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# Best Buys and Discounts II

Lesson Everyone loves a great deal or a 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 device. 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 shops advertising, "$25%$25% off," "pay less when you pay cash" and so on. These are all examples of shops offering discounts. A discount is basically a reduction in price, whether it be on goods or services. Businesses often use discount sales as a way of enticing people so it's important to be able to calculate discounts to make sure you're getting a great deal.

### Definitions

Regular price: The price before any discount is given (i.e. 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%

### A common mistake

When working with percentages, students often subtract the percentage from the regular price. For example, if a t-shirt cost $\$30$$30 and had 20%20% off, some students may write: 30-20%3020% But this doesn't specify what they're finding 20%20% of and as such, can cause problems doing calculations. Remember we need to find 20%20% of the original price, which is maths speak for "use multiplication." ### Calculating discounts to find sales prices There are two ways of calculating discounts to find the sales price of item. We will look at both way using the tablet example above. You can use either method and they should both give you the same answer. • ##### Calculate the discount, then subtract it from the regular price Store A was charging \350$$350 with a $20%$20% discount. So to calculate the discount, we need to find $20%$20% of $\$350$$350: 350\times20%=\70350×20%=70 Then we can take the discount amount of \70$$70 away from the regular price to give us the sale price at store A:

$350-70=\$280$35070=$280

So you could buy the tablet at store A for $\$280$$280. • ##### Calculate the sales price as a percentage of the regular price This way seems a bit theoretical at first but can make for quick calculations once you get the hang of it. The regular price represents the whole or 100%100% of the price. A sale price can be expressed as the remaining percentage after a discount has been taken away. So if an item had a 10%10% discount, 100-10=9010010=90, so the sale price would be 90%90% of the original price. Let's use this method to calculate the sale price at store B, which was offering the tablet for \335$$335 with a $15%$15% discount.

So firstly, how much of the regular price would remain after a $15%$15% discount?

$100-15=85%$10015=85%

So now we can calculate $85%$85% of the regular price to give us the sale price.

$335\times85%=\$284.75$335×85%=$284.75

So even though store B had a cheaper regular price, store A offered a slightly better sales price. Switch calculation methods between the stores and you will see that you get the same answer regardless of what method you use.

### Amount per dollar

If Bill's Brand sells $200$200 grams of cement for $\$24$$24 and Bob's Brand sells 150150 grams of the same cement for \13$$13, which is better value? Let's see how much cement we would get for one dollar at each shop:

Bill's:

 $\$24$$24 == 200200 grams (divide both sides by 2424) \1$$1 $=$= $8.33$8.33... grams

So at Bill's, we'd get approximately $8.3$8.3 grams of cement for a dollar.

Bob's:

 $\$13$$13 == 150150 grams (divide both sides by 1313) \1$$1 $=$= $11.53$11.53... grams

We get approximately $11.5$11.5 grams of cement at Bob's, which is much better value than at Bill's.

#### More examples

##### Question 1

At a growers' market, durians are sold at stand A for $80$80 cents per kilogram. At stand B, each $4.5$4.5kg durian is sold for $\$4.00$$4.00. A) First, calculate the price of 4.54.5kg of durian from store A. Give your answer correct to two decimal places. Think: How do we change the price per kilo to work out the price for 4.54.5kg? Do: 4.5\times0.8=\3.604.5×0.8=3.60 B) Which store is the best buy? Think: In other words, at which store would you pay less for 4.54.5kg of durian? Do: Choose Store A, as you pay less for a 4,5kg in this store. C) What should the owner of store C charge for his 4.54.5kg durians if he wants to beat the best buy by 6%6%. Give your answer correct to two decimal places. Think: This means he wants to charge 94% of the lowest price. Do:  0.94\times3.600.94×3.60 == 3.3843.384 == \3.38$$3.38 (to $2$2 d.p.)

The owner of store C should charge $\$3.38$$3.38 for his 4.54.5kg durians. ##### Question 2 At the end of the financial year, a television, originally advertised at \3800$$3800, is discounted by $5%$5%. Valentina negotiates a further discount for paying cash, which brings the price down to $\$3285.10$$3285.10. What further percentage discount did she receive for paying cash? A) Firstly, calculate the price Valentina would have paid after receiving only the first discount. Think: How do we calculate a 5%5% discount? Do:  3800\times5%3800×5% == \190$$190 $3800-190$3800−190 $=$= $\$3610$$3610 Valentina would have paid \3610$$3610.

B) Hence calculate the percentage value of the second discount.

Think: What is the difference between the prices after the first discount and the second discount? Convert this difference into a percentage of the first reduced price.

Do:

 $3610-3285.10$3610−3285.10 $=$= $\$324.90$$324.90 \frac{324.90}{3610}\times100%324.903610​×100% == 9%9% ##### Question 3 Which offers a greater discount: a single discount of 13%13% or successive discounts of 6%6% and 11%11%? Think: Successive discounts of 6%6% and 11%11% are NOT the same as a single 17%17% discount. Let's think about this theoretically by letting the original amount be xx. In other words, xx would be 100%100%. Do: A single discount of 13%13% of xx can be expressed as a decimal: \left(1-0.13\right)x=0.87x(10.13)x=0.87x In other words, it leaves 87%87% of the original amount. Now let's consider the successive discounts: \text{6% discount}=0.94x6% discount=0.94x The 11%11% discount will be calculated on this new amount: 0.89\times0.94x=0.8366x0.89×0.94x=0.8366x In other words, these successive discounts leave about 83.66%83.66% of the original amount, so this option offers a greater discount. #### More Worked Examples ##### QUESTION 4 A TV normally sells for \1792.94$$1792.94, but is currently on sale.

In each of the following scenarios, calculate the percentage discount correct to two decimal places.

1. The TV is discounted by $\$149.50$$149.50. 2. The TV is on sale for \1428.74$$1428.74.

##### QUESTION 5

The price of a heater selling for $\$234234 is initially discounted by $14%$14% and later marked up by $14%$14%.

1. Choose the expression that correctly represents the final sales price of the heater.

$234-14%+14%$23414%+14%

A

$234\times\left(\left(-14\right)%\right)\times14%$234×((14)%)×14%

B

$234\times86%\times114%$234×86%×114%

C

$234\div86%\times114%$234÷​86%×114%

D
2. What is the final sales price to the nearest cent?