6.05 Discounts and markups
Everyone loves a great deal or sale when they're shopping! However, most products today come in different varieties and are sold at more than one store. For example, say you were going to buy a new tablet. Store A is charging $\$350$$350, with a $20%$20% discount and Store B is charging $\$335$$335 with a $15%$15% discount. How do you work out which store will give you the best price?
Discounts
Let's start by talking about discounts. Everyone will have seen stores advertising, "$25%$25% off", "pay less when you pay cash" and so on. These are all examples of discounts. A discount is basically a reduction in price (often a percent decrease), whether it be on goods or services. Businesses often use discount sales to encourage people to buy from them so it's important to be able to calculate discounts to make sure you're getting a great deal.
Regular price: The price before any discount is given (the non-sale price).
Sale price: The price after a discount is given.
Percentage discount: The amount of discount received expressed as a percentage (usually of the regular price but this may change depending on the question).
To work out the percentage discount, we use the formula:
$\text{Percentage discount}=\frac{\text{discount }}{\text{regular price}}\times100%$Percentage discount=discount regular price×100%
When working with percentages, students often subtract the percentage from the regular price. For example, if a t-shirt cost $\$30$$30 and had $20%$20% off, some students may write:
$30-20%$30−20%
But this doesn't specify what they're finding $20%$20% of and as such, can cause problems doing calculations. Remember we need to find $20%$20% of the original price, which is math speak for "use multiplication."
Calculating discounts to find sales prices
There are two methods for calculating discounts to find the sales price of item. We will look at both ways using the example of the tablet from above. You can use either method and they will both give you the same answer.
Recall that store A was charging $\$350$$350 with a $20%$20% discount.
| Method 1: | Calculate the sales price and subtract it from the regular price |
|---|---|
| Find $20%$20% of $\$350$$350 | $350\times20%=\$70$350×20%=$70 |
| Now we know the sale is for $\$70$$70 off | |
| Subtract $\$70$$70 from the original price of $\$350$$350 | $350-70=\$280$350−70=$280 |
| So the sale price of the tablet at store A is: | $\$280$$280 |
The next method can make for quick calculations once you get the hang of it. For this method you need to think of the regular price as the whole or $100%$100% of the price. The sale price can be expressed as the remaining percentage after a discount has been taken away. So if an item had a $10%$10% discount, $100-10=90$100−10=90, so the sale price would be $90%$90% of the original price. Let's use this method to calculate the sale price at store A and compare it with our result from Method 1.
| Method 2: | Calculate the sales price as a percentage of the regular price |
|---|---|
| With a $20%$20% discount what percentage remains? | $100-20=80%$100−20=80% |
| Calculate $80%$80% of the regular price. | $350\times80%=\$280$350×80%=$280 |
| Again we find the sale price of the tablet at store A is: | $\$280$$280 |
Now calculate the sale price of the tablet at store B using the method you prefer. Recall store B was offering the tablet for $\$335$$335 with a $15%$15% discount.
Did you find that the sale price of the tablet at store B to be $\$284.75$$284.75? So even though store B had a cheaper regular price, store A offered a slightly better sales price resulting in a better deal overall.
Worked examples
Question 1
At the end of the year, a television, originally advertised at $\$3800$$3800. Valentina pays $\$3610$$3610for the television. What percentage discount did she receive?
First, calculate what percentage the sale price is of the regular price.
Think: How do we calculate a percentage?
Do:
| Percentage | $=$= | $\frac{\text{Part }}{\text{Whole }}$Part Whole | $=$= | $\frac{\text{Sale Price }}{\text{Regular Price}}$Sale Price Regular Price |
| Percentage | $=$= | $\frac{\$3610}{\$3800}$$3610$3800 | $=$= | $0.95$0.95 or $95%$95% |
Think: If the sale price is $95%$95% of the regular price, what was the percent discount?
Do:
| $100%-95%$100%−95% | $=$= | $5%$5% |
The television was sold at a $5%$5% discount.
Question 2
Two stores are offering a new pair of $\$200$$200 sneakers at a discounted price. Store A is offering a $13%$13%. Store B originally offered a $6%$6% but when the shoes didn't sell they took off an additional $7%$7%. Which store is offering the shoes at a lower price?
Think: Successive discounts of $6%$6% and $7%$7% are not the same as a single $13%$13% discount. So we will have to calculate the sale price of each item before we can make a decision.
Do:
A single discount of $13%$13% of $\$200$$200 can be calculated using one of the two methods discussed above.
$200-200\times0.13=\$174$200−200×0.13=$174 or $\frac{100-13}{100\times200}=\$174$100−13100×200=$174
Now let's consider the successive discounts.
$6%$6% discount: $200-200\times0.06=\$188$200−200×0.06=$188
The $7%$7% discount will be calculated using the new amount of $\$188$$188.
$7%$7% discount: $188-188\times0.07=\$174.84$188−188×0.07=$174.84
We can see here that taking $13%$13% off of the original price provides a better deal than taking successive discounts of $6%$6% and $7%$7%.
Practice questions
Question 3
A tennis racket marked at a price of $\$90$$90 is advertised to be selling at $45%$45% off the marked price. Find the discounted price.
QUESTION 4
A TV normally sells for $\$1792.94$$1792.94, but is currently on sale.
In each of the following scenarios, calculate the percentage discount correct to two decimal places.
The TV is discounted by $\$149.50$$149.50.
The TV is on sale for $\$1428.74$$1428.74.