Percent difference formula finds the change of any quantity in terms of percentage. It is the ratio of the absolute difference of the initial and final value divided by their average and multiplied by 100.
Properties:
- It calculates the change of the value in terms of 100.
- It measures the absolute value of the difference between two quantities
- It is always a positive quantity.
- It is a dimensionless quantity.
Formula
Percentage difference = Absolute difference/Average × 100
Derivation:
Lets, initial value be a and the final value is b.
Then average value is (a + b) / 2
The difference stands as (b – a).
The absolute value of this difference = |b – a|
So, the ratio of the above two is |b – a| / ((a + b) / 2) = 2|b – a| / (b + a)
So, the percentage difference stands as 2|b – a| / (b + a) × 100
Thus, the formula stands as:
Percentage difference = 2|b – a| / (b + a) × 100
Sample Problems
Question 1. Find the percentage difference of 20 and 40.
Solution:
The average value is equal to (20 + 40) / 2 = 60/2 = 30
The absolute value of difference = |40 – 20| = 20
Thus, percentage difference = 20 / 30 × 100 = 66.67 %
Question 2. If the initial value of the speed of a car is 30 km/hr and the final value is 60 km/hr, find the value of the percentage difference of speed of the car.
Solution:
The average value of the speed = (30 + 60)/2 = 45 km/hr
The absolute value of the difference of the speeds = |60 – 30| = 30 km/hr
Thus, the value of the percentage difference = 30 / 45 × 100 = 66.67%
Question 3. Find the other value if one value is 70 and the percentage difference is 50%.
Solution:
Let’s another value be ‘a’.
Given the other value is 70 and the percentage difference is 50%.
The average value stands as (a + 70) / 2.
The absolute value of the difference = |70 – a|
The percentage difference is |70 – a| / (a + 70)/2 = 2|70 – a| / (a + 70) × 100
Thus, we can write,
2|70 – a| / (a + 70) × 100 = 50
|70 – a| / (a + 70) = 0.25
There are two cases possible:
Case 1: If a > 70,
Then, |70 – a| = a – 70
So, we can write:,
(a – 70) / (a + 70) = 0.25
a – 70 = 0.25a + 70 × 0.25 = 0.25a + 17.5
a – 0.25a = 70 + 17.5
0.75a = 87.5
a = 116.67
Case 2: If a < 70,
Then, |70 – a| = 70 – a
(70 – a) / (a + 70) = 0.25
70 – a = 0.25a + 70 × 0.25 = 0.25a + 17.5
a + 0.25a = 70 – 17.5
1.25a = 52.5
a = 42
Question 4. If a person wrote 45 as 54 by mistake, find the value of the percentage difference.
Solution:
The average value of the two numbers = (45 + 54)/2 = 99/2 = 49.5
The absolute value of the difference of the numbers = |45 – 54| = 9
Thus, the value of the percentage difference = 9/49.5 × 100 = 18.18 %