Finding the Better Deal: Discounts and Coupons | Grade 7 Math | LevelKite

A store puts two offers on the same jacket. One offer is 20% off. The other is a coupon for 15 dollars off. You may use only one. Which offer saves you more money?

You can already find a sale price from a percent discount. In this lesson, you use that skill to compare offers. A few terms will help:

The original price is the price before any offer is used.
A percent discount takes away a percent of the original price.
A coupon, in this lesson, takes away a fixed number of dollars.
The final price is what you pay after one offer is used.
The better deal is the offer with the lower final price for the same item.

Every money amount in this lesson is in dollars.

Start with what each number means. A percent is a rate per 100 of a stated whole. The whole is the amount the percent is taken of, and it must be positive. For a discount, the whole is the item's original price. So 20% off means that 20 dollars come off for every 100 dollars of original price.

That is why 20% off is worth a different number of dollars at different original prices. On an item with an original price of 50 dollars, 20% off takes away 10 dollars. On an item with an original price of 200 dollars, it takes away 40 dollars. Here is that arithmetic:

\frac{20}{100} \times 50 = 10 \qquad \frac{20}{100} \times 200 = 40

A coupon does not change that way. A coupon for 15 dollars off takes away 15 dollars from any original price of 15 dollars or more.

So the 20 in one offer and the 15 in the other are in different units. You cannot tell which offer takes away more until the percent has a price to work on. The table tries both offers at four original prices.

20% off or a 15-dollar coupon at four original prices

Original price (dollars)	20% off takes away (dollars)	Final price with 20% off (dollars)	Final price with coupon (dollars)	Better deal
$40$	$\frac{20}{100} \times 40 = 8$	$40 - 8 = 32$	$40 - 15 = 25$	Coupon
$60$	$\frac{20}{100} \times 60 = 12$	$60 - 12 = 48$	$60 - 15 = 45$	Coupon
$100$	$\frac{20}{100} \times 100 = 20$	$100 - 20 = 80$	$100 - 15 = 85$	20% off
$150$	$\frac{20}{100} \times 150 = 30$	$150 - 30 = 120$	$150 - 15 = 135$	20% off

Read across each row. The coupon takes away 15 dollars every time. The 20% discount takes away more dollars as the original price gets larger, because it is always 20 dollars per 100 dollars of that price.

In each row, both offers start from the same original price, so the offer that takes away more dollars leaves the lower final price. At 40 and 60 dollars, that is the coupon. At 100 and 150 dollars, it is the 20% discount. The same two offers trade places.

So the jacket question has no answer until you know the jacket's original price. Somewhere between 60 and 100 dollars, the better deal flips. You will find the exact price in the last example.

Key concept

Compare final prices, not the numbers in the offers

A percent is a rate per 100 of a stated positive whole. For a discount, that whole is the item's original price, not a sale price. So the same percent discount is worth a different number of dollars at each original price. A coupon is already in dollars.

To compare offers on the same item, find each offer's final price in dollars. The formula gives each final price, using these quantities:

$P$ is the original price in dollars. It must be positive.
$r$ is the percent taken off. It is more than $0$ and at most $100$ .
$c$ is the coupon amount in dollars. It is more than $0$ and at most $P$ .

Use each offer's own original price. The lower final price is the better deal. Equal final prices mean a tie.

\text{discount: } P - \frac{r}{100} \times P \qquad \text{coupon: } P - c

Worked example

One item, two offers

A backpack has an original price of 60 dollars. Offer A takes 25% off. Offer B is a coupon for 12 dollars off. You may use only one offer. Which offer is the better deal, and by how many dollars?

Name the whole. Both offers start from the same original price, 60 dollars, which is positive. The 25% in Offer A is a rate per 100 of that original price, so you can turn it into dollars.
Find the dollars Offer A takes away: 25 per 100 of the 60-dollar original price.
$\frac{25}{100} \times 60 = \frac{1500}{100} = 15$
Subtract that amount from the original price, because a discount comes off the original price. The result is Offer A's final price.
$60 - 15 = 45$
Offer B is already in dollars. Subtract the coupon from the same original price to get Offer B's final price.
$60 - 12 = 48$
Compare the two final prices. Both are in dollars for the same backpack, so the smaller one is the better deal. Subtract to see how far apart they are.
$45 < 48, \qquad 48 - 45 = 3$

Answer

Offer A, the 25% discount, is the better deal. You pay 45 dollars instead of 48 dollars, so Offer A costs 3 dollars less.

Worked example

Two stores, two original prices

Two stores sell the same headphones. Store A has an original price of 80 dollars and takes 25% off. Store B has an original price of 70 dollars and takes 20% off. Which store has the better deal?

Each percent is a rate per 100 of its own store's original price, and both prices are positive. For Store A, find 25 per 100 of 80 dollars, then subtract it from 80.
$80 - \frac{25}{100} \times 80 = 80 - 20 = 60$
For Store B, find 20 per 100 of 70 dollars, then subtract it from 70.
$70 - \frac{20}{100} \times 70 = 70 - 14 = 56$
Compare the final prices, because the final price is what you pay for the same headphones. Subtract to find the difference.
$56 < 60, \qquad 60 - 56 = 4$
Notice what did not decide it. Store A has the bigger percent (25 against 20) and takes away more dollars (20 against 14), but it starts from a higher original price. When original prices differ, a bigger percent or a bigger saving does not guarantee a lower final price.

Answer

Store B has the better deal. You pay 56 dollars there and 60 dollars at Store A, so Store B costs 4 dollars less, even though its percent discount is smaller.

Worked example

Find where the better deal flips

Go back to the two offers on the jacket: 20% off, or a coupon for 15 dollars off. Suppose the jacket's original price is at least 15 dollars. At which original price do the two offers tie, and which offer is the better deal on each side of that price?

Define the variable. Let P be the original price in dollars, with P at least 15 so the coupon never takes away more than the price. The 20% discount is 20 per 100 of P. The coupon is a flat 15 dollars. Write the dollars each offer takes away.
$\text{discount: } \frac{20}{100} \times P \qquad \text{coupon: } 15$
Both offers start from the same P. So their final prices are equal exactly when they take away the same number of dollars. Write that as an equation.
$\frac{20}{100} \times P = 15$
Multiply both sides by 100 to clear the fraction. Doing the same operation to both sides keeps them equal. On the left, 100 times 20/100 is 20. On the right, 100 times 15 is 1500.
$20 \times P = 1500$
Divide both sides by 20, which is allowed because 20 is not zero. This leaves P by itself.
$P = 1500 \div 20 = 75$
Check the tie at 75 dollars. The discount takes away 20 per 100 of 75, which is 15 dollars, the same as the coupon. Either way, the final price is 60 dollars. This fits the table, where the better deal flipped between 60 and 100 dollars.
$\frac{20}{100} \times 75 = 15, \qquad 75 - 15 = 60$
Now look at each side of 75. Multiplying by the positive number 20/100 gives a larger result when P is larger. So above 75 the discount takes away more than 15 dollars, and below 75 it takes away less. From the same P, the offer that takes away more leaves the lower final price. Test one price on each side.
$\frac{20}{100} \times 100 = 20 > 15, \qquad \frac{20}{100} \times 40 = 8 < 15$

Answer

The offers tie at an original price of 75 dollars, where each final price is 60 dollars. Above 75 dollars, the 20% discount is the better deal. For original prices of at least 15 dollars but less than 75 dollars, the coupon is the better deal.

Here is the idea to keep. A percent discount and a coupon are in different units, so the bigger number is not always the better deal. A percent discount is a rate per 100 of the positive original price, and it turns into dollars only when you multiply it by that price.

To find the better deal on an item, use each offer's own original price, work out each final price in dollars, and choose the lower one. Change the original price, and the better deal can change too.

Next, you will put several percent steps together in multi-step percent problems, and you will practice finding and fixing mistakes in percent work.