the checkout. The coupon is taken from the already reduced price.

50% off and then an extra 50% off is not free: $0.5 \times 0.5 = 0.25$, so you still pay 25% of the price.

Reading a Sale Tag

Outlet stores often show only the sale price and the discount. Working backwards tells you the original price and how much you really save.

A tag says 25% off, now 30 dollars. The original price was $30 \div 0.75 = 40$ dollars.

Selling price = cost price × (1 + markup rate); the markup percent is a percent of the cost price

Sale price = original price × (1 - discount rate); the markdown percent is a percent of the original price

Percent of markup = markup ÷ cost × 100%; percent of markdown = discount ÷ original price × 100%

To find the original price from a sale price, divide by the multiplier (63 ÷ 0.70 = 90), never add the percent back

Successive discounts multiply: 20% then 10% off is 0.80 × 0.90 = 0.72, a 28% discount, not 30%

A markup followed by a markdown of the same percent ends below the start: 1.25 × 0.75 = 0.9375

Q: Is markup the same as profit margin?

A: No. Markup is a percent of the cost price, margin is a percent of the selling price. An item bought for 80 and sold for 100 has a 25% markup (20 of 80) but a 20% margin (20 of 100).

Q: Can a markup be more than 100%?

A: Yes. A 100% markup doubles the cost price, and a 200% markup makes the selling price three times the cost. Some shops use markups this large on cheap items.

Q: Can a markdown be more than 100%?

A: No. A 100% markdown already makes the price 0. A store cannot take off more than the whole price.

Q: Does the order of two discounts matter?

A: No. Multiplication can be done in any order: $0.80 \times 0.90 = 0.90 \times 0.80 = 0.72$. 20% then 10% off gives the same price as 10% then 20% off.

Exit Presentation

Markup and Markdown

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Decimals & Percents / Percentages

Markup and Markdown

Learn how stores set prices with a markup on the cost price and lower them with a markdown, and how to find the percent, the original price and the result of stacked discounts.

Intermediate25 min

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