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Discounts and sale prices

Ratio, rates and money · level 4, lesson 5 of 6

A question to start

Two shops are having a sale.

  • Shop A: 30% off everything.
  • Shop B: 20 dollars off everything.

You want a jacket that costs 80 dollars at both shops. Which shop should you go to? Now your friend wants a shirt that costs 50 dollars at both shops. Which shop should she go to?

Here is the surprise: the answers are different. One offer is better for the jacket and the other is better for the shirt. There is even a price at which the two offers are exactly equal. Finding it is part of this lesson.

The idea

Words of a sale

  • The original price (or regular price) is the price before the sale.
  • The discount is the amount taken off. It may be given in dollars ("20 dollars off") or as a percent ("30% off").
  • The sale price is what you pay: original price minus discount.

A percent discount is always a percent of the original price.

Method 1: find the discount, then subtract

30% off an 80 dollar jacket:

  • Discount: 10% of 80 is 8, so 30% is 24 dollars.
  • Sale price: \(80 - 24 = 56\) dollars.

Method 2: find what you pay

If 30% is taken off, what is left? The whole price is 100%, so you pay \(100\% - 30\% = 70\%\) of it.

you pay: 70% off: 30% original price: 100%

So the sale price is 70% of 80: \(0.70 \times 80 = 56\) dollars. Same answer, one step.

In general: for \(p\%\) off, multiply by \((100 - p)\%\) as a decimal. 30% off is \(\times 0.70\); 15% off is \(\times 0.85\); 25% off is \(\times 0.75\); half price (50% off) is \(\times 0.5\).

This is the same trick as sales tax in lesson 4, where adding 13% was \(\times 1.13\). A number you multiply by to make a percent change is called a multiplier. Taking a percent off gives a multiplier less than 1; adding a percent on gives a multiplier more than 1.

Comparing a percent with an amount

Now the two shops. For the 80 dollar jacket:

  • Shop A: 30% of 80 is 24 dollars off.
  • Shop B: 20 dollars off.

Shop A saves more. For the 50 dollar shirt:

  • Shop A: 30% of 50 is 15 dollars off.
  • Shop B: 20 dollars off.

Shop B saves more. Why the switch? A percent discount grows with the price: the dearer the item, the bigger the discount. A fixed discount is the same 20 dollars whatever the price. For cheap items the fixed discount wins; for dear items the percent wins.

Where do they cross? When 30% of the price is exactly 20 dollars. 10% of that price would be a third of 20, about 6.67 dollars, so the price is about 66.67 dollars. Below that, take the 20 dollars off; above it, take the 30%.

"Buy one, get one" deals

Some deals are percents in disguise.

  • "Buy 2, get 1 free": you get 3 items for the price of 2. You pay \(\tfrac23\) of the full price, so it is \(\tfrac13\) off, about 33.3% off, but only if you wanted 3.
  • "Second one half price": two items for \(1 + \tfrac12 = 1.5\) times the price, instead of 2 times. You pay \(1.5 \div 2 = 0.75\) of the full price: 25% off the pair.

Discount and tax: does the order matter?

In a shop, the discount comes off first and then the tax is added. With multipliers, a 25% discount and 13% tax on a 120 dollar bike is

\[120 \times 0.75 \times 1.13 = 101.70 \text{ dollars}.\]

What if the tax were added first and then 25% taken off? \(120 \times 1.13 \times 0.75\) is also 101.70 dollars, because you can multiply numbers in any order. That is a nice fact to know when you check a receipt: the order does not change the total.

Reading sale signs carefully

  • "Up to 50% off" means some items are 50% off. Many may be less.
  • "Save 40%" is a percent of the original price.
  • "30% off, then take an extra 20% off" is not 50% off. The extra 20% is taken from the already reduced price, so it is a smaller amount. On a 100 dollar item: \(100 \times 0.70 = 70\), then \(70 \times 0.80 = 56\). That is 44% off, not 50%. Changes one after another are the subject of level 6.

What percent off was it?

If you know the original and sale prices, find the discount in dollars and write it as a percent of the original, as in lesson 3. A game that was 60 dollars and is now 45 dollars has 15 dollars off, and \(15 \div 60 = 0.25\), so it is 25% off.

Worked examples

1. A simple discount. Shoes cost 45 dollars. They are 15% off. What is the sale price?

  • You pay 85%: \(0.85 \times 45 = 38.25\) dollars.
  • Check by method 1: 10% is 4.50, 5% is 2.25, so 15% is 6.75. \(45 - 6.75 = 38.25\). ✓

2. Half price and a quarter off. A 64 dollar board game is 25% off. Another 64 dollar game is half price. Find both sale prices.

  • 25% off: \(0.75 \times 64 = 48\) dollars.
  • Half price: \(0.5 \times 64 = 32\) dollars.

3. Which offer? A 90 dollar coat. Shop A: 25% off. Shop B: 20 dollars off.

  • Shop A: 25% of 90 is 22.50 dollars off, so 67.50 dollars.
  • Shop B: \(90 - 20 = 70\) dollars.
  • Shop A is cheaper by 2.50 dollars.

4. A "buy 2, get 1 free" deal. Notebooks cost 6 dollars each, buy 2 get 1 free. What is the price per notebook if you buy 3?

  • You pay for 2: 12 dollars for 3 notebooks.
  • Per notebook: \(12 \div 3 = 4\) dollars, instead of 6. That is 2 dollars off each, and \(2 \div 6 \approx 0.333\), so about 33.3% off.

5. Discount, then tax. A 120 dollar bike is 25% off, and then 13% HST is added.

  • Sale price: \(0.75 \times 120 = 90\) dollars.
  • With tax: \(90 \times 1.13 = 101.70\) dollars.

6. Finding the percent off. A tablet was 250 dollars and is now 200 dollars.

  • Discount: \(250 - 200 = 50\) dollars.
  • As a percent of the original: \(50 \div 250 = 0.2\), so 20% off.

Try it

  1. Find the sale price of a 35 dollar shirt at 20% off.
  2. Find the sale price of a 65 dollar item at 40% off.
  3. Use a multiplier to find the price of a 220 dollar bike at 15% off.
  4. A hoodie was 50 dollars and is now 42 dollars. What percent off is that?
  5. For a 45 dollar item, which is better: 20% off or 10 dollars off? For a 60 dollar item? At what price are the two offers the same?
  6. A shop has "second item half price" on 30 dollar shirts. What do two shirts cost? What percent off is that for the pair?
  7. A 96 dollar coat is 25% off, and then 13% HST is added. What do you pay?
  8. Sam says: "30% off and then an extra 20% off means 50% off." Test his claim on a 100 dollar item.
Answers
  1. \(0.80 \times 35 = 28\) dollars. (20% of 35 is 7, and \(35 - 7 = 28\).)
  2. \(0.60 \times 65 = 39\) dollars.
  3. \(0.85 \times 220 = 187\) dollars.
  4. Discount: \(50 - 42 = 8\) dollars. \(8 \div 50 = 0.16\), so 16% off.
  5. On 45 dollars: 20% is 9 dollars off, so 10 dollars off is better. On 60 dollars: 20% is 12 dollars off, so 20% off is better. They are the same when 20% of the price is 10 dollars, at a price of 50 dollars (10% would be 5, so the whole is 50).
  6. \(30 + 15 = 45\) dollars instead of 60. Saving: 15 dollars, and \(15 \div 60 = 0.25\), so 25% off the pair.
  7. Sale price: \(0.75 \times 96 = 72\) dollars. With tax: \(72 \times 1.13 = 81.36\) dollars.
  8. \(100 \times 0.70 = 70\), then \(70 \times 0.80 = 56\) dollars. That is 44 dollars off, 44%, not 50%. The extra 20% was taken from 70 dollars, not from 100.

Common mistake

Taking the discount but forgetting to subtract it. 30% of 80 is 24, but the sale price is not 24 dollars; that is the discount. The sale price is \(80 - 24 = 56\). Using the multiplier 0.70 avoids this.

Adding percent discounts together. Two discounts one after another do not add up, because the second one is a percent of a smaller price. Multiply the multipliers instead.

Where it leads