Explainer: Percentage Change

In this explainer, we will learn how to apply percentage change and work out the percentage increase or decrease.

You should already be familiar with proportions and percentages.

We are here considering a quantity, called the original quantity, that undergoes a change (i.e., it either increases or decreases).

We want to express the amount of change (either the increase or the decrease) as a percentage of the original quantity. This means that we are comparing the increase or the decrease to the original quantity as we do in any proportion that compares a part to a whole. (The only difference is that the increase here is not a part of the original quantity. The rest of the mathematical comparison goes exactly the same way.)

How to Find the Percentage of Change

If we are given the original quantity and the quantity after change, we first need to find the amount by which the quantity has either increased or decreased, the increase or decrease. For this, we simply take the absolute value of the difference between the quantity after change and the original quantity: increaseordecreasequantityafterchangeoriginalquantity=||.

Then, we compare the increase or decrease to the original quantity as a percentage: percentofchangeincreaseordecreaseoriginalquantity=×100.

Combining the two steps in one gives percentofchangequantityafterchangeoriginalquantityoriginalquantity=||×100.

Let us look at the first example to check our understanding of the percentage of change.

Example 1: Increasing a Quantity by a Given Percentage

Increase 45 by 12%.

Answer

Here, the original quantity is 45, and we want to increase it by 12%. So, we need to find 12% of 45. This is given by multiplying 45 by 12%, or 12100. We find that 12% of 45 is 1210045=6109=5.4.

We have found the value of the increase. Then the final quantity is 45+5.4=50.4.

In the previous example, we have found that when we increase a quantity 𝑁 by 𝑝%, the final quantity is then 𝑁+𝑝100𝑁, which can be rewritten as 𝑁1+𝑝100. The same is true if a quantity 𝑁 is decreased by 𝑝%; it is given by 𝑁𝑝100𝑁, which can be rewritten as 𝑁1𝑝100.

Now we are going to look at examples where we are given the quantity after the change (the increase or decrease) and the percent of change, and we need to find the original quantity.

Example 2: Calculating the Percentage of Change

The costs of two different video games decreased by $9. The original cost of the first game was $49, and the original cost of the second one was $43. Which game had the greater percent of decrease?

Answer

We see that the decrease was the same in absolute value: $9. But as the original prices are different, if we express the decrease as a percentage of the original price, they will be different. Without performing the calculation, we see that the same amount ($9) is a greater fraction of the smaller original price ($43).

Our answer is, the second game had the greater percent of decrease.

Example 3: Finding the Original Quantity Knowing Its Value after a given Percentage of Change

A bottle of hand lotion is on sale for $5.01. If this price represents a 28% discount from the original price, determine the original price to the nearest cent.

Answer

Here we have the price of the bottle after the discount, which means that this price is the price after change. The bottle is on sale with a 28% discount. This means that the price has been reduced by 28%. This can be summarized with this diagram.

Therefore, the sale price ($5.01) is (10028)%=72% of the original price.

We can see on the diagram that dividing 5.01 into 72 equal shares will give the value of 1% of the original price. The original price is then 100 times this amount: 5.0172×1006.96.

We can also write an equation that describes that 72% of the original price is the sale price: 72100𝑥=5.01, where 𝑥 is the original price.

We have 0.72𝑥=5.01.

Dividing both sides of the equation by 0.72, we find that 𝑥=5.01÷0.726.96.

Therefore, the original price was $6.96.

Example 4: Finding a Whole Knowing a Decrease and the Percentage of Decrease

William was ill this year, and he missed 4.5 days of school. This lowered his annual attendance from 100% to 97.5%. How many school days are there in a year at William’s school?

Answer

William missed 4.5 days of school, and this lowered his annual attendance from 100% to 97.5%. This means that the percent decrease is 10097.5=2.5%. Therefore, 4.5 days of school are 2.5% of the total number of school days in a year. Let us write an equation that describes this with 𝑥 being the number of school days in a year: 2.5100𝑥=4.50.025𝑥=4.5.or

By dividing both sides of the equation by 0.025, we find that 𝑥=4.5÷0.025=180.

The answer is that there are 180 school days in a year in William’s school.

Example 5: Finding a Whole Knowing an Increase and the Percentage of Increase

Natalie sold her house at 116.6% of the price she had bought it for six years before, which made her a profit of $39,000. How much had Natalie bought her house for? Give your answer to the nearest thousand.

Answer

Natalie sold her house at 116.6% of the price she had bought it. She made a profit of $39,000. This means that the price increase between the time she had bought the house and the time she sold it is $39,000 and corresponds to a percent change of 116.6100=16.6%. Therefore, 39,000 is 16.6% of the price at which Natalie had bought her house. Let us write an equation that describes this, with 𝑥 being the price at which Natalie bought her house: 0.166𝑥=39,000.

By dividing both sides of the equation by 0.166, we find that 𝑥=39,000÷0.166235,000.

Our answer is that Natalie had bought her house for $235,000.

In the last example, we are going to look at a particular situation where a quantity is increased and then decreased by the same percentage.

Example 6: Finding a Final Price after a Decrease and an Increase by the Same Percentage

An electronics store had a one-day sale of 14%. Given that, on the following day, the sale price was increased by 14%, determine, to the nearest cent, the price of a pair of headphones that originally cost $216.

Answer

In this question, the original price of a pair of headphones ($216) underwent two changes: first it decreased by 14%, and then the new price increased by 14%.

After the first change, the price was 21614100×216=216×0.86185.76.

The new price, $185.76, is then increased by 14%. The final price is then 185.76×1.14211.77.

The final price of the pair of headphones is $211.77.

Key Points

  1. When a quantity 𝑁 is increased by 𝑝% (𝑝 is the percent of change), the final quantity is then 𝑁1+𝑝100.
  2. When a quantity 𝑁 is decreased by 𝑝%, it is given by 𝑁1𝑝100.
  3. The percent of change, 𝑝, that corresponds to a quantity 𝑁 changing to 𝑁nal is given by 𝑝=𝑁𝑁𝑁100.nal
  4. The sign of 𝑝 indicates whether the change is an increase (𝑝 is positive) or a decrease (𝑝 is negative).

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