Percentages
Percentage represents part of or fraction of the whole. It is a way to describe a number as a fraction with denominator 100. Percent implies “for every hundred” and is denoted by the symbol ‘%’.
This concept is developed to facilitate easier comparison of fractions by equalizing the denominators of all fractions to hundred.
To write a fraction or decimal as a percentage, convert it to an equivalent fraction with a denominator of 100.
For example, if a student scored 20 marks out of 50, then his percentage can be calculated as follows:
Using simple unitary method, we get,
More examples:
0.52 = 0.52 
100 = 52%
We can use percentages to understand many situations in our daily lives. For example, the percentage of students that score a first class in an exam and the percentage increase in the cost of a product.
If in a class, 15 out of 25 students are girls, then it is equivalent to saying that the proportion of girls is 15/25 which is equivalent to 60%. In ratios, we say that the ratio of the number of girls to boys = 3 : 2
Questions on percentage can be asked in 3 different ways. Let’s find the value of x in the following cases:
Case 1: x is 15% of 70
x = 10.5
Case 2: 484 is 40% of x
Case 3: x% of 85 is 15
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An agent sells goods of value of Rs. 15,000.
[SNAP 2009]
(1) Rs. 1875(2) Rs. 2125(3) Rs. 2000(4) Rs. 2700
Solution: Value of goods = Rs. 15,000
= 1875 Hence, option 1. |
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Student A scores 30 out of 75 marks. Student B scores 25 out of 60. Who performed better?
Solution:
Hence, student B performed better. |
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Sonali invests 15% of her monthly salary in insurance policies. She spends 55% of her monthly salary in shopping and on household expenses. She saves the remaining amount of Rs. 12,750. What is Sonali’s monthly income? [SNAP 2009]
(1) Rs. 42,500(2) Rs. 38,800(3) Rs. 40,000(4) Rs. 35,500
Solution: Percentage of Sonali’s salary invested in insurance policies = 15% Percentage of salary spent in shopping and household expenses = 55%
Let Sonali’s monthly salary be x
Hence, option 1. |
ALTERNATIVE METHOD FOR CALCULATING PERCENTAGES
This is an alternative method of finding the percentage when a ratio is given.
The basic idea of this method is to remove certain percentages of the denominator from the numerator preferably multiples of 10. This can be better understood when illustrated using an example.
This can be written as:
Now we know that 10% of 4626 = 462.6.
Also, 400
6 = 2400
462.6
6 = 2775.6
60% of 4626 = 2775.6
(By now, you know that the required
percentage lies in the range 160% 
170%. Depending on the accuracy required, you could either have stopped here, or
move on.)
Now, 462.6/4 = 115.65
∴ 115.65 = 2.5% of 4626
(You now know that it lies between 162%
163%. For most of the questions, you could have stopped here.)
0.1% of 4626 = 4.626
We can see that,
2
4.626 > 8.75
Also, 1.5
4.626 = 6.939
6.939 = 0.15% of 4626
This method gives you the liberty of stopping calculation as soon as the required accuracy is arrived at.
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If Roshini scored 71 marks out of 93 marks in mathematics, then what was her percentage score?
Solution: We know that, 10% of 93 = 9.3 Also, 9.3
Now, 9.3
1% of 93 = 0.93 1/3% of 93 = 0.93/3 =0.31
Hence, Roshini scored 76.3% in mathematics. |
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Percentages are often used to indicate changes in the quantity.
For example, if the cost of an article changes from 6 to 8, the percentage change is given by,
If a quantity increases by a%, then its value gets multiplied by (100 + a)/100
Similarly, if a quantity decreases by a% then its value gets multiplied by (100 – a)/100
For example,
If there is a 25% increase on an article
worth Rs. 464, its new price = 464
1.25 = Rs. 580
If there is a 25% decrease on an article
worth Rs. 464, its new price = 464
0.75 = Rs. 348
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If the sales of a company grew from Rs. 200 Crores to Rs. 450 Crores, then what is the growth registered by the company in the given time period?
Solution:
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A’s salary is 20% more than B’s salary. By what percentage is B’s salary less than A’s salary?
Solution: Let B’s salary be 100. Then, A’s salary is 120.
Hence, B’s salary is 16.67% less than A’s salary. |
The absolute value change denotes the actual change that occurs in the measure of a quantity, whereas percentage change is the absolute change with respect to the measure of the original quantity (unless otherwise stated).
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If the cost of a product increases from Rs. 500 in 2000 to Rs. 750 in 2001, then calculate the absolute value change and the percentage change of the product between the two years.
Solution: The absolute value change = |Final Value – Original Value| = 750 – 500 = Rs. 250
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Consider the following example: The interest rate of a bank increased from 11% in 2003 to 12.5% in 2004. In such a case,
The percentage point change from ‘03 to ‘04 = 12.5% – 11% = 1.5%
- CHANGES IN THE NUMERATOR
The numerator is directly proportional to the value of the ratio. In fact, the percentage change in the value of the numerator is equal to the percentage change in the value of the ratio.
For example, if the value of an item increases from 20/10 to 30/10, then the percentage change in the value of the numerator is (30 – 20)/20 = 50%. Also, the percentage change in the value of the ratio is
(3 – 2)/2 = 50%.
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A student took a certain entrance test and scored 180 marks on his first attempt and 250 marks on his second attempt. What was the percentage change of his marks between the two attempts? Assume that the total marks remain the same for both the attempts.
Solution: Let the total amount of marks for the test be X. Hence, the student scored 180/X in his first attempt and 250/X in his second attempt. Since the denominator doesn’t change, the percentage change will be equal to the change in the numerator. Hence, Percentage Change
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If there is threefold increase in all the sides of a cyclic quadrilateral, then the percentage increase in its area will be: [IIFT 2009]
(1) 81%(2) 9%(3) 900%(4) None of the above Solution: A square is a cyclic quadrilateral. So we can find the answer using a square. The area of a square with side x is x2 The area of a square with side 3x is 9x2
Hence, option 4. |
- CHANGES IN THE DENOMINATOR
The denominator is inversely proportional to the value of the ratio; i.e. if the value of the denominator increases, then that of the ratio decreases and vice versa.
For example, consider that the value of an
item decreased from 20/10 to 20/12 (i.e. the denominator has increased by 20%).
The value of the ratio has decreased by 16.67%. That is, a 20% increase in the
value of the denominator has resulted in a 16.67% decrease in the value of the
ratio. This property can also be used in product consistency. So, if a product
(a
b) is to remain constant, and a has increased by 20%, then
b has to decrease by 16.67%.
A real-life example could be that if the price of a commodity increases by 20%, then in order for the expenditure (which is the product of the price and consumption) to remain constant, the percentage reduction in the consumption should be 16.67%.
In general, if the price of a commodity increases by a%, then the percentage reduction in the consumption, so that the expenditure remains the same is:
Similarly, if the price of a commodity decreases by b%, then the percentage increase in consumption, so that the expenditure remains the same is:
Explanation:
Let the initial price and consumption of the commodity be p and v respectively.
Initial expenditure = pv
After an increase of a%, its new price becomes
p
(100 + a)/100.
Let the consumption after the price increase be v’.
Since expenditure remains same
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Wheat is now being sold at Rs. 27 per kg. During last month its cost was Rs. 24 per kg. Find by how much per cent a family reduces its consumption so as to keep the expenditure fixed. [SNAP 2009]
(1) 10.2%(2) 12.1%(3) 12.3%(4) 11.1
Solution: Assume the family consumes 1 kg usually. To keep expenditure at Rs. 24, its new
Hence, option 4. |
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The price of a commodity decreases by 20%. By what percentage should the quantity increase so as to keep the revenue constant?
Solution:
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Sumaiya generally spent Rs. 800 for buying a month’s provision of potatoes. However, lower yield this year caused the cost of one kilogram of potatoes to be increased by 60%. Owing to this, Sumaiya had to buy 30 kg less potatoes than usual. What was the cost of potatoes this year?
Solution: The cost of potatoes increased by 60%.
This percentage decrease in consumption is equivalent to a decrease of 30 kg.
(i.e. this year), she will buy 80 – 30 = 50 kg
Since her expenditure is Rs. 800, the cost of potatoes this year = 800/50 = Rs. 16 |
- CHANGES IN BOTH THE NUMERATOR AND DENOMINATOR
If the numerator increases and the denominator decreases, then it will be clear that the ratio will increase. Similarly, if the numerator decreases and the denominator increases, then it is apparent that the ratio will decrease.
On the other hand, if both the numerator and denominator simultaneously increase/decrease, then it is not quite so apparent how the ratio will change.
We have already seen how individual changes in the numerator and denominator affect the ratio. Now, we will try to combine these concepts together.
Consider that the numerator increases by a% and the denominator increases by b%. Since the numerator increases by a%, we can conclude that the ratio will also increase by a%. Also, since the denominator increases by b%, we can conclude that the ratio will decrease by b/(100 + b)%.
Applying successive percentage changes of a% (increase) and b/(100 + b)% (decrease) will give us the required net percentage change (Or you could use the method shown in the section
CALCULATING THE PERCENTAGE CHANGE IN A
QUANTITY (A
B), WHEN BOTH A AND B CHANGE, (which appears later in
this lesson).
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A student took a certain entrance exam and scored 180 out of 250 marks in the first attempt. However, during the second attempt, the total marks were increased to 330, of which the student scored 250. Find the percentage change in the percentage marks of the student in the two attempts.
Solution: There are several ways to solve this; however, we will be solving it using the technique shown above in order to better explain the concept. The numerator increases by (250 – 180)/180
Hence, the ratio will increase by a%. The denominator increases by (330 – 250)/250 Hence, the ratio will decrease by 32/(100 + 32) = 24.24% (= b%)
Applying the formula for successive changes, we get,
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The numerator of the ratio M/N is increased by 20% and its denominator is increased by 25%. Then, the numerator of the new ratio formed is increased 10% to get the fraction P/Q. What is the net percentage change between the ratios M/N and P/Q?
Solution: An increase of 25% in the denominator results in a
decrease of 25/125
An increase of 10% in the numerator results in an increase of 10% in the ratio. Hence,
Hence, Net Percentage change = 105.6 – 100 = 5.6% increase |
B), WHEN BOTH A AND B
CHANGE
Earlier we saw how to find the percentage
change in one of the variables (say b), when the other (say a)
changes and the product (a
b) remains constant. Now, we will see how to calculate the percentage
change in (a
b) when both a and b change.
Let the original value of an item be
A = a
b. This changes to B = x
y in the next year. So, to find the percentage change between A and
B, we will find the percentage changes between a and x (say
p%) and b and y (say q%). So,
(Here, p and q will be negative if there is a percentage decrease.)
We will now assume A to be 100. We will then calculate B’s value (when A = 100) by applying successive percentage changes of p and q to A (i.e. 100). Using the values obtained for A and B, we can then find the percentage change between the two.
Consider an example. Let A = 10
12 and B = 9
16 (Hence, the percentage change should be 24/120
100 = 20%)
Now, p = –10% and q = 33.33%
Therefore,
Hence, net percentage change = (120 – 100) = 20%
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Due to the erosion of soil from some parts of his field, a farmer considered increasing the length of his rectangular field by 25% and reducing its breadth by 12%. What will be the percentage change in the area of his plot?
Solution: Since area of a rectangle is the product of its length and breadth, we can use the above method to solve this problem.
Here, p = 25% and q = –12%. Thus, we get,
Hence, the percentage change in the area of the plot = 110 – 100 = 10% |
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Ernest won a large amount of money in a lottery. Overcome with impulse, he blew off 40% of his winnings on a brand new Ferrari and Rs. 1,20,000 on jewellery for his wife. Then, realizing that he may have been too rash with the money, he decided to buy a house, spending 30% of the remaining amount on that. He then deposited Rs. 1,00,000 in a trust for his daughter’s education. He gave away the remaining amount to various charities. If the money he gave to charities amounted to Rs. 26,000, then how much money did Ernest win in the lottery?
Solution: Let the amount he won in the lottery be Rs. X. Then,
Since, this amounted to Rs. 26,000, hence 0.42X – 1,84,000 = 26,000
Hence, Ernest won Rs. 5 lakh in the lottery.
Alternatively, It is generally easier to solve these kind of problems if you work with the given options. For example, if Rs. 6 lakhs was one of the options, then you could have proceeded as follows:
From this, you would’ve known that the required amount should be less than Rs. 6 lakh, and could have tried out other options that were lesser in value. Soon, you would have got the following:
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SUCCESSIVE PERCENTAGE CHANGES
Two successive increases on a particular value of a% and b% would be equal to a net increase of
In case of decline in growth or a discount, the value of a, b or both is negative.
In general, if there are successive increases of p%, q% and r% in 3 stages, then:
Explanation:
Let the initial value be X, and let a% and b% be two successive increments on X.
Then,
After applying an increment of a% to X, we get,
Applying an increment of b% to the above result, we get,
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A shopkeeper increases the price of his new product by 20%. He makes a loss and decreases the price by 35%. Find the total percentage change.
Solution:
Hence, the total decrease is 22%. |
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Ravi’s income has increased by 10% over the last year and will be 20% higher next year. If last year his salary was Rs. 15,000, what will it be next year?
Solution:
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At the end of year 1998, Shepard bought nine dozen goats. Henceforth, every year he added p% of the goats at the beginning of the year and sold q% of the goats at the end of the year where p > 0 and q > 0. If Shepard had nine dozen goats at the end of year 2002, after making the sales for that year, which of the following is true? [CAT 2003 Leaked Test]
(1) p = q(2) p < q(3) p > q(4) p = q/2
Solution: Shepard bought 9 dozen goats at the end of 1998. Consider that he added 1 dozen goats to it, i.e. 11.11% of 9 dozen And he sold 1 dozen to get back the same 9 dozen i.e. 10% of 10 dozen.
Hence, option 3. |
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A student took five papers in an examination, where the full marks were the same for each paper. His marks in these papers were in the proportion of 6 : 7 : 8 : 9 : 10. In all papers together, the candidate obtained 60% of the total marks. Then the number of papers in which he got more than 50% marks is [CAT 2001]
(1) 2(2) 3(3) 4(4) 5
Solution: Let the maximum marks for each of the paper = 100
Let the marks obtained by the student be 6x, 7x, 8x, 9x and 10x in each of the paper.
Hence, option 3.
Alternatively,
= 6.67
Hence, option 3. |
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A college has raised 75% of the amount it needs for a new building by receiving an average donation of Rs. 600 from the people already solicited. The people already solicited represent 60% of the people the college will ask for donations. If the college is to raise exactly the amount needed for the new building, what should be the average donation from the remaining people to be solicited? [CAT 2001]
(1) Rs 300(2) Rs 250(3) Rs 400(4) Rs 500
Solution: Let the total population be p. Then the amount already received = 0.6p This is 75% (3/4th) of the amount.
Hence, option 1. |
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Fresh grapes contain 90% water by weight while dry grapes contain 20% water by weight. What is the weight of dry grapes available from 20 kg of fresh grapes? [CAT 2001]
(1) 2 kg(2) 2.4 kg(3) 2.5 kg(4) None of these
Solution: Total weight of fresh grapes = 20 kg
In dried grapes, water is 20%.
Hence, option 3. |
= 2.5 kg
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The owner of an art shop conducts his business in the following manner: Every once in a while he raises his prices by X%, then a while later he reduces all the new prices by X%. After one such up-down cycle, the price of a painting decreased by Rs. 441. After a second up-down cycle the painting was sold for Rs. 1,944.81. What was the original price of the painting? [CAT 2001]
(1) Rs 2,756.25(2) Rs 2,256.25(3) Rs 2,500(4) Rs 2,000
Solution: As the price decreases after the first cycle, it has to decrease after the second cycle too. Also the decrease in the second cycle will be less than 441 as the original price for the second cycle is less than the original price for the first cycle.
Now we consider options.
So, options 2 and 4 are eliminated.
As the percentage change in the price in the first and second cycles is equal,
Only option 1 satisfies this. Hence, option 1.
Alternatively, Let the original price of the painting be p. After the first cycle,
After the second cycle,
Solving, p = 2756.25 Hence, option 1. |
should be equal to
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The table below shows the age-wise distribution of the population of Reposia. The number of people aged below 35 years is 400 million.
If the ratio of females to males in the ‘below 15 years’ age group is 0.96, then what is the number of females (in millions) in that age group? [CAT 2000]
(1) 82.8(2) 90.8(3) 80.0(4) 90.0
Solution: Population below 35 years of age = 30 + 17.75 + 17 = 64.75% of the total population = 400 million
The ratio of females to males in the ‘below 15 years’ age group is 0.96. i.e. if the total population is 196, then there are 96 females.
Approximately, the number of females (in millions) in the ‘below 15 years’ age group
Hence, option 2. |
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In view of the present global financial crisis, the Finance Minister decided to slash the excise duties to boost demand and propel economic growth. The excise duty on cement was reduced by 30% of its present amount to boost the spending in the infrastructure. What should be the percentage increase in the consumption of cement so that the revenue of the government remains unchanged? [FMS 2009]
Solution: The revenue of the government can be calculated as the product of the cement consumption and price of the cement. If the excise duty is reduced by 30%, then the price of the cement also reduces by 30%. If the original price of the cement is Rs. P, then the new price of the cement is Rs. 0.7P. Let the quantity of cement consumed when the price of the cement was Rs. P be Q. And when the price reduces to Rs. 0.7P, then let the quantity of cement consumed be Q1. Since revenue remains constant,
Percentage increase in quantity consumed
Hence, option 2.
Note: In general, whenever the price of commodity decreases by a%, then the percentage increase in consumption, so that the expenditure remains the same is:
Here, since the price of cement decreased by 30%, the increase in its consumption so that the revenue remains constant = 30/(100 – 30) |
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The present value of an optical instrument is Rs. 20,000. If its value will depreciate 5% in the first year, 4% in the second year and 2% in the third year, what will be its value after three years? [FMS 2009]
(1) Rs. 16,534.5(2) Rs. 16,756.5(3) Rs. 17,875.2(4) Rs. 17,556.8
Solution: The value of the object depreciates by 5% in the first year, 4% in the second year and 2% in the third year.
= 0.95 = Rs. 17,875.2 Hence, option 3. |
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BSNL offers its share at a premium of Rs. 40, whereas its par value is Rs. 160. Parul Mehra invested Rs. 50,000 in this stock. After one year, BSNL declared a dividend of 19%. What rate of interest did Ms. Mehra receive on her investment? [FMS 2009]
(1) 15.2%(2) 16.2%(3) 19%(4) 19.2%
Solution: Share premium is the excess amount of money that a company receives for a share, over its par value. BSNL offers a share of par value 160 at a premium of Rs. 40.
Dividend is the amount per share that the shareholders receive from a company. Parul receives a dividend of 19% per share. Hence her total income from dividend is given by: 0.19
= 15.2% Hence, option 1. |
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Delhi Metro Corporation engaged 25,000 workers to complete the project of IP state to Dwarka Metro Line in 4 years. At the end of the first year, 10% workers were shifted to the other projects of Delhi Metro. At the end of the second year, again 5% workers were reduced. However, the number of workers increased by 10% at the end of the third year to complete the above project in time. What was the size of work force during the fourth year? [FMS 2009] (1) 23145(2) 23131(3) 23512(4) 23513
Solution: The work force was first reduced by 10%, then by 5% and ultimately it was increased by 10% at the end of the 3rd year to complete the work on time.
= 1.1
Hence, option 4. |
