2.07 Repeated percentage change

In a previous lesson we looked at calculating amounts after a single percentage increase or decrease. In this lesson we take a look at successive increases and decreases and determining the overall percentage change.

Repeated percentage change can seem a bit misleading. If I wanted to buy a TV for $\$1000$$1000 that had been reduced by $20%$20% and then $50%$50% it would be easy to think you save $20%$20% of $1000$1000 ($\$200$$200) and then $50%$50% of $1000$1000 ($\$500$$500), which would be a saving of $\$700$$700 or $70%$70%. But this is not the case! 

The reduction happens successively. So at first you save the $\$200$$200. So the TV was then priced at $\$800$$800. You can then save $50%$50% of $\$800$$800 - which is $\$400$$400. So in total this is a saving of $\$600$$600 or only $60%$60%.  

 

Worked example

Example 1

A set of tools is on sale for $20%$20% off and then has a further $15%$15% trade discount applied. If the original price of the tools is $\$500$$500, what is the discounted price?

Think: This is not equivalent to a $35%$35% discount, we must apply the two discounts in series. To obtain the price after the first discount we need to multiply by $80%$80% and then to apply the second discount we need to multiply by $85%$85%. This can be done in two steps, or in one as follows:

Do:

$\text{Discounted price}$Discounted price	$=$=	$\$500\times80%\times85%$$500×80%×85%
	$=$=	$\$500\times0.8\times0.85$$500×0.8×0.85
	$=$=	$\$340$$340

Reflect: Notice we are multiplying by both changes and we can reorder a multiplication without changing the result. Hence, the order of the discounts is not important - an $80%$80% discount followed by a $20%$20% discount is the same as a $20%$20% discount followed by an $80%$80% discount.

 

Practice question

Question 1

 

Finding overall percentage change

To find the overall percentage change after applying successive percentage decreases and/or increases we could find the find the change from the original amount and then express this as a percentage of the original amount. For the example above the original price was $\$500$$500, the final discounted price was $\$340$$340, so we have a discount of $\$160$$160 which is equivalent to a percentage change of:

$\text{Percentage discount}$Percentage discount	$=$=	$\frac{\$160}{\$500}\times100%$$160$500​×100%
	$=$=	$32%$32%

So we can see a series discount of $20%$20% and $15%$15% is equivalent to a single discount of $32%$32%.

However, often we are not given an original price and asked for the overall percentage change. To find this: 

• Calculate the single multiplier equivalent to the chain of increases/decreases
• Convert the multiplier to a percentage
• Find the difference between the multiplier and $100%$100%

 

Worked example

Example 2

Find the overall percentage change equivalent to:

(a) A mark-up of $30%$30% followed by a discount of $20%$20%.

Think: To mark-up(increase) by $30%$30% we need to multiply by $1.3$1.3 and to discount by $20%$20% we need to multiply by $0.8$0.8. Let's find what single multiplier this is equivalent to:

Do:

$\text{Net price}$Net price	$=$=	$\text{Original price}\times1.3\times0.8$Original price×1.3×0.8
	$=$=	$\text{Original price}\times1.04$Original price×1.04

Our single multiplier is $1.04$1.04 which as a percentage is $104%$104%. We can see the difference between this and $100%$100% is a $4%$4% increase $\left(104%-100%=4%\right)$(104%−100%=4%).

(b) A discount of $25%$25% followed by a further discount of $10%$10%.

Think: To discount by $25%$25% we need to multiply by $0.75$0.75 and to discount by $10%$10% we need to multiply by $0.9$0.9. Let's find what single multiplier this is equivalent to:

Do:

$\text{Net price}$Net price	$=$=	$\text{Original price}\times0.75\times0.9$Original price×0.75×0.9
	$=$=	$\text{Original price}\times0.675$Original price×0.675

Our single multiplier is $0.675$0.675 which as a percentage is $67.5%$67.5%. We can see the difference between this and $100%$100% is a $32.5%$32.5% decrease $\left(100%-67.5%=32.5%\right)$(100%−67.5%=32.5%).

 

Practice questions

Question 2

Question 3

 

Finding original amount

The process of finding the final (net) amount after a series of percentage increases/decreases can be reversed through division to find the original amount. 

 

Worked example

Example 3

A computer is on sale for $\$957$$957 after being discounted by $25%$25% and then a further $12%$12%, what was the original selling price of the computer?

Think: To obtain the sale(net) price the original price would need to be multiplied by $0.75$0.75 to discount by $25%$25% and also multiplied by $0.88$0.88 to discount by $12%$12%. Set this up as an equation and then undo the multiplications by division.

Do:

$\text{Net price}$Net price	$=$=	$\text{original price}\times0.75\times0.88$original price×0.75×0.88
$\$957$$957	$=$=	$\text{original price}\times0.75\times0.88$original price×0.75×0.88	Substitute in sale price.
$\therefore\text{original price}$∴original price	$=$=	$\$957\div0.75\div0.88$$957÷0.75÷0.88	Find the original price by dividing both sides by the discount multipliers.
	$=$=	$\$1450$$1450

The original selling price of the computer was $\$1450$$1450.

 

Practice questions

Question 4

Question 5