COURSE MODULE NAME PERCENTAGES

COURSE MODULE NAME
PERCENTAGES
Percentages
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Module Objective:
PERCENTAGES
What is percentage?
The word percent can be understood as follows:
Per cent => for every 100.
So, when percentage is calculated for any value, it means that you calculate the value for every
100 of the reference value
Why Percentage?
Percentage is a concept evolved so that there can be a uniform platform for comparison
of various things. (Since each value is taken to a common platform of 100.)
Eg: To compare three different students depending on the marks they scored we cannot directly
compare their marks until we know the maximum marks for which they took the test. But by
calculating percentages they can directly be compared with one another.
Before going deeper into the concept of percentage, let u have a look at some basics and tips for
faster calculations:
Prerequesties(Important facts and formulae):
1. Concept of Percentage:
By a certain percent, we mean that many hundredths. Thus x percent means x hundredths,
written as x%.
To express x% as a fraction: We have , x% = x/100.
Thus, 20% =20/100 =1/5; 48% =48/100 =12/25, etc.
To express a/b as a percent: We have, a/b =((a/b)*100)%.
Thus, ¼ =[(1/4)*100] = 25%; 0.6 =6/10 =3/5 =[(3/5)*100]% =60%.
2. If the price of a commodity increases by R%, then the reduction in consumption so asnot to
increase the expenditure is
[R/(100+R))*100]%.
If the price of the commodity decreases by R%, then the increase in consumption so as to
decrease the expenditure is
[(R/(100-R)*100]%.
3. Results on Population: Let the population of the town be P now and suppose it increases at the
rate of R% per annum, then :
1. Population after nyeras = P [1+(R/100)]^n.
2. Population n years ago = P /[1+(R/100)]^n.
4. Results on Depreciation: Let the present value of a machine be P. Suppose it depreciates at the
rate
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R% per annum. Then,
1. Value of the machine after n years = P[1-(R/100)]n.
2. Value of the machine n years ago = P/[1-(R/100)]n.
5. If A is R% more than B, then B is less than A by
[(R/(100+R))*100]%.
If A is R% less than B , then B is more than A by
[(R/(100-R))*100]%.
Calculation of Percentage:
Percentage = (Value / Total value) X 100
Eg: 50 is what % of 200?
Soln: Percentage = (50/200) X 100 = 25%.
Calculation of Value:
Value = (Percentage/100) X total value
Eg: What is 20% of 200?
Soln: Value = (20/100) X 200
Note: Percentage is denoted by “%”, which means “/100”.
Eg: What is the decimal notation for 35%?
Soln: 35% = 35/100 = 0.35.
For faster calculations we can convert the percentages or decimal equivalents into their
respective fraction notations.
Percentages – Fractions Conversions:
The following is a table showing the conversions of percentages and decimals into
fractions:
Percentage Decimal Fraction
10% 0.1 1/10
12.5% 0.125 1/8
16.66% 0.1666 1/6
20% 0.2 1/5
25% 0.25 1/4
30% 0.3 3/10
33.33% 0.3333 1/3
40% 0.4 2/5
50% 0.5 1/2
60% 0.6 3/5
62.5% 0.625 5/8
66.66% 0.6666 2/3
70% 0.7 7/10
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75% 0.75 3/4
80% 0.8 4/5
83.33% 0.8333 5/6
90% 0.9 9/10
100% 1.0 1
Similarly we can go for converting decimals more than 1 from the knowledge of the above
cited conversions as follows:
We know that 12.5% = 0.125 = 1/8
Then, 1.125 = [8(1)+1]/8 = 9/8 (i.e., the denominator will add to numerator once, denominator
remaining the same.
Also, 2.125 = [8(2)+1]/8 = 17/8 (here the denominator is added to numerator twice)
3.125 = [8(3)+1]/8 = 25/8 and so on.
Thus we can derive the fractions for decimals more than 1 by using those les than 1.
We will see how use of fractions will reduce the time for calculations:
Eg: What is 62.5% of 320?
Soln: Value = (5/8) X 320 (since 62.5% = 5/8)= 200.
Percent change:
A change can be of two types – an increase or a decrease.
When a value is changed from initial value to a final value,
% change = (Difference between initial and final value/initial value) X 100
Eg: If 20 changes to 40, what is the % increase?
Soln: % increase = (40-20)/20 X 100 = 100%.
Note:
If a value is doubled the percentage increase is 100.
If a value is tripled, the percentage change is 200 and so on.
Percentage Difference:
% Difference = (Difference between values/value compared with) X 100.
Eg: By what percent is 40 more than 30?
Soln: % difference = (40-30)/30 X 100 = 33.33%
(Here 40 is compared with 30. So 30 is taken as denominator)
Eg: By what % is 60 more than 30?
Soln: % difference = (60-30)/30 X 100 = 100%.
(Here is 60 is compared with 30.)
Hint: To calculate percentage difference the value that occurs after the word “than” in the
question can directly be used as the denominator in the formula.
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Important Points to Note:
When any value increases by
10%, it becomes 1.1 times of itself. (since 100+10 = 110% = 1.1)
20%, it becomes 1.2 times of itself.
36%, it becomes 1.36 times of itself.
4%, it becomes 1.04 times of itself.
Thus we can see the effects on the values due to various percentage increases.
When any value decreases by
10%, it becomes 0.9 times of itself. (Since 100-10 = 90% = 0.9)
20%, it becomes 0.8 times of itself
36%, it becomes 0.64 times of itself
4%, it becomes 0.96 times of itself.
Thus we can see the effects on a value due to various percentage decreases.
Note:
1. When a value is multiplied by a decimal more than 1 it will be increased and when
multiplied by less than 1 it will be decreased.
2. The percentage increase or decrease depends on the decimal multiplied.
Eg: When the actual value is x, find the value when it is 30% decreased.
Soln: 30% decrease => 0.7 x.
Eg: A value after an increase of 20% became 600. What is the value?
Soln: 1.2x = 600 (since 20% increase)
ð x = 500.
Eg: If 600 is decrease by 20%, what is the new value?
Soln: new value = 0.8 X 600 = 480. (Since 20% decrease)
Thus depending on the decimal we can decide the % change and vice versa.
Eg: When a value is increased by 20%, by what percent should it be reduced to get the actual
value?
Soln: (It is equivalent to 1.2 reduced to 1 and we can use % decrease formula)
% decrease = (1.2 – 1)/1.2 X 100 = 16.66%.
When a value is subjected multiple changes, the overall effect of all the changes can be obtained
by multiplying all the individual factors of the changes.
Eg: The population of a town increased by 10%, 20% and then decreased by 30%. The new
population is what % of the original?
Soln: The overall effect = 1.1 X 1.2 X 0.7 (Since 10%, 20% increase and 30% decrease)
= 0.924 = 92.4%.
Eg: Two successive discounts of 10% and 20% are equal to a single discount of ___
Soln: Discount is same as decrease of price.
So, decrease = 0.9 X 0.8 = 0.72 => 28% decrease (Since only 72% is remaining)
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SOLVED EXAMPLES:
1. A batsman scored 110 runs which included 3 boundaries and 8 sixes. What percent of his total
score did he make by running between the wickets?
A.45% B.45 5 %
11
C.54 6 %
D.55%
11
Explanation:
Number of runs made by running = 110 - (3 x 4 + 8 x 6)
= 110 - (60)
= 50.
Required percentage = 50 x 100 % = 45
5 %
110 11
2. Two students appeared at an examination. One of them secured 9 marks more than the other
and his marks was 56% of the sum of their marks. The marks obtained by them are:
A.39, 30 B.41, 32
C.42, 33 D.43, 34
Answer: Option C
Explanation:
Let their marks be (x + 9) and x.
Then, x + 9 = 56 (x + 9 + x)
100
25(x + 9) = 14(2x + 9)
3x = 99
x = 33
So, their marks are 42 and 33.
3. A fruit seller had some apples. He sells 40% apples and still has 420 apples. Originally, he
had:
A.588 apples B.600 apples
C.672 apples D.700 apples
Answer: Option D
Explanation:
Suppose originally he had x apples.
Then, (100 - 40)% of x = 420.
60 x x = 420
100
x = 420 x 100 60 = 700.
4. What percentage of numbers from 1 to 70 has 1 or 9 in the unit's digit?
Answer: Option C
Explanation:
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Clearly, the numbers which have 1 or 9 in the unit's digit, have squares that end in the digit 1.
Such numbers from 1 to 70 are 1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69.
Number of such number =14
Required percentage = 14 x 100
70 % = 20%.
5. If A = x% of y and B = y% of x, then which of the following is true?
A.A is smaller than B.
B.A is greater than B
Relationship between A and B cannot be If x is smaller than y, then A is greater than
C. D.
determined.
B.
E.None of these
Answer: Option E
Explanation:
x% of y = x x y = y x x = y% of x
100 100
A = B.
6. If 20% of a = b, then b% of 20 is the same as:
A.4% of a
B.5% of a
C.20% of a
D.None of these
Answer & Explanation
Answer: Option A
Explanation:
20
20% of a = b a = b.
100
b% of 20 = b x 20 = 20 a x 1 x 20 = 4 a = 4% of a.
100 100 100 100
7. In a certain school, 20% of students are below 8 years of age. The number of students
above 8 years of age is of the number of students of 8 years of age which is 48. What is
the total number of students in the school?
A.72 B.80
C.120 D.150
E.100
Answer & Explanation
Answer: Option E
Explanation:
Let the number of students be x. Then,
Number of students above 8 years of age = (100 - 20)% of x = 80% of x.
80% of x = 48 + 2 of 48
3
80 x = 80
100
x = 100.
8. Two numbers A and B are such that the sum of 5% of A and 4% of B is two-third of the
sum of 6% of A and 8% of B. Find the ratio of A : B.
A.2 : 3 B.1 : 1
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9.
C.3 : 4 D.4 : 3
Answer & Explanation
Answer: Option D
Explanation:
5% of A + 4% of B = 2 (6% of A + 8% of B)
3
5
A + 4 B= 2 6 A + 8 B
100 100 3 100 100
1 1 1 4 A + B= A + B
20 25 25 75
1 1 4 1
- -
20 25 A = 75 25 B
1
A = 1 B
100 75
A 100 4 = = .
B 75 3
Required ratio = 4 : 3
A student multiplied a number by 3 instead of 5 .
5 3
What is the percentage error in the calculation?
A.34%
B.44%
C.54%
D.64%
Answer & Explanation
Answer: Option D
Explanation:
Let the number be x.
Then, error = 5 x - 3 x = 16 x.
3 5 15
Error% = 16x x 3 x 100
15 5x % = 64%.
10. In an election between two candidates, one got 55% of the total valid votes, 20% of the
votes were invalid. If the total number of votes was 7500, the number of valid votes that
the other candidate got, was:
A.2700 B.2900
C.3000 D.3100
Answer & Explanation
Answer: Option A
Explanation:
Number of valid votes = 80% of 7500 = 6000.
Valid votes polled by other candidate = 45% of 6000
= 45 x 6000 = 2700.
100
11. Three candidates contested an election and received 1136, 7636 and 11628 votes respectively.
What percentage of the total votes did the winning candidate get?
A.57%
B.60%
C.65%
D.90%
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Answer & Explanation
Answer: Option A
Explanation:
Total number of votes polled = (1136 + 7636 + 11628) = 20400.
Required percentage = 11628 x 100
20400 % = 57%
12. Two tailers X and Y are paid a total of Rs. 550 per week by their employer. If X is paid 120
percent of the sum paid to Y, how much is Y paid per week?
A.Rs. 200 B.Rs. 250
C.Rs. 300
D.None of these
Answer: Option B
Explanation:
Let the sum paid to Y per week be Rs. z.
Then, z + 120% of z = 550.
z + 120 z = 550
100
11 z = 550
5
z = 550 x 5 11 = 250
13. Gauri went to the stationers and bought things worth Rs. 25, out of which 30 paise went on
sales tax on taxable purchases. If the tax rate was 6%, then what was the cost of the tax free
items?
A.Rs. 15 B.Rs. 15.70
C.Rs. 19.70 D.Rs. 20
Answer: Option C
Explanation:
Let the amount taxable purchases be Rs. x.
30
Then, 6% of x =
100
x =
= 5.
30 100 x
100 6
= 5.
Cost of tax free items = Rs. [25 - (5 + 0.30)] = Rs. 19.70
14. Rajeev buys good worth Rs. 6650. He gets a rebate of 6% on it. After getting the rebate, he
pays sales tax @ 10%. Find the amount he will have to pay for the goods.
A.Rs. 6876.10 B.Rs. 6999.20
C.Rs. 6654 D.Rs. 7000
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Answer: Option A
Explanation:
Rebate = 6% of Rs. 6650 = Rs.
6 x 6650 = Rs. 399.
100
10
Sales tax = 10% of Rs. (6650 - 399) = Rs. x 6251 = Rs. 625.10
100
Final amount = Rs. (6251 + 625.10) = Rs. 6876.10
15. The population of a town increased from 1,75,000 to 2,62,500 in a decade. The average
percent increase of population per year is:
A.4.37%
B.5%
C.6%
D.8.75%
Answer & Explanation
Answer: Option B
Explanation:
Increase in 10 years = (262500 - 175000) = 87500.
Increase% = 87500 x 100
175000 % = 50%.
Required average = 50 10 % = 5%.
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EXERCISE PROBLEMS:
1.The original price of a shirt was $20. It was decreased to $15 . What is the percent decrease of
the price of this shirt.
Solution to Problem 1:
The absolute decrease is
20 - 15 = $5
The percent decrease is the absolute decrease divided by the the original price
(part/whole).
percent decease = 5 / 20 = 0.25
Multiply and divide 0.25 to obtain percent.
percent decease = 0.25 = 0.25 * 100 / 100 = 25 / 100 = 25%
2.Mary has a monthly salary of $1200. She spends $280 per month on food. What percent of her
monthly salary does she spend on food?
Solution to Problem 2:
The part of her salary that is spent on food is $280 out of her monthly salary of
$1200
percent = part / whole = 280 / 1200 = 0.23 (rounded to 2 decimal places)
Multiply and divide 0.23 by 100 to convert in percent
percent = 0.23 * 100 / 100 = 23 / 100 = 23%
3.The price of a pair of trousers was decreased by 22% to $30. What was the original price of
the trousers?
Solution to Problem 3:
Let x be the original price and y be the absolute decrease. If the price was
decreased to $30, then
x - y = 30
y is given by
y = 22% of x = (22 / 100) * x = 0.22 x
Substitute y by 0.22 x in the equation x - y = 30 and solve for x which the original
price.
x - 0.22 x = 30
0.78 x = 30
x = $38.5
Check the solution to this problem by reducing the original price found $38.5 by
22% and see if it gives $30.
4.The price of an item changed from $120 to $100. Then later the price decreased again from
$100 to $80. Which of the two decreases was larger in percentage term?
Solution to Problem 4:
First decrease in percent
part / whole = (120 - 100) / 120 = 0.17 = 17%
Second decrease in percent
part / whole = (100 - 80) / 100 = 0.20 = 20%
The second decrease was larger in percent term. The part were the same in both
cases but the whole was smaller in the second decrease.
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5.The price of an item decreased by 20% to $200. Then later the price decreased again from
$200 to $150. What is the percent of decrease from the original price to the final price of $150?
Solution to Problem 5:
We first need to find the original price x. The first decrease gives
x - 20% x = 200
0.8 x = 200
x = 200 / 0.8 = 250
The percentage decrease fro the original price 250 to 150 is given by
part / whole = (250 - 150) / 250 = 0.4 = 40%
6.A number increases from 30 to 40 and then decreases from 40 to 30. Compare the percent of
increase from 30 to 40 and that of the decrease from 40 to 30.
Solution to Problem 6:
Percent increase from 30 to 40 is given by
(40 - 30) / 30 = 10 / 30 = 0.33 = 33% (2 significant digits)
Percent decrease from 40 to 30 is given by
(40 - 30) / 40 = 0.25 = 25%
In absolute term, the percent decrease is less than the percent increase.
7.A family had dinner in a restaurant and paid $30 for food. They also had to pay 9.5% sale tax
and 10% for the tip. How much did they pay for the dinner?
Solution to Problem 7:
They paid for food, sales tax and tip, hence
total paid = $30 + 9.5% * 30 + 10% * 30 = $35.85
8.A shop is offering discounts on shirts costing $20 each. If someone buys 2 shirts, he will be
offered a discount of 15% on the first shirt and another 10% discount on the reduced price for the
second shirt. How much would one pay for two shirts at this shop?
Solution to Problem 8:
The reduced price for the first shirt
20 - 15% * 20 = $17
The reduced price for the second shirt. The 10% discount will be on the already
reduced price, hence the price of the second shirt is given by
17 - 10% * 17 = $15.3 The total cost for the two shirts is
17 + 15.3 = $32.3
9.Smith invested $5000 for two years. For the first year, the rate of interest was 7% and the
second year it was 8.5%. How much interest did he earn at the end of the two year period?
Solution to Problem 9:
Interest at the end of the first year
7% * 5000 = $350
Interest at the end of the second year
8.5% * (5000 + 350) = $454.75
Total interest at the end of the two year period is
$350 + $454.75 = $804.75
10.Janette invested $2000 at 5% compounded annually for 5 years. How much interest did she
earn at the end of the 5 year period?
Solution to Problem 10:
At the of the first year, she has the principal plus the interest on the principal
P1 = 2000 + 5% * 2000 = 2000(1 + 5%)
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At the of the second year, she has the principal P1 plus the interest on P1
P2 = P1 + 5% * P1 = P1(1 + 5%)
Substitute P1 by 2000(1 + 5%) found above to find
P2 = 2000 * (1 + 5%) 2
Continuing with this process, it can easily be shown that a the end of the 5th year,
the principal is given by
P5 = 2000 * (1 + 5%) 5
= 2000 * (1 + 0.05) = $2552.56
The interest earned at the end of 5 years is
$2552.56 - $2000 = $552.56
11.Tom borrowed $600 at 10% per year, simple interest, for 3 years. How much did he have to
repay (principal + interest) at the end of the 3 year period?
Solution to Problem 11:
The interest to pay is given by
Interest = 600 * 10% * 3 = $180
Total to repay
600 + 180 = $780
12.Out of a world population of approximately 6.6 billion, 1.2 billion people live in the richer
countries of Europe, North America, Japan and Oceania and is growing at the rate of 0.25% per
year, while the other 5.4 billion people live in the lees developed countries and is growing at the
rate of 1.5%. What will be the world population in 5 years if we assume that these rates of
increase will stay constant for the next 5 years. (round answer to 3 significant digits)
Solution to Problem 12:
Let us first calculate the population PR in 5 years in the richer countries
PR = (1.2 + 0.25% * 1.2) = 1.2(1 + 0.25%) after one year
PR = 1.2(1 + 0.25%) + 0.25% * 1.2(1 + 0.25%)
= 1.2(1 + 0.25%) 2 after two years
Continue with the above and after 5 years, PR will be
PR = 1.2(1 + 0.25%) 5 after 5 years
Similar calculations can be used to find the population PL in less developed
countries after 5 years.
PL = 5.4(1 + 1.5%) 5 after 5 years
The world population P after 5 years will be
P = PR + PL = 1.2(1 + 0.25%) 5 + 5.4(1 + 1.5%) 5 = 7.03 billion.
13.Cassandra invested one part of her $10,000 at 7.5% per year and the other part at 8.5% per
year. Her income from the two investment was $820. How much did she invest at each rate?
Solution to Problem 13:
Let x and y be the amount invested at 7.5% and 8.5% respectively
Income = $820 = 7.5% * x + 8.5% * y
The total amount invested is also known
10,000 = x + y
Solve the system of the equations to find x and y.
x = $3000 and y = $7000
As a practice check that 7.5% of $3000 and 8.5% of $7000 gives $820.
14.The monthly salary S of a shop assistant is the sum of a fixed salary of $500 plus 5% of all
monthly sales. What should the monthly sales be so that her monthly salary reaches $1500?
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Solution to Problem 14:
Let S be the total monthly salary and x be the monthly sales, hence
S = 500 + 5% * x
Find sales x so that S = 1500, hence
1500 = 500 + 5% * x = 500 + 0.05 x
Solve for x
x = (1500 - 500) / 0.05 = $20000
15.A chemist has a 20% and a 40% acid solutions. What amount of each solution should be used
in order to make 300 ml of a 28% acid solution?
Solution to Problem 15:
Let x be the solution at 20% and y be the solution at 40%, hence
x + y = 300 ml
We now write an equation that expresses that the total acid in the final 300 ml is
equal to the sum of the amounts of acid in x and y
28% * 300 = 20% * x + 40% * y
Solve the above system of equations to find
x = 180 and y = 120
16.What percent of the total area of the circular disk is colored red?
Solution to Problem 16:
Total area of disk
Ad = pi * r 2
Angle t in radians of central angle of red sector
t = (360-120)* pi / 180 = (4/3) pi
Area of red sector
As = (1/2) t * r 2
Percentage of total area in red
P = [ (1/2) t * r 2 ] / [ pi * r 2 ]
= 4 / 6 = 66.7% (3 significant digits)
THINK: compare 66.7% to 240 / 360, why are they equal?
17.What percent of the total area of the rectangle is colored red?
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Solution to Problem 17:
Total area of rectangle
Ar = L * W
Area of triangle
At = (1/2) base * height = (1/2) [ L * (1/2) W ]
Percentage of area in red
P = (1/2) [ L * (1/2) W ] / [L*W] = 1/4
= 25%
18. In Chicago in the year 2000, there were approximately 1.053 million African Americans,
907 thousand whites (non-Hispanic), and 754 thousand Hispanics, and 181 thousand others
(other races or two or more races). What percent of Chicagoans in 2000 were of Hispanic
origin?
Solution:
% of Hispanics = # of Hispanics / Total Population
% of Hispanics = 754,000 / (1,053,000 + 907,000 + 754,000 + 181,000)
% of Hispanics = 0.26 * 100%
% of Hispanics = 26%
19. 54% of DePaul’s student body of approximately 21,000 is female. Approximate how
many females attend DePaul?
Solution:
% of female students = # of female students / 21,000
# of female students = % of female students * 21,000
# of female students = 54% of female students * 21,000
# of female students = 0.54 * 21,000
# of female students = 11,340
Percentage Change (Note: second problem is over 100%)
20. Chicago’s population grew from 2.78 million in 1990 to 2.90 million in 2000. By how
many percent did it grow?
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Solution:
Percentage Change = (new – old ) / old
Percentage Change = (2.90 – 2.78) / 2.78
Percentage Change = 0.0432
Percentage Change = 0.0432 * 100%
Percentage Change = 4.32%
21. In 1996, 94,007 tons of waste was recycled in Chicago’s blue bag recycling program. In
the year 2000, 296,363 tons were recycled. By how many percent did it increase from 1996 to
2000?
Solution:
Percentage Change = (296,363 tons – 94,007 tons) / 94,007 tons
Percentage Change = 202,356 / 94,007
Percentage Change = 2.15 * 100%
Percentage Change = 215%
Percent More Than; Times More Than
22. The life expectancy in Canada is 79.1 years; the life expectancy in the US is 76.0 years.
By how many percent is the life expectancy of Canada higher than the life expectancy in the US?
Solution:
The key word for this kind of problems is "than". Whatever is after "than" should go in the
denominator (bottom part).
How many percent more = (Canada – US) / US
How many percent more = (79.1 – 76) / 76
How many percent more = 3.1 / 76
How many percent more = 0.0407 * 100%
How many percent more = 4.07%
23. By how many percent is the life expectancy of people in the US lower than the life
expectancy in Canada?
Solution:
How many percent less = (US – Canada) / Canada
How many percent less = (76 – 79.1) / 79.1
How many percent less = –3.1 / 79.1
How many percent less = –0.0392 * 100%
How many percent less = –3.92%
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24. How many times more is the life expectancy of Canada higher than the life expectancy in
the US?
Solution:
We expect a number here!
How many times more = Canada / US
How many times more = 79.1 / 76
How many times more = 1.041
Successive Percentage Change
25. Spot prices for crude oil are rather volatile. From 1998 to 1999, spot prices for crude oil
decreased by 28%. From 1999 to 2000, they increased by 106%. What was the percentage
change over the two year period from 1998 to 2000?
Solution:
A short way to do this kind of problem is
Percentage change over two year period = (1 ± %)(1 ± %) – 1
Percentage change over two year period = (1 – 28%)(1 + 106%) – 1
Percentage change over two year period = (1 – 0.28)(1 + 1.06) – 1
Percentage change over two year period = (0.72)(2.06%) – 1
Percentage change over two year period = 1.4832 – 1
Percentage change over two year period = 0.4832 * 100%
Percentage change over two year period = 48.32%
An increase of 48.32% over the original price!
26. Suppose you are earning a salary of $1,000 per week. Your company experiences a
slowdown in earnings, and asks all workers to take a 20% pay cut. What is your salary after the
cut? Six months later, the company recovers and it offers you a 20% increase on your current
salary. How much will you be earning after the increase?
Solution:
Reverse Percentage Change
New salary = (1 – 20%)(1 + 20%) – 1
New salary = (1 – 0.20)(1 + 0.20) – 1
New salary = (0.80)(1.20) – 1
New salary = 0.96 – 1
New salary = –0.04 * 100%
New salary will have a decrease of 4%, or ($1000 – $1000 * 0.04) = $960
Percentages
17

27. According to the official 2000 census, the Hispanic population of the US in 2000 was
35,305,818. It rose by an astounding 57.9% from 1990 to 2000. What was the Hispanic
population in 1990?
Solution:
1990 → 2000
old new
old 35,305,818
If we use the percentage change formula, then
% change = (new – old) / old
57.9% = (35,305,818 – old) / old
doing some algebra
0.579 * old = 35,305,818 – old
0.579 * old + old = 35,305,818
1.579 * old = 35,305,818
old = 35,305,818 / 1.579
old = 22,359,606
28. From a news article: "In the first six months of 1996, the number of deaths [from AIDS] fell
12% to 22,000 [compared to the same period a year earlier]." How many AIDS deaths were there
during the first six months of 1995?
Solution:
1995 → 1996
old new
old 22,000
Because there was a 12% decrease, we have to use –12%
–12% = (22,000 – old) / old
–0.12 * old = 22,000 – old
–0.12 * old + old = 22,000
0.88 * old = 22,000
old = 22,000 / 0.88
old = 25,000
29. If 75% of a number is added to 75, the result is the number itself. Then the number is
Ans.300
Basic formula:
X% = x / 100
a/ b as percent : (a/b x 100) %
Answer with explanation
Take the number as x
Percentages
18

75% of x + 75 = x
75/100 + 75 = x
75 = x – 75/100 x
75 = x – ¾ x
75 = x/4
x = 300
30. Subtracting 40% of a number from the number, we get the result as 30, the number is
Ans: 50
Answer with explanation:
Consider the number as x
x – 40/100 x = 30
x - 2/5 x = 30
3/5 x = 30
3x = 150
x = 150/3
x = 50
1. If three fifth of a number is 40 more than 40% of the same number
2. If three fifth of a number is 40 more than 40% of the same number. What
is the number Ans: 200
Answer with explanation
Consider the number as x
3/5 x = 40 + 40 % of x
= 40 + 40/100x
3/5 x – 40/100 x = 40
3/5x - 2/5 x = 40
1/5 x = 40
x = 200
31.A number on subtracting 15 from it, reduces to its 80% what is 40% of that number?
Answer with explanation:
Let the number be x
x-15 = 80%x
x – 15 – 80/100 x
x – 15 = 4/5 x
x – 4/5 x = 15
x / 5 = 15
Percentages
19

Percentages
Now,
Hint:
75
40% of 75
= 40 /100 x 75
=2/5 x 15
=30
30% (75) = 80/100 x 75 = 60 ---(1)
= 75- 15 = 60 ---(2)
(1)=(2)
32. Calculation shows that an angle is 37 ½ %. The size obtained by drawing and measurement is
36%. The error percent is
Let the number be x
X – 15 = 80% x
X – 15 = 80% x
X-15 = 4/5 x
X – 4/5 = 15
X / 5 = 15
X = 75
Now,
40% of 75
=40 / 100 x 75
= 2 / 5 x 15
= 30
Hint:
30% (75) = 80/100 x 15 = 60 --- >(1)
75-15 = 60 --(2)
1=2
percentage, BU
33. Calculation shows that an angle is 37 ½ % The size obtained by drawing and measurement is
36%. The error percent is
Ans: 4
Error for 37 ½
37 ½ - 36 =75 / 2 = 36
= 75- 72 / 2 = 3/2 (=1 ½)
error for 100
= 3/2 x 2/75 x 100 = 4%
20

34. If x is 90% of Y, what percent of x is y?
ans : 111.1
Answer with explanation:
X = 90 % of y
X = 90/ 100 y
X = 9/10 y
y = 10/9 x
y / x = 10/9
required percentage = y / x x 100
= 10/9 x 100
= 1.111 x 100
= 111.1 %
35. If x % of y is the 4/5 of 80, then the value of xy is
ans : 6400
Answer with explanation:
X % of y = 4/5 x 80
X / 100 x y = 4/5 x 80
xy = 4 x 16 x 100
xy = 6400
36. Subtracting 6% of % from x is equivalent to multiplying % by how much?
Ans : 94
Answer with explanation:
Consider the number as x
X – 6/100 = xy
100 / 100 x – 6 / 100 x = xy
100-6 / 100 x = xy
94 /100 x = xy
94% of x = xy
= y = 94%
37. If 8% of x = 4% of y, then 20% of x is
ans: 10
Answer with explanation:
8% of x = 4% of y
8x /100 = 4 /100 y 8x = 4y
2x = y
x = ½ y
20% of x = 20/100 x
= 20/100 x ½ y
= 10/100 y
= 10% of y
Percentages
21

38. (X% of Y + Y % of X) = ?
Ans: 2% of xy
Answer with explanation:
X % of y + y % of X
= xy / 100 + yx / 100
= 2 x y / 100
= 2% xy
39. If 90% of A=30% of B and b=x% of A, then x is equal to
ans: 300
Answer with explanation:
90% of A = 30 % of B
90 / 100 A = 30 / 100 B
90A = 30B
9A = 3B
3A = B
B = x/100 A
3A = x / 100 A
x = 300
40.. If 70% of students in a school are boys and the number of girls is 504, the number of boys is
ans: 1176
Answer:
70% of students in a school are boys:
(100-70)% = 30% girls percentage
30% of x = 504
30/100x = 504
3/10 x = 504
x = 504 x 10 / 3
= 1680
No of boys = 70% of 1680
= 70/100 x 1680
= 1176
41. A number increased 37 ½ % gives 33. The number is
Ans: 24
(100 + 37 ½ ) % of x = 33
(100+ 75/2 ) % of x = 33
(200+75/2) % of x = 33
= 275 / 2 x 1 / 100 x = 33
x = (33 x 100 x 2 ) / 275
= 3 x 4 x 2
x = 24
Percentages
22

42. The number which when decreased by 27 ½ % given 87 is
Ans: 120
Answer
(100 – 27 ½ ) % of x = 87
200 - 55 / 2 x 1/100x = 87
145 / 2 x 100 = 87
x = 87 x 2 x 20 / 145
= 3 x 2 x20
= 120
43. 40 Quintal is what percent of 2 metric tones?
Ans: 200%
1 metric tonee= 10 quintals
2 metric tonee = 20 quintals
= (40 / 20 x 100) %
= (2 x 100) %
= 200%
44. It is known that 20% of the mangoes are rotten. If the number of rotten mangoes is 35, then
the total number of mangoes is
Ans: 175
20% x = 35
20 / 100 = 35
1/5 = 35
x = 175
45.A student has to secure 40% marks topass. He gets 178 marks and failed by 22 marks. The
maximum marks are :
40% x = 178 +22
40/100 x = 200
2/5 x = 200
x = 100 x 5/2
x = 500
46. A house owner was having his house painted. He was advised that he would require 25kg of
paint. Allowing for 15% wastage and assuming that the paint is available in 2kg cans, what
would be the cost of pain purchased, if one can cost Rs.16?
Ans: 240Rs.
Answer:
Required no of kgs 25kgs
10% 2.5
5% / 15% 1.25 / 28.75kg
1 can contains 2kg paint
Percentages
23

= 28.75
14 can 1 can
= 15 cans
= 15 x 16 = 240
1 can = 16 Rs.
47. A reduction of 12.5% in the price of a dining table brought down its price to Rs.4375, the
original price (in Rs.)of the table was:
Ans: Rs.5000
Answers:
The original price to be x
X – 12.5% of x =4375
X – 12.5/ 100 x = 4375
X – 125 / 1000 x =4375
1000-125 / 1000 x = 4375
875/ 1000 x = 4375
x = 4375 x 1000 / 875
=Rs.5000
48. Of the total amount received by Kiran, 20% was spent on purchases and 5% of the
remaining on transportation. If he is left with Rs.1520, the initial amount was
Ans: 2000
Answer:
For purchase 20% of x => 20/100 x = 1/5 x
Remaining x – 1/5 = 4/5 x
For transportation == 5 % 4/5 x
= 5/100 x 4/5 x
= x/25
Balance = 4x /5 – x / 25 = 20x / 25 – x/25 = 19/25 x
19x / 25 = 1520
x = 1520 x 25 / 19
x =2000
49. In a library 20% books are in Hindi, 50% of the remaining are in English and the remaining
9000 are in various other languages. What is the total number of books in English.
Answers:
Let the no of books be x
20% of x = 20/100 x /5
Remaining x – x /5 = 4x / 5
For English 56/ 100 x 4x / 5
2x/ 5
Remaining 4 x / 5 – 2x / 5 = 2 x / 5
For other languages
2x / 5 = 9000
2x = 45000
x = 22500
Percentages
24

Another method
Let it be 100 %
Hindi 20% (-20)/80
Eng 50% of remaining ( -40) / 40 = 9000
= 100
9000 x 100/ 40 = 22500
50. Avinash spends 30% of his income on scooter petrol, ¼ o the remaining on house rent and
the balance on food. If he spends Rs.300 on petrol, then what is the expenditure on house rent?
Ans : 175
Answer :
From the given information,
Method: 1
For scooter petrol
30x / 100 = 3x / 10
remaining x – 3x / 10 = 7x / 10
for house ¼ x 7x / 10 = 7x / 40
remaining for food -- > 7x / 10 – 7 x / 40
=> 28 x – 7x / 40 = 21x /40
Amount for petrol = 30/100 x = 30
= 1000
House rent,
7x / 40 = 7 x 1000 / 40 = 175
Method :2
Total income x
X - 30/100 x = x – 300
(in %) = (in Rs.)
30 / 100 x = 300
x = 1000
rem
¼ x 100 = 175Rs.
Percentages
25

QUESTION BANK:
1. If 20% of 40% of a = 25% of a% of b, then what is b?
a. 8/5 b. 16/25 c. 8/25 d. None
2. By what % is 200 more than 50?
a. 100 b. 200 c. 300 d. None
3. A value changes from 30 to 80. What is the percentage change?
a. 125 b. 166.66 c. 156 d. None
4. The population of a city is increased by 30% and thus became 78000. What is the original
population?
a. 76000 b. 64200 c. 60000 d. None
5. In a theatre, the number of seats is increased by 20% and the price per ticket is increased by
10% but the public response decreased by 30%. What is the net effect on the economy of the
theatre?
a.10% rise b. 7% fall c. 7% rise d. None
6. A saves 20% of his income. His income is increased by 20% and so he increased his
expenditure by 30%. What is the percentage change in his savings?
a. 20% fall b. 4% fall c. 20% rise d. 4% rise
7. The price of petrol is increased by 25%. By what percent the consumption be reduced to make
the expenditure remain the same?
a. 25% b. 33.33% c. 20% d. None
8. The side of a square is increased by 20%. The percentage change in its area is ___
a. 20% b. 44% c. 36% d. None
9. If the length of a rectangle is increased by 33.33%, by what percentage should the breadth be
reduced to make the area same?
a. 20% b. 33.33% c. 25% d. None
10. In an election between two candidates, A and B, A secured 56% of the votes and won by
48000 votes. Find the total number of votes polled if 20% of the votes were declared invalid.
a. 500000 b. 400000 c. 600000 d. None
11. A reduction of 10% in price of sugar enables a housewife to buy 5 kg more for Rs. 300/-.
Find the reduced price per kg of sugar.
a. 5/- b. 4.5/- c. 6/- d. None
12. From a 20lt solution of alt and water with 20% salt, 2lt of water is evaporated. Find the new
% concentration of salt.
a. 20% b. 23% c. 25% d. None
13. In a list of weights of candidates appearing for police selections, the weight of A is marked as
58 kg instead of 46.4 kg. Find the percentage of correction required.
a. 30 b. 20 c. 24 d. None
14. A person spends 20% of his income on rent, 20% of the rest on food, 10% of the remaining
on clothes and 10% on groceries. If he is left with Rs. 9520/- find his income.
a. 10000/- b. 15000/- c. 20000/- d. None
15. A shopkeeper offers three successive discounts of 10%, 20% and 30% to a customer. If the
actual price of the item is Rs. 10000, find the price the custome has to pay to the shopkeeper.
a. 5040/- b. 4000/- c. 6000/- d. None
16. If 10lt solution of water and alcohol containing 10% alcohol is to be made 20% alcohol
solution, find the volume of alcohol to be added.
Percentages
26

a. 1 lt b. 1.25 lt c. 1.5 lt d. 2 lt
17. A is twice B and B is 200% more than C. By what percent is A more than C?
a. 400 b. 600 c. 500 d. 200
18. In an examination, a student secures 40% and fails by 10 marks. If he scored 50%, he would
pass by 15 marks. Find the minimum marks required to pass the exam.
a. 250 b. 100 c. 110 d. 125
19. If A is 20% taller than B, by what percent is B shorter than A?
a. 20% b. 25% c. 16.66% d. None
20. The population of a town increases at a rate of 10% for every year. If the present population
is 12100, find the population two years ago.
a. 11000 b. 9800 c. 10000 d. 10120
21. A solution of salt and water contains 15% salt. If 30 lt water is evaporated from the solution
the concentration becomes 20% salt. Find the original volume of the liquid before water
evaporated.
a. 100 lt b. 120 lt c. 200 lt d. None
22. If 240 lt of oil is poured into a tank, it is still 20% empty. How much more oil is to be poured
to fill the tank?
a. 300 lt b. 60 lt c. 120 lt d. None
23. A and B were hired for the same salary. A got two 40% hikes whereas B got a 90% hike.
What is the percentage difference in the hikes thay got?
a. 16% b. 6% c. 10% d. 8%
24. The population of a town doubled every 5 years from 1960 to 1975. What is the percentage
increase in population in this period?
a. 800 b. 400 c. 700 d. 600
25. In a test of 80 questions, Jyothsna answered 75% of the first 60 questions correctly. What %
of the remaining questions she has to answer correctly so that she can secure an overall
percentage of 80 in the test?
a. 80% b. 90% c. 85% D. 95%
26.A person allows discount of 20% on the already discounted MP. What is the initial discount
given if he totally gains 20% and MP is double the CP?
a. 25% b. 20% c. Data inadequate d. none
27.If a group of 24 men working 8 hrs a day can complete building a wall in 12 days then in how
many days 16 men working 6 hrs a day can complete 3 such walls?
a. 24 b. 48 c. 72 d. none
(28-31)Read the following information and answer the questions given below it :
(i) ‘A + B’ means ‘A is the father of B’
(ii) ‘A – B’ means ‘A is the wife of B’
(iii) ‘A ´ B’ means ‘ A is the brother of B’
(iv) ‘A ¸ B’ means ‘A is the daughter of B’
28. If P ¸ R + S + Q, which of the following is true?
a. P is the daughter of Q b. P is the mother of Q
c. P is the aunt of Q d. Q is the ant of P
29. If P – R + Q, which of the following statement is true?
a.P is the mother of Q
b. P is the sister of Q
c. Q is the daughter of P d. P is the aunt of Q
30. If ‘P ´ R ¸Q’, which of the following is true?
Percentages
27

a.P is the uncle of Q b. P is the father of Q
c. P is the son of Q d. P is the brother of Q
31. If ‘P ´ R – Q’, which of the following is true?
a.P is the brother-in-law of Q
b. P is the father of Q
c. P is the brother of Q d. P is the uncle of Q
32. The average of 13 papers is 40. The average of the first 7 papers is 42 and of the last seven
papers is 35. Find the marks obtained in the 7 th paper?
a. 23 b. 38 c.19 d. None of these
33. In a mixture of 40 liters, the ratio of milk and water is 4:1. How much water much be
added to this mixture so that the ratio of milk and water becomes 2:3
a. 20 liters b. 32 liters c. 40 liters d. 30 liters
34. A shop owner sells two puppies at the same price making a profit of 20% on one and a loss
of 20% on the other. Find his loss or gain percent on the whole transaction.
a. Gain of 4% b. No profit no loss c. Loss of 10% d. Loss of 4%
35. N, at a speed of 20 km/h reaches his office 10 min late. Next time he increases his speed by 5
km/h, but is still late by 4 min. What is the distance of office from his house?
20 km b. 6 km c. 12 km d. None of these
36. 4 men and 3 women finish a job in6 days, and 5 men and 7 women can do the same job in 4
days. How long will 1 man and 1 woman take to do the work?
a. 22(2/7) days b. 19(1/2) days c. 5(1/7) days d. 12(7/22) days
37. If the code for BLUE is 240, that for TEA is 18 then the code for MATCH is _____
a. 3100 b. 3120 c. 3210 d. none
38. Two equal sums were borrowed at 8% simple interest per annum for 2 years and 3 years
respectively. The difference in the interest was Rs.56. the sum borrowed were
a. 690/- b. 700/- c. 740/- d. 780/-
39. If A is 25% of C and B is 30% of C, then what percentage of A is B?
a)83 1/3% of B b)83 9/3% of B
c)81 1/3% of B d)83 2/3% of B
39 .Two numbers are respectively 25% and 50% more than a third number. What percent is the
first of the second?
a)26 2/3 % of second b)26 3/4 % of second
c)83 1/3 % of second d)83 2/3 % of second
40. Two numbers are respectively 20% and 32% less than a third number. What percent is the
second of the first?
a)15% of the first
b)85% of the first
c)25% of the first
d)none of these
41. If the price of a commodity increases by 50%, find how much percent its consumption be
reduced so as not increase the expenditure
a)33 1/3 % b)66 3/4 %
c)75%
d)none of these
42. If the price of a commodity decreases by 50%, find how much percent its consumption be
increased so as not decrease the expenditure
a)100%
b)0%
c)10%
d)none of these
43. If the salary of Mr. Shashi is first increased by 18% and thereafter decreased by 15%, what is
the net change in his salary?
Percentages
28

a)7%
b)3%
c)0.3%
d)o.7%
44. The population of a town is decreased by 20% and 40% in two successive years. What
percent population is decreased after two years?
a)55%
b)45%
c)48%
d)52%
45. If the side of a square is increased by 10%, its area increased by k%. Find the value of k
a)21 b)15
c)42 d)12
46. The radius of a circle is increased by 4%. Find the percentage increase in its area
a)9% b)8 4/25%
c)2 2/25% d)none of these
47. The population of a town increases by 4% annually. If its present population is 12500, what
will it be in 2 years time?
a)13520 b)14520
c)11520 d)none of these
48. In a group of 80 boys, 60% play chess, 75% play cricket and 55% play both. How many of
them do not play any of these two games?
a) 24 b) 20
c) 18 d) 16
49. In a bookstore 25% of books are in English, 60% of the remaining are in Hindi, % of the
remaining are in Telugu and remaining 64,000 are in other languages. What is the total number
of books in that bookstore?
a) 2,56,000 b) 3,20,000
c) 4,50,000 d) 6,40,000
50. Three numbers are in the ratio of 3:4:8 respectively. If the first is increased by 25%, the
second is decreased by 20% and the third is unaltered, respectively their ratio will be?
a) 75:64:180 b) 75:56:160
c) 75:64:160 d) 125:64:160
Solutions:
1.1/5 X 2/5 X a = ¼ X a X b => b = 8/25
2.% difference = (200-50)/50 X 100 = 300 %
3.% increase = (80-30)/30 X 100 = 166.66 %
4.1.3 x = 78000 => x = 60000.
5.Net effect = 1.2 X 1.1 X 0.7
= 0.924 => 7.6% decrease.
6.Let I be the income.
Expenditure = 0.8I Savings = 0.2I => 20%
New income = 1.2I (since 20% rise)
New expenditure = (0.8I) X 1.3 (Since 30% rise)
= 1.04I
So, new savings = 1.2I – 1.04I = 0.16I => 16%
(So income decreased form 20% to 16%)
% decrease = (20-16)/20 X 100 = 20%.
7.It is equivalent to 1.25 decreased to 1.
Percentages
29

% decrease = (1.25-1)/1.25 X 100 = 20%
8. % change in area = 1.2 X 1.2 (since area = side X side)
= 1.44 => 44%.
9.It is equivalent to 1.25 decreased to 1. So 20% decrease.
10. Valid Votes:
A got 56% => B got 44%
Difference = 12% = 48000
So, 100% = 400000. These are valid votes.
But valid votes are only 80% of total votes.
So, 80% of total votes = 400000 => total votes = 500000
11.Total money = Rs. 300.
Saving of the lady = 10% of 300 = 30/-
With 30/- she bought 5 kg sugar => each kg costs Rs. 6/-
12.In 20lt, salt = 20% => 4 lt.
New volume = 18 lt (2 lt evaporated)
So, new % = 4/18 X 100 = 22.22%
13.% correction = (58-46.4)/58 X 100 = 20%
14.Three successive decreases of 20%, 20% and 10% => 0.8 X 0.8 X 0.9 = 0.576
Again 10% decrease => 0.576 – 0.1 = 0.476.
So, 0.476 x = 9520 => x = 20000.
15.Total discount = 0.9 X 0.8 X 0.7 = 0.504 of actual price.
So, price = 0.504 X 10000 = 5040.
16.In 10 lt, alcohol is 10% = 1 lt.
Let x lt alcohol is added.
So, (1+x)/(10+x) = 20% = 1/5 => x = 1.25 lt.
17.A = 2B and B = 3C (since 200% more)
ð A = 6C => 500 % more.
18.50% of max marks – 40% of max marks = 25
ð max marks = 250
Pass marks = 40% of max + 10 => 100 + 10 = 110.
19.A = 1.2 B => B = A/1.2 => 0.8333A => 16.66%.
(OR) Decrease from 1.2 to 1 => 16.66%.
20.1.1 X 1.1 X x = 12100 => x = 10000.
21.Salt = 15% of x = 0.15x (x = volume of solution)
Now, 0.15x/(x-30) = 20% = 1/5 (since 30 lt evaporated)
ð x = 120 lt
22.20% empty => 80 % full = 240 lt => 20% = 60 lt
23.A => 1.4 X 1.4 = 1.96
B => 1.9 => 6% difference.
24. From 1960 to 1975, in 15 years population doubled every 5 yrs => three times
So, 2 X 2 X 2 = 8 times => 700% more.
25.[(75% X 60) + (x% X 20)] / 80 = 80% => x = 95. (since required is 80%)
(OR) 60 out of 80 is 3/4. So, (3/4 X 75) + (1/4 X x) = 80 => x =95.
Percentages
30

Percentages
31