Percentages

Hong Kong

Stage 1 - Stage 3

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?

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.

**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%

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 maths speak for "use multiplication."

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.

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%=\$70$350×20%=$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$350−70=$280

So you could buy the tablet at store A for $\$280$$280.

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=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 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%$100−15=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.

We may also want to calculate which item is better value. To do this, we need to find a common amount to compare. Often it's easiest to find the unit price of each item, where we find the cost per unit of measurement, for example per litre, per kilogram or per item. We can also compare the amount per dollar, where we work out how much of something we would get for $\$1$$1.

For example, are you better off paying $\$10.50$$10.50 for $3$3kg of apples or $\$6.20$$6.20 for $2$2kg of apples? An easy way to compare the two options is to find the price per kilogram for each option.

$\$10.50$$10.50 for $3$3kg | $\$6.20$$6.20 for $2$2kg |

Divide by $3$3 to get the price per kilo | Divide by $2$2 to get the price per kilo |

$\$3.50$$3.50/kg | $\$3.10$$3.10/kg |

So you're better off paying $\$6.20$$6.20 for $2$2kg of apples because it's a cheaper price per kilogram.

If Bill's Brand sells $200$200 grams of cement for $\$24$$24 and Bob's Brand sells $150$150 grams of the same cement for $\$13$$13, which is better value? Let's see how much cement we would get for one dollar at each shop:

Bill's:

$\$24$$24 | $=$= | $200$200 grams | (divide both sides by $24$24) |

$\$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 | $=$= | $150$150 grams | (divide both sides by $13$13) |

$\$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.

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=\$334.30$4488.70−4154.40=$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% | $=$= | $7.447$7.447.. |

$=$= | $7.45%$7.45% |

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$100−6) 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=\$762.08$866×0.88=$762.08

**B)** Express the initial price as a percentage of the discounted price. Write your answer as a percentage correct to $2$2 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 $2$2 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 $2$2 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%$113.64−100=13.64%.

Calculate the amount per dollar to two decimal places if you can buy:

$349$349 g for $\$9$$9

$709$709 mL for $\$5$$5

$27$27 m for $\$15$$15

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

a discount of $18%$18% to its regular price of $100

a discount of $95.64 to its regular price of $546.59