What is Percentage?
Before we talk of the utility of the concept â€˜percentageâ€™, we should be clear regarding when should we read (%) as per cent and when as percentage. Whenever (%) symbol is attached to any value, known or unknown, the word which we use for symbol (%) is per cent, e.g., x% is read as x per cent or 20% is read as 20 per cent, not 20 percentage. The word percentage is used whenever (%) is not attached to any value.
This whole nomenclature can be understood with the help of the following example:
The salary of Abhishek is 20% more than the salary of Abhinav. By how much percentage is the salary of Abhinav less than the salary of Abhishek?
Let us first have some standard meaning of percentage:
This whole nomenclature can be understood with the help of the following example:
The salary of Abhishek is 20% more than the salary of Abhinav. By how much percentage is the salary of Abhinav less than the salary of Abhishek?
Let us first have some standard meaning of percentage:
- Percentage as a common platform
- Percentage as a rate measurer
Percentage as a Common Platform
The concept of percentage can be further understood with the help of the following table, which gives us the marks obtained by different students in their Class 10 exam:
Student in country |
Marks obtained |
America |
100 |
India |
25 |
China |
45 |
France |
50 |
Right now, by just having this piece of information, we cannot decide that students of which country have performed the best and the data required here is the total marks obtained in each country.
Now, suppose if we add the following data to this question, then the whole situation looks like:
Actually, here we are using percentage as a common platform to compare all the given values.
Student in country | Marks obtained | Total marks | Marks obtained/100 marks |
America | 100 | 1000 | 10% |
India | 25 | 25 | 100% |
China | 45 | 300 | 15% |
France | 50 | 100 | 50% |
Percentage as a Rate Measurer
The concept of percentage is best-suited to find out the rate of change when two or more than two quantities are attached to each other by any common relationship. In this case, we get the respective change which should be brought in the value of quantities, when the value of other quantities are changing. We will see the application of percentage as a rate measurer while doing product stability ratio.Before we move ahead, it is important to understand the basic statements used in percentage.
Basic statement 1
What is x% of y?
It can also be seen that x% of y = y% of x
For example 4.5% of 20,000 = 20,000% of 4.5
This one simple fact can be used to divide or multiply any number by 50 or 25 or so. Let us see this with the help of an example: We are trying to find out the value of 25 Ã— 32, which is nothing but 32 Ã— 100/4 = 800. Similarly if we have to divide any number by 50, we should be multiplying the number by 100 and dividing it by 2 finally.
Using this, we can see that if we have to calculate 24% of 25 (or any other calculation of similar nature), it is better to find out 25% of 24 (= 24 Ã— 25/100) = 6.
This one simple fact can be used to divide or multiply any number by 50 or 25 or so. Let us see this with the help of an example: We are trying to find out the value of 25 Ã— 32, which is nothing but 32 Ã— 100/4 = 800. Similarly if we have to divide any number by 50, we should be multiplying the number by 100 and dividing it by 2 finally.
Using this, we can see that if we have to calculate 24% of 25 (or any other calculation of similar nature), it is better to find out 25% of 24 (= 24 Ã— 25/100) = 6.
Example-1
What is 20% of 50% of 40% of 20?
Solution
Percentage means â€˜per hundredâ€™.
So, 20% of 50% of 40% of 20 =
(20/100) Ã— (50/100) Ã— (40/100) Ã— 20 = 0.8
What we can observe here is that even if we change the order of values here, the final result will be the same.
So, 20% of 50% of 40% of 20 =
(20/100) Ã— (50/100) Ã— (40/100) Ã— 20 = 0.8
Example-2
Statistics show that 20% of smokers get lung cancer and 80% of lung cancer patients are smokers. If 30% of the population smokes, then the percentage of population having lung cancer is
- 4
- 3
- 8
- 7.5
Solution
Assume that the total population is 100.
30 people smoke, and out of them 6 people will be having lung cancer. This 6 represents 80% of lung cancer patients (because they smoke).
Basic statement 2
What percentage of x is y?
(This can be easily deduced by assuming Z% of x is y and then proceeding as given in statement 1.)
Some more example of the same kind are given below:
Some more example of the same kind are given below:
- What is the percentage growth in the sales value of the year 2002-03 over the sales value of 2003-04?
- What is the percentage hike in the salary of Manoj this year compared to his salary hike last year?
Basic statement 3
Change in value
- Percentage change =
- Percentage point change â€“ It is the numerical difference between the values for which we have to calculate change.
Let us assume some values to understand the above written concept
Percentage change in the market share of Maruti over the years
Percentage point change in the market share of Maruti over the years = 48% â€“ 40% = 8%
Similarly, if we have to increase any quantity N by S%, then it is equal to and when the same quantity N is to be decreased by S%, then final quantity =
It is worth mentioning here that
So, if the final value and percentage increase or percentage decrease is given and we have to find out the initial value, then it can be done in similar way.
Using S â†’ 30%â†‘â†’ S Ã— 1.3 = 195
Example-3
My Reliance India phone bill for the month of May is Rs B. Moreover, there is a service tax of S% which is to be levied upon this value. But since they are overcharging their customers they offer a discount of D% on it. So, now I have two options to make the payment
Rs B â†’ S%â†‘â†’ D% â†“â†’ Final bill
Rs B â†’ D% â†“â†’ S%â†‘â†’ Final bill
Which option is beneficial for me if S > D?
Rs B â†’ S%â†‘â†’ D% â†“â†’ Final bill
Rs B â†’ D% â†“â†’ S%â†‘â†’ Final bill
Solution
Prima facie, it might appear that the 1st one is better than the 2nd one or the 2nd one is better than 1st one, but a close and deep inspection will reveal that final bill is same in both the cases.
It can be checked with the help of assuming values also.
B = Rs 100, S% = 20% and D% = 10%
Rs 100 â€“ (20%â†‘) âˆ’ Rs 120â†’ (10%â†“) â€“ Rs 108 (Final bill)
Rs 100 â€“ (10%â†“) âˆ’ Rs 90â†’ (20%â†‘) â€“ Rs 108 (Final bill)
So, both the values are same at the end.
Importance of base Whenever we are talking about percentage, it is important to specify what it is relative to, i.e., what is that total which corresponds to 100%. The following situation illustrates this point:
In a certain school 60% of all students are male, and 10% of all students are computer science majors. If 5% of males are computer science majors, what percentage of computer science majors are male?
Here, we are asked to calculate the ratio of male computer science majors to all computer science majors. We know that 60% of all students are male, and among hese 5% are computer science majors, so we conclude that .6 Ã— .05 = .03 or 3% of all students are male computer science majors. Dividing this by the 10% of all students that are computer science majors, we arrive at the answer: 3%/10% = .3 or 30% of all computer science majors are male.
While in QA, choosing the right denominator is often not a problem but sometimes it becomes very problematic in Data Inerpretation (DI), because we are unsure as to which value should be taken as denominator or base. However, if we go by some keywords viz., of/than/over/to, the quantity, which is attached to these keywords will be the denominator.
It can be seen through the examples also.
Example-4
A student multiplies a number x by 5 instead of dividing it by 5. What is the percentage change in the result due to this mistake?
Solution
Let us assume x = 5
So, actual result should have been 100 Ã· 5 = 20
But the result obtained = 100 Ã— 5 = 500
So, percentage change = (500 â€“ 20) Ã— 100/20 = 2400%