Using Percentages to Compare Proportions
Comparing Proportions :
Sometimes, we need to compare two quantities that belong to different wholes.
In such situtations, simply comparing the numbers may not give a fair result.
For example, suppose two students answered questions correctly in a test.
- Student A answered 36 questions correctly out of 45.
- Student B answered 32 questions correctly out of 40.
Although Student A answered more questions correctly, the total number of questions attempted by both students is different. Therefore, we compare their proportions by expressing each result as a percentage.
Student A answered 80 percent of the questions correctly.
Student B also answered 80 percent of the questions correctly.
Therefore, both students have performed equally well.
Thus, Percentages help us compare proportions fairly when the wholes are different.
Percentage Increase:
When a quantity becomes larger than its Original value, there is an Increase.
So, the percenatge increase tells us how much a quantity has increased in comparision with its original or base value
Percentage Increase:
Let the original value be \(P\) and the new value be \(Q\).
If \(Q > P\), then the quantity has increased.
Increase \(= Q - P\)
Percentage Increase \(= \frac{Q - P}{P} \times 100\%\)
So, \(\text{Percenatge Increase} = \text{Increase} \div \text{Original value} \times 100\%\)
Percentage Decrease:
When a quantity becomes smaller than its original value, there is a decrease.
So, the percentage decrease tells us how much of a quantity has decreased in comparision with its original or base value.
Percentage Decrease:
Let the original value be \(P\) and the new value be \(Q\).
If \(Q < P\), then the quantity has decreased.
Decrease \(= P - Q\)
Percentage Decrease \(= \frac{P - Q}{P} \times 100\%\)
So, \(\text{Percenatge Decrease} = \text{Decrease} \div \text{Original value} \times 100\%\)
Important!
- The original value is always taken as the base value for comparison.
- If the new value is greater than the original value:
- If the new value is smaller than the original value:
For Finding the Original Number,
(i) After a Percentage Increase
- If a number is increased by \(r\%\) and the new value is \(\text{N}\), then
(ii) After a Percentage Decrease
- If a number is decreased by \(r\%\) and the new value is \(\text{N}\), then