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# Best Buys and Discounts I

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 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.

#### Examples

##### Question 1

A TV normally sells for $\$4488.70$$4488.70. If it is on sale, calculate the discount as a percentage, to two decimal places, of the normal price when: A) the discount is \180.60$$180.60

Think: How do we express the discount as a percentage of the original price?

Do:

 $\frac{180.60}{4480.70}\times100%$180.604480.70​×100% $=$= $4.023$4.023 $=$= $4.02%$4.02%

B) the sale price is $\$4154.40$$4154.40 Think: Firstly, we need to determine the value of the discount (in dollars), then we can use it to calculate the percentage discount. Do: 4488.70-4154.40=\334.304488.704154.40=334.30 (this is the discount amount. Now let's calculate it as a percentage of the regular price).  \frac{334.30}{4488.70}\times100%334.304488.70​×100% == 7.4477.447.. == 7.45%7.45% ##### Question 2 There is a 6%6% off sale in store. Calculate the regular price, to the nearest cent, of an item that sells for \9778$$9778.

Think: The sale price is $94%$94% ($100-6$1006) of the original price. Let the original price be $x$x.

Do:

 $0.94x$0.94x $=$= $9778$9778 $x$x $=$= $9778\div0.94$9778÷​0.94 $x$x $=$= $\$10402.13$$10402.13 The original price was \10402.13$$10402.13.

A washing machine initially selling for $\$866$$866 is discounted by 12%12%. A)What is the discounted price? Think: A 12%12% discount leaves 88%88% of the original amount. Do: 866\times0.88=\762.08866×0.88=762.08 B) Express the initial price as a percentage of the discounted price. Write your answer as a percentage correct to 22 decimal places. Think: How would we express these amounts as a percentage? Do: \frac{866}{762.08}\times100%=113.64%866762.08×100%=113.64% (to 22 d.p.) C) Hence what is the percentage increase in price needed to restore the discounted price back to its original? Give your answer as a percentage correct to 22 decimal places. Think: What percentage represents the initial price? Do: Because we are thinking in terms of discounted price back to initial price, we must think of the change as a percentage of the discounted price. We know that if the discounted price represents 100%100% then the initial price would be 113.64%113.64%. This means that the percentage increase is 113.64-100=13.64%113.64100=13.64%. #### Further Examples ##### Question 1 Calculate the amount per dollar to two decimal places if you can buy: 1. 349349 g for \9$$9

2. $709$709 mL for $\$5$$5 3. 2727 m for \15$$15

##### Question 2

Calculate the sales price of an item which is sold at:

1. a discount of $18%$18% to its regular price of $100 2. a discount of$95.64 to its regular price of \$546.59