‘Percent’ as the name suggests ‘per’ means every and ‘cent’ means- hundred i.e. “for every hundred”.
It is an important tool for comparison of data and information.
Example: Ana got 20 marks out of 40. Then, how much marks did she get when compared to 100?
Solution - By unitary method. In total of 40 marks Ana got 20 marks, Then, In total of 100 marks Ana would have got = 20/40 *100 (Unitary method)
= 50 marks
It is just that when expressed in terms of hundred it becomes percent.
That means to say Ana got 50%.
Concept of percentage
A percentage can only be calculated if it has base value or denominator.
In the above example the base value or denominator was 40, which means to say that Ana secured 20 marks out of 40 (which is the base value of calculation of marks secured.)
Concept of percent change = (Final value – initial value)/ initial value * 100
Note: Initial value is taken as base or ‘denominator’ while calculating percent change.
Percent point change
It is very important to understand the difference between the percent change and percent point change.
Let us understand the difference between then by a following example.
Ana secured 70 percent marks in her last term exam.
Now, when we say that her marks increased by 5 percent then this means that there is an increase of 5% marks when compared to the last marks she secured, which is 5% of 70% = 3.5%
Her new secured marks will be 70 + 3.5 = 73.5%
Now, when we say that her marks increased by 5 percentage points then this means that there is an absolute increase of 5% marks in her marks, irrespective of the the last marks she secured, which is
Her new secured marks will be 70 + 5% = 75%
Percentage point change = Difference of two percentage figures
Example 1. A report consists of 20 sheets each of 55 lines and each such line consists of 65 characters. This report is reduced onto sheets each of 65 lines such that each line consists of 70 characters. The percentage reduction in number of sheets is closest to:
No. of Characters in one line = 65
No. of characters in one sheet = No. of lines × No. of characters per line = 55 × 65
Total number of characters = No. of sheets × No. of characters in one sheet
= 20 × 55 × 65 = 71500
If the report is retyped –
New sheets have 65 lines, with 70 characters per line
No. of characters in one sheet = 65 × 70
Number of pages required,
Hence, 16 pages will be required if report is retyped.
Hence, reduction of (20 – 16) = 4 pages
% reduction is = (4/20) × 100 = 20%
Example 2. The number of votes not cast for the PNC Party increased by 25% in the National General Election over those not cast for it in the previous Assembly Polls, and the PNC Party lost by a majority twice as large as that by which it had won the Assembly Polls. If a total 2,60,000 people voted each time, how many voted for the PNC Party in the previous Assembly Polls?
Total Votes = 2,60,000
Let x voters voted against the party in the Assembly Poll.
Then votes in favour = 260000 – x
Therefore, majority of votes by which party won in previous poll = 260000– x – x = 260000 – 2x
Next year votes against the PNC party increase by 25%
So, votes against the party in general election = 1.25x
And votes polled in favour of the party = total votes – votes against = 260000 – 1.25x
Therefore, majority of votes by which party lost in general election
= 1.25x – (260000 – 1.25x) = 2.5x – 260000
It is given that, PNC Party lost by a majority twice as large as that by which it had won the Assembly Polls, Therefore
2.5x – 260000 = 2(260000 – 2x)
⇒ 2.5x – 260000 = 2 260000 – 4x
⇒ 6.5x = 3260000
⇒x = 1,20,000
Therefore, votes polled by the voters for the party in Assembly Polls for previous year
= (2,60,000 – x) = (2,60,000 – 1,20,000) = 1,40,000.
Example 3. 2/5th of the voters promise to vote for A and the rest promised to vote for B. Of these, on the last day 15% of the voters went back of their promise to vote for A and 25% of voters went back of their promise to vote for B, and A lost by 200 votes. Then, the total number of voters is:
Let x be the total number of voters
Voters promised to A = 2/5 x
Voters backed out = 15% of 2/5 x
Voters promised to B = 3/5 x
Voters backed out = 25% of 3/5 x
Total Number of votes for A = 2/5 x – 15% of 2/5 x + 25% of 3/5 x
Total Number of votes for B = 3/5 x – 25% 0f 3/5 x + 15% of 2/5x
Give Difference in votes is 200
So, there were 10000 voters.
Example 4. A person who has a certain amount with him goes to market. He can buy 50 oranges or 40 mangoes. He retains 10% of the amount for taxi fares and buys 20 mangoes and of the balance, he purchases oranges. Number of oranges he can purchase is:
The person can buy 50 oranges or 40 mangoes.
Let the price of one orange be Rs. x
Total amount the person has = Rs. 50x
40 mangoes cost 50x, So one mango costs 1.25x
10% of the total amount is retained for taxi fare = 10% of 50x = 5x
20 mangoes bought for 20 x 1.25x = 25x
Money left with the person = 50x – (Taxi fare) – (Mangoes cost)
= 50x – 5x – 25x = 20x
One Orange was for Rs. x, Therefore, 20 oranges can be bought with Rs. 20 x
Thus, the person bought 20 oranges.
Example 5. Forty per cent of the employees of a certain company are men and 75% of the men earn more than Rs. 25,000 per year. If 45% of the company’s employees earn more than Rs. 25,000 per year, what fraction of the women employed by the company earn Rs. 25,000 or less per year?
Let the total number of employees in the company be x
Then the number of men and women be 0.4x and 0.6x respectively.
75% of men earn more than Rs. 25000 => 0.75 x 0.4 x = 0.3 x
Total number of employees earning more than Rs. 25000 = 45% x = 0.45 x
Number of women earning more than Rs. 25000 =
Total employees earning more than Rs. 25000 – total number of Men earning more than Rs. 25000
= 0.45 x – 0.30 x = 0.15 x
Number of the women earning Rs. 25000 or less = 0.60 x – 0.15 x = 0.45 x
Fraction of the women employed by the company who earn Rs. 25000 or less
(0.45x/0.60x) = 45/60 = ¾