Percentage Increase and Decrease

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5.3Percentage Increase and Decrease

In this Math tutorial, you will learn:

  • What is percentage change? How do you calculate it?
  • How to express a percentage increase/decrease?
  • What is the formula of percentage increase/decrease calculation?
  • How do you find the original value after a percentage change?
  • What is percentage distribution? Why do we use it?
  • What is the rule governing the percentage distribution in a data set?

Introduction

So far, we have discussed percentages and their analogue methods of expressing numbers (fractions & decimals). Now, we will explain in more detail how to deal with percentage changes - a very important and useful concept that has many practical applications, especially in finance and banking. Pay increases, tax cuts, and changes in interest rates - all represent percentage changes.

This tutorial is in line with the general opinion of scientists who support the idea that it is not very important to know the actual value of a quantity; rather, it is the way the quantity changes that really matters.

Percentage Change

Imagine you were paid $16/hour and got a $3/hour salary increase. Obviously, now you earn $19/hour. How do you respond to somebody who asks you what percent your salary increased by?

It is clear that you have to express the change in salary as a percentage. For this, we use the general formula

% change = change in value/original value × 100%
= ∆x/x × 100%

where x is the original value while Δx = xfinal - xinitial (initial = original) is the difference between final and initial value of the quantity involved in the study.

The change Δx may be positive or negative. If it is positive, then the final value is greater than the initial one. In this case, we have a percentage increase; otherwise, if the final value is smaller than the initial (original) one, we have a percentage decrease, We will discuss percentage increase and percentage decrease extensively in the following paragraphs.

In our example, we have an increase in salary by $3 (Δx = $3), so we don't need to know the initial and final values, as we already have the change (increase) provided. In addition, we know that the original value is x = $16. Thus, applying the formula above, yields

% change in salary = ∆x/x × 100%
= $3/$16 × 100%
= 0.1875 × 100%
= 18.75%

Although $3 may seem a small amount of daily salary increase, it is actually high when considered against the original daily salary. Therefore, knowing the net increase or decrease of an amount is not always sufficient to draw a conclusion on how the situation changes. Let's consider an example to clarify this point.

Example 1

Two friends, Sam and Jack had a simultaneous increase in salary while working for two different companies. Sam had an increase of $3/hour while Jack had an increase of $4/hour. Sam was original paid by $15/hour while Jack was paid by $22/hour. Who had a higher percentage of salary increase? Write the answers at three significant figures.

Solution 1

We denote the values related to Sam by x and those related to Jack by y. Therefore, we have

x = 15,
Δx = 3,
y = 22, and
Δy = 4

Thus, applying the formula of percentage change, we obtain for the percentage of salary increase for Sam:

% change (x) = ∆x/x × 100%
= 3/15 × 100%
= 0.200 × 100%
= 20.0%

On the other hand, the percentage increase in salary for Jack is

% change (y) = ∆y/y × 100%
= 4/22 × 100%
= 0.1818 × 100%
= 18.8%

Therefore, despite Jack receiving a higher net increase in salary than Sam ($4 > $3), Jack had a higher percentage increase in salary than Sam (20.0% > 18.8%).

Percentage Increase

Now, let's focus on percentage increase and start with a mind blowing example:

Example 2

An employee has excellent performance in his work, so the boss decided to triple his salary. By how what percent did the employee's salary increase?

Solution 2

I am sure many people will respond immediately: "The employee's salary increased by 300% since the boss tripled his previous salary." Although this may seem reasonable, it is a wrong answer because the term "triples" means three times the original value. Therefore, if we express the initial salary by S, the actual salary after the increase becomes 3S. Therefore, the increase in salary, which represents the difference between the current and the previous salary is obtained through subtraction, i.e.

Increase in salary = Current salary - Previous salary
= 3S - S
= 2S

Therefore, since S represents 100% of the previous salary, we have a percentage increase by

% increase = 2 × S
= 2 × 100%
= 200%

and not 300% as you may initially have thought.

In this way, we may induce the formula of percentage increase, that is

Percentage increase = Actual value-Initial value/Initial value × 100%

If the old and the new percentage (i.e. initial and actual) of an amount are known, the percentage increase is obtained through the formula

Percentage increase = New percentage - Old percentage

or

Percentage increase = Actual percentage - Initial percentage

Example 3

The number of bacteria in a sample of contaminated water increased from 2,000 to 15,000 in one day. What is the percentage increase of the bacteria?

Solution 3

Clues:

Initial value = 2,000
Actual value = 15,000

Thus, we have

Percentage increase = Actual value - Initial value/Initial value × 100%
= 15,000 - 2,000/2,000 × 100%
= 13,000/2,000 × 100%
= 6.5 × 100%
= 650%

As we stated earlier, we can also work out the percentage increase by taking the initial value as 100% and finding the actual value in terms of the initial value as percentage (100%). Then, the initial percentage is subtracted from the actual, in order to obtain the percentage increase. Thus, in the previous example, we may express the actual number of bacteria in terms of 100% (the initial value), i.e.

2,000 bacteria = 100%
15,000 bacteria = x

Thus, applying the ratio approach, we obtain

2,000∶15,000 = 100%∶x
2,000/15,000 = 100%/x

Taking the cross product, we obtain

2,000 × x = 15,000 × 100%
2,000 × x = 1,500,000%
x = 1,500,000%/2,000
= 750%

Therefore, the percentage increase is

% increase = Actual percentage - Initial percentage
= 750% - 100%
= 650%

Percentage Decrease

As we mentioned at the beginning of this tutorial, percentage decrease is the inverse of percentage increase of an amount. Thus, to obtain the value of a given percentage decrease, we must subtract the new percentage from the old one. Mathematically, we have

Percentage decrease = Old percentage - New percentage

For example, if there is a 30% discount on the price of an item, the new price will be 70% of the original (since the original price represents the whole, which when written as a percentage gives 100%). Here, the discount represents the percentage decrease while the new price is expressed as a percentage of the original.

On the other hand, when we know the new and the old values expressed in numbers and not in percentages, we can use the formula

Percentage decrease = Initial value - Actual value/Initial value × 100%

Example 4

An item cost $125 but after a discount, it became $90. What is the percentage decrease of this item?

Solution 4

We can use two approaches to solve this exercise. The first is to express the new price as a percentage of the original (old percentage), which on the other hand represents the percentage of the whole (100%). Hence since,

New percentage = Old percentage - Percentage decrease

and giving that

New percentage = Actual value/Initial value × 100%

we obtain

Actual value/Initial value × 100% = Old percentage - Percentage decrease

Thus,

Percentage decrease = Old percentage - Actual value/Initial value × 100%
= 100% - $90/$125 × 100%
= 100% - 0.72 × 100%
= 100% - 72%
= 28%

This result can be obtained using a shorter approach, i.e. by directly using the formula

Percentage increase = Initial value - Actual value/Initial value × 100%

that yields

Percentage increase = $125 - $90/$125 × 100%
= $35/$125 × 100%
= 0.28 × 100%
= 28%

Finding the Original Value after a Percentage Change

If we have the percentage change and the new amount given, we can find the original amount by rearranging the formula

Percentage change = Actual value - Initial value/Initial value × 100%

(If the percentage change is positive, we have a percentage increase involved; otherwise, we have a percentage decrease).

Rearranging the above formula to isolate the original (initial) value, yields

Percentage change = Actual value/Initial value × 100% - Initial value/Initial value × 100%
= Actual value/Initial value × 100% - 100%

Thus,

Actual value/Initial value × 100% = 100% + Percentage change
Initial value = Actual value × 100%/100% + Percentage change

Example 5

The actual price of an item is $270 after a discount of 10%. What was the original price of the item?

Solution 5

We have the actual value = $270 and the percentage change = -10% as the discount involves a percentage decrease. Thus, applying the above formula, yields

Initial value = Actual value × 100%/100% + Percentage change
= $270 × 100%/100% + (-10)%
= $270 × 100%/90%
= $270 × 10/9
= $30 × 10
= $300

We can calculate the original value when we have a percentage increase as well. Let's use another example to illustrate this point.

Example 6

A student obtained 84 points in an exam, increasing her result by 5% compared to the previous exam. What was her previous result?

Solution 6

We have the following clues:

Actual value = 84
Percentage change = +5% (as there is a percentage increase)
Initial value = ?

Using the formula

Initial value = Actual value × 100%/100% + Percentage change

yields

Initial value = 84 × 100%/100% + 5%
= 84 × 100%/105%
= 84 × 20/21
= 4 × 20
= 80 points

Percentage Distribution

Percentage distribution tells you the number per hundred that is represented by each group in a larger whole. In other words, percentage distribution occurs when we have a data set presented as percentages, where each percentage represents different groups of the whole (100%).

It is clear that the total of all individual percentages in these data sets must be 100%. However, due to rounding made when expressing a number as a percentage of another number, we may have slight deflections from 100%, which is acceptable.

Thus, if we have four data groups A, B, C and D out of the whole, we have for the percentage distribution:

%A + %B + %C + %D = 100%

Example 7

A school has 1247 students where 134 of them (category A) receive scholarships due to their high results while 261 students (category B) receive scholarships because of low family income. The rest (category C) are full tuition students. Calculate the percentage distribution in this data set.

Solution 7

We can calculate the individual percentages of each category by applying the known formula

%(Y) = Value of Y/Total × 100%

As there are three student categories A, B and C, we obtain

%(A) = Value of A/Total × 100%
= 134/1247 × 100%
= 0.1074578 × 100%
≈ 0.1075 × 100%
= 10.75%
%(B) = Value of B/Total × 100%
= 261/1247 × 100%
= 0.2093023 × 100%
≈ 0.209 × 100%
= 20.93%

Now, we have to find the number of students N(C) included in the category C (full tuition) before calculating the percentage of the school students they represent. We have:

N(C) = N(total) - N(A) - N(B)
= 1247 - 134 - 261
= 852 students

This category makes

%(C) = Value of C/Total × 100%
= 852/1247 × 100%
= 0.68323977 × 100%
≈ 0.6832 × 100%
= 68.32%

Thus, adding the above percentages yields

Total percentage = 10.75% + 20.93% + 68.32%
= 100.00%

as it should be.

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