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Preface
The Digi-notes (Percentage) as presented here is prepared strictly
according to IBPS-PO, IBPS-Clerk, IBPS-SO and other PO/Clerk level
competitive exams. The main aim of the Digi-notes is to develop the
notion of the percentage so that candidate would be able to handle the
quantitative problems of the real world easily and conveniently and
hence in their competitive exams. Candidates are advised to handle the
Digi-notes by pre-assuming that they are going to learn the rule of the
language of the percentage so that you; yourself would be able to
minimize the steps required to conclude the answer of the question
and hence develop the SHORT TRICKS.
All the best
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Contents
1.Preface ... 2
2.Contents . 3
3.
Introduction . 4
4.What is Percentage? .. 5
5.Strategy...... 6
6.The Percentage calculation Techniques ...... 7
7.The Application of Unitary Method 15
8.The Application of Venn diagram . 22
9.Percentage Increase/Percentage Decrease..... 31
10. The Technique of Net Effect of Successive
Percentages.. 40
11. The Percentage Change in the Product of Two or More
Quantities .... 43
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Introduction:
Everything is RELATIVE in this world; nothing is ABSOLUTE. Therefore;
to analyse and solve the quantitative problems in day to day life;
percentage plays an important role as it provides a convenient method
to relate two or more entities in the real world. It is extensively used in
data analysis in any organization.
Percentage is the backbone of the entire Quantitative Aptitude. It is
spread everywhere in the entire Quantitative Aptitude. If anyone
analyses a paper of 50 questions; he/she would find that; nearly 80%
questions (I stated it in terms of percentage) involves the concept of
percentage calculation extending from profit/loss/discount; simple
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interest; compound interest; its application in Alligation; in time work
and so on to Data Interpretation. So candidates should be well
equipped with how to speak the language of percentage while dealing
with the question.
What is Percent?
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From the mathematically perception of the percent; we could conclude
that the mathematical meaning of the sign % is . When we
would see the sign % we would thinkas there is a multiplication by
.Strategy:
We would handle the questions of the percentage according to the
following strategy (Our main aim would be to develop those logical
steps which would make us psychologically strong enough to evolve
the so called SHORT TRICKS of a question):
1.The percentage calculation technique (Most candidates do not
know that they are weak in the percentage calculation technique
which is extensively used in Data Analysis; and they think that
Data Analysis is very tough).
2.The application of Unitary Method; when the base of the given
percentages are same.
3.
The Application of Venn diagram; in questions in which the data
overlaps.
4.The Percentage Increase and/or Percentage Decrease. When the
base of the given percentages are different; the application of
initial/Final value, again it is another form of Unitary Method.
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5.The Technique of Net Effect of Successive Percentages.
6.The percentage change in PRODUCT of two or more quantities.
The Percentage Calculation Technique:
`
Just Keep one thing in mind; while calculating the percentage value of
a given quantity (Q); try to extract the information that from which
quantity it is compared (C) and then put it in the denominator and then
multiply it by 100 and finally put % sign after that (just to nullify the
multiplication by 100, its equivalent % should be multiplied.).To sum up:
From the above discussion we could conclude and easily digest the
following in our mind:
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Let us discuss it with the help of some examples:
Q.1. Find 15% of Rs.250?
Sol:
= .=..
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Q.2: Change %into equivalent fraction?
Q.3: Change
into equivalent percentage?
Q.4: 2 days is what percent of 2 weeks?
Q.5: The salary of a person is increased from Rs.4050 to
Rs.4500. Find the percentage increased in his salary?
Sol:
= =
Sol:
= %= %
Sol:
= %= %
Sol:% = %= %
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Q.6: The salary of a person is decreased from Rs.4500 to
Rs.4050. Find the percentage decreased in his salary?
Q.7: The ratio of two numbers P and Q is 3:5. Find:
1.
P is what percent of Q?
2.
Q is what percent of P?
3.
P is what percent of (P+Q)?
4.
Q is what percent of (P+Q)?
5.
P is what percent less than Q?
6.
Q is what percent more than P?
Sol:
% = %=%
Sol 1: % = %Sol 2: % = %Sol 3: % = %Sol 4: % = %Sol 5: % = %Sol 6: % = %
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Q8: P, Q and R are three numbers. The ratio of P and Q is 2:3
and that of Q and R 2:1. Find:
1.
(P+Q) is what percent of (Q+R)?
2.
(P+R) is what percent less than (P+Q+R)?
Sol: Learn the rule for combining two different ratios having
one quantity common in the given ratios:
1. = =
= % =
%
2. = = = % = %
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Q.9: The salary of Ram is 20% more than the salary of Mohan.
Find the salary of Mohan is what percent less than the salary of
Ram?
Q.10: The salary of Ram is 20% less than the salary of Mohan.
Find the salary of Mohan is what percent more than the salary
of Ram?
Q.11: A, B and C are three persons. The income of A is 40% more
than the income of B. While the income of B is 20% less than the
income of C. Find:
1.The income of C is what percent less than the income of A?
2.The income of C is what percent less than the income of A?
Short Trick: : = := : = %= %
Short Trick:
: =
:
=
:
= %=%
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Q.12: 20% of the salary of Amit is equal to 30% of the salary of
Bunty. While 80% of the salary of Bunty is equal to 40% of the
salary of Chintu. Find the total salary of Amit and Chintu is what
percent less than the total salary of Amit, Bunty and Chintu?
Short Trick: = %= %
= %
=%
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Q.13: A positive numbers is by mistake multiplied by 5 instead
of being divided by 5.
1.
Find the result obtained is what percent of the required
correct value?
2.
Find the result obtained is what percent more than the
required correct value (OR Find the percentage error in the
calculation)?
Q.14: A positive numbers is by mistake divided by 3/5 instead
of being multiplied by 5/3. Find the percentage error in the
calculation.
Short Trick:
=
= = ==Sol 1:
= %=%
Sol 2:
= %=%
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The Application of Unitary Method:
Recall your unitary method which we are used to apply to solve the
questions involving the quantities having either direct or inverse
relation among them. Here we have to apply unitary method involving
the quantities having direct relation. Let us discuss it with the help of
the following example:
The cost of 13 pen is 520. Find the cost of 15 pens.
We were used to approach the question as below in our childhood
classes:
Short Trick:
=
= = = = Sol:
=
%= %
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13 .520
1 .40
15 .600
Pen Rs
Pen Rs
Pen Rs
We are used to concise it in the following form:
.52015 .600
13
RsRs
i.e.
Given ValueCorresponding No of Unit Asked
Corresponding No of Unit
The same approach we will apply in the percentage questions in which
the base (from which comparison is made) of all the given percentagesare same. That is:
Let us solve some examples based on the discussion above:
Q 1: 59% of a number is 760 more than the 40% of the same number.
Find 25% of the number.
Q. 2: The sum of 35% and 24% of a number is 472. Find 3/4thof the
number.
x (Percentage value asked)
Short Trick:Sol:
% %
=
% %
=
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Q. 3: The difference between a number and its 39% is 732; then find
the number.
Q. 4: Find the difference of 28.50% and 37.50% of 75.
Q. 5: A fruit seller had some apples. On a particular day he sold 36% of
the apples and he still remains with 448 apples. Find how many
apples had he sold on that day?
Short Trick:Sol:
=%
= % %= +% %=
Short Trick:Sol:
=
% = % %=
Short Trick:Sol:
= % %
= % ..%= .
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Q.6: In an exam; 35% of the appeared students failed while 1625
students passed. Find the difference between the number of the
passed and the number of the failed students in the exam.
Q.7: In a village; 45% of the total population are males while 30% of the
total population are females. If the number of the children in the village
is 1500; find the difference of the number of the males and the number
of the females in the village?
Short Trick:Sol:
=
% %
= % %=
Short Trick:Sol: % = %% = % % = % % %
=
% %=
Short Trick:Sol:
%=% %% =%
% %= % %=
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Q.8: Raju spends 15% of his monthly salary on house rent; 20% on food;
10% on travelling; 25% on childrens education and 5% on
miscellaneous. After all these expenditures he deposited Rs.6250 in his
savings bank account. Find how much amount did he spend on foodand house rent per month?
Q.9: In an election between two candidates; a candidate who got 84%
of the total voters on the voting list; won by a majority of 476 votes.
Find the total number of the voters on the voting list?
Q.10: In an exam, 6% candidates passed from class 11th. From class 12th,
an equal number of candidates appeared for the exam and 7%
Short Trick:Sol: %=% % % % % % =% % %
= % %=.
Short Trick:Sol: % =%% = % % = % % %
= % %=
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candidates got passed with 80 more candidates got passed than class
11th. Find the number of candidates appeared from each class.
Q.11: In an examination, a candidate must get 80% marks to pass. If a
candidate, who gets 210 marks, fails by 50 marks, find the maximum
marks.
Q.12: A student has to secure 40% marks to pass. If he gets 90 marks
which is 40 marks more than the passing marks, find the maximum
marks set for the examination.
Short Trick:Sol: = % %
= % %=
Short Trick:
Sol: = % %= +% %=
Short Trick:
Sol: = % %= % %=
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Q.13: In an examination it is required to get 290 of the aggregate
marks to pass. A student gets 209 marks and is failed by 12% marks.
What are the maximum aggregate marks a student can get?
Q.14: A candidate who gets 30% of the marks in a test fails by 50 marks.
Another candidate who gets 320 marks fails by 30 marks. Find themaximum marks.
Q.15: A candidate scores 25% and fails by 30 marks, while another
candidate who scores 50% marks, gets 20 marks more than the
minimum required marks to pass the examination. Find the maximum
marks for the examination.
Short Trick:
Sol: = % %= % %=
Short Trick:
Sol: .= % %
=+
% %=
Short Trick:
Sol: .= % %= %% %=
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Q.16: In an exam, A scores 280 marks which are 20 more than passing
marks. B got 80% marks which are 60 more than the passing marks.
What is the passing percentage of the exam?
Q.17: A student got a total of 70% marks in an exam of 4 subjects. In
the three subjects he scores 81, 66 and 60 marks. If the maximum marks
of each subject are 100, then find the marks obtained in 4 thsubject.
The Application of Venn diagram:
Short Trick:
Sol: % = % = %+ =%
Short Trick:Sol: =
= % % =
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There are some questions involving the percentage in which a certain
keywords (Only; Either-or; Both; Either-ornot both etc.) are used
which tests our ability to perceive these keywords in practical
problems.You are aware from your childhood classes about the Venn diagram;
and learned a formula related to it for two sets A and B having a
common value as described below: = First of all you forget about this formula. If you will try to apply it in the
questions you will involve yourself only in mathematical equations and
calculations. You would not be able to perceive the question logically
that what is going on psychologically behind the scene.
Let us equip ourselves with the psychological steps to conclude the
question easily and with the help of mental calculation:
First; let us learn how to read a Venn diagram according to the given
keywords as described below:
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Which Portion
Represent
Description Answer
A Entire Portion x + y
Only A A butnot B Remove B from A xA and B Always theIntersection
(Common)
Portion
y
Either A or B (OR
simply A or B)
Only A + Only B +
Both
x + z +
y
A or B but notboth
Exclude Bothfrom A or B
x + z
You could perceive and digest it easily into your mind with the help of
the following diagram:
For the better perception; consider the following example:
Q.1: In an Office; 35 people like tea, 55 people like coffee while 15
people like both tea and coffee.
1.Find the number of the people who:
1)Like Tea
2)Like Coffee
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3)Like Only Tea
4)Like Only Coffee
5)Like Both Tea and Coffee
6)
Like Either Tea or Coffee7)Like Either Tea or Coffee but not Both
2.Find the Total number of the people in the office?
Solution: As there is a case of overlapping in the question: it would be
easy to analyse the situation with the help of the Venn diagram:
1.See the solution below extracted from the Venn diagram:
1)20 + 15 = 35
2)40 + 15 = 55
3)20
4)
40
5)15
6)20 + 40 + 15 = 75
7)20 + 40 = 60
2.Data Inadequate as there could be some people in the office who
might do not like both tea and coffee; about which no information
is given in the question.
Q.2: In an Office; 35 people like tea, 55 people like coffee while 15
people like both tea and coffee. 10 people like neither tea nor coffee
(i.e. do not like both)
1.Find the number of the people who:
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1)DO NOT LIKE Tea
2)DO NOT LIKE Coffee
3)DO NOT LIKE Only Tea
4)
DO NOT LIKE Only Coffee5)DO NOT LIKE Both Tea and Coffee
6)DO NOT LIKE Either Tea or Coffee
7)DO NOT LIKE Either Tea or Coffee but not Both
2.Find the Total number of the people in the office?
Sol: Now its turn to talk about the complement part about the Venn
diagram. This discussion would give us the idea about the pivot point
for the entire Double Venn diagram questions. Let us discuss it (Try toperceive the following diagram):
2.See the solution below extracted from the DO NOT LIKE Venn
diagram:
1)40 + 10 = 50
2)20 + 10 = 30
3)
404)20
5)10
6)40 + 20 + 10 = 70
7)40 + 20 = 60
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3.Now this time we are given the information about how many
people are there in the office who DO NOT LIKE both tea and
coffee which is 10.
Now from the LIKE and DO NOT LIKE Venn diagram:Total = [LIKE only T + LIKE only C +LIKE both] + DO NOT LIKE both
Total = Either-or LIKE + DO NOT LIKE both = (20 + 40 + 15) + 10 =
85
Or we can also say that (Try to convince yourself by visualizing the
situation practically)
Total = Either-or DO NOT LIKE + LIKE both = (40 + 20 + 10) + 15 =
85
We could diagrammatically fit it in our mind as:
Conclusion:
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Now the learning is complete; let us apply the learning in percentage
questions:
Q.3: In an exam; 50% of the appeared students failed in English, 45%
failed in Hindi while 15% failed in both the subjects. Find thepercentage of the students who PASSED in:
1.Both English and Hindi.
2.Either English or Hindi.
Sol: From the above Discussion (Attention: Pass is the complement of
Fail and vice-versa):
1.Pass in both = 100% - Failed in Either E or H = 100% - 80% = 20%
2.Pass in Either E or H = 100% - Failed in Both = 100% - 15% = 85%
Q.4: In a school; 55% of the students play Tennis, 50% play Hockey
while 20% play both the games.If 90 students play neither Tennis nor
Hockey; find the number of the students who:
1.DO NOT PLAY Tennis
2.DO NOT PLAY Hockey
3.DO NOT PLAY Only Tennis
4.DO NOT PLAY Only Hockey
5.DO NOT PLAY Both the games
6.DO NOT PLAY Either Tennis or Hockey
7.DO NOT PLAY Either Tennis or Hockey but not both the games.
8.TOTAL number of the students in the hostel.
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9.PLAY Tennis
10. PLAY Hockey
11. PLAY Only Tennis
12.
PLAY Only Hockey13. PLAY Both the games
14. PLAY Either Tennis or Hockey
15. PLAY Either Tennis or Hockey but not both the games.
Sol: We could conclude the following from the discussion till now:
Now Consider the DO NOT PLAY Diagram and conclude the answers
using unitary Method:
1. = % % = 2. = % % = 3. = % % = 4.
=
% % =
5. = % % =
6. = % % =
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7. % % = 8.
=
% %=
9. = % % = 10. = % % = 11. = % % = 12. = % % = 13.
=
% % =
14. = % % =
15. =% % = Q5: In a factory, 72% people like tea while 44% people like coffee. If
20% people like neither tea nor coffee and144 people like both tea andcoffee; find the total number of the people in the office; if.
Solution:
= % %=
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Q.6: In a factory, 72% people like tea while 44% people like coffee. If
EACH of them like either tea or coffee and 64 people like both tea and
coffee; find the number of the people who like only coffee.
Sol:EACH of them like either T or C There are NO people in the officewho DO NOT LIKE both T and C
= % % = Percentage Increase and/or Percentage Decrease:
We will apply the concept of initial and final value of given fraction.What we would mean by the INITIAL and FINAL vale of a fraction :
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For Example
:
= : =
So for quick calculation you would have to learn the fractional value
of some percentages. Some are as below:
Initial = D
Final = D
N
+ Increase
- Decrease
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100% = 1 50 % = 25 % =
1 % = 3 % = 62% = 6
% = 16% = 3 % = 66% = 1 % = 9 % = 1 % = 2 % = 8 % =
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The rest is the story of unitary method as discussed below with the
help of atomic examples (Later we will discuss the combination of
these first four cases):
Q.1: Rs.640 is increased by %. Find the amount after increase.
Q.2: An amount is increased by %and becomes Rs.6250.Find the initial amount.
Q.3: Rs.275 is decreased by %. Find the amount after decrease.
Short Trick:
I = Rs.640 % + F = =
Short Trick:
Sol: I % + F = Rs.6250 = =
Short Trick:
Sol: I = Rs.275 % F = =
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Q.4: An amount is decreased by %and becomes Rs.250. Find theinitial amount.
Learning complete, now Let us apply it as a combination in differenttypes of questions:
Q.5: A man loses %of his money and after spending % ofthe remaining; he is left with Rs.1050. Find how much had he at first?
Q.6: A man had Rs.39200 in his locker. During first year; he deposited
25% of the amount in his locker; while during next year he deposited
%
of the increased amount in his locker. Find the total amount
deposited by him during a span of 2 years?
Short Trick:
Sol: I % F = Rs.250 = =
Short Trick:
Sol: Amount Remainder Rs.1050= . =.
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Q.7: The present population of a town is 108000. During first year; the
population of the town increases by % while decreases by % during second year. During third year the population increasesby %. Find the population of the town after 3 years.
Q.8: 10% people of a village having died of cholera. Due to fear; 25% ofthe remaining people left the village; then the population of the town
reduces to 4050. Find the decrease in the number of the people in the
town?
Short Trick:
Sol: 39200
+ Increased
+ Amount
Amount after 2 years
=
. =.
=..=.
Short Trick:
Sol:
108000+ ? ? + After 3 years= =
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Q.9: The value of a machine depreciates at the rate of 10% per
annum. If its present worth is Rs.3645000; find its worth
1.After 3 years.
2.Before 3 years.
Short Trick:
Sol: Initially
Remaining
4050
= . ===
Short Trick:
Sol:1
Rs.3645000 ? ? After 3 years= =.
Short Trick:
Sol:2
Before 3 years ? ? Rs.3645000
=
=
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Q.10: The value of a land appreciates at the rate of %per annum.If its present worth is Rs.1984500; find its worth
1.After 2 years.
2.
Before 2 years.
Q.11: Amit received a certain amount from his father. He
spends 20% of the amount on hostel expenses; 1/3 rdof the
remaining on stationary;
%
off the amount on food and
25% of the remaining amount he spends on other expenses.
After all these expenditures he is left with Rs1200.Find the
total expenditure made by Amit.
Short Trick:
Sol:1 Rs.2381400+ ? + After 2 years
=
. =.Short Trick:
Sol:2 Before 2 years+ ? + Rs.2381400
= . =.
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Q.12: Raju spends 40% of his monthly salary on house rent; 10%
of the remaining on travelling; %of the remaining. Afterall these expenditures he saves Rs.2250. Find how muchamount did he spends on food per month?
Q.13: Shalini spends
%
of his monthly income on house
rent; 30% on food and 20% of the remaining on other expenses.
After all these expenditures he deposited Rs.16000 in his
savings bank account. Find her monthly salary?
Short Trick:
Sol:
Received R R R Rs.1200Amount = . =.=..=.
Short Trick:
Sol: No use of 40% and 10% to conclude the answer
Remaining Saves=Rs.2250Amount % After Spending on Travelling= . =.
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Net Effect of Successive Percentages:
Let us discuss it with the help of an example;
The salary of a person is first increased by 10% and thereafter it is
reduced by 5%. Find the net percentage change in his salary.By conventional method:
%
+
%
%5.4
1005.104
100100
95
100
110
100%
S
SS
i
ifN
Short Trick:
Sol:
Salary % % R % Rs.16000= % = . =.
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Assume that the net effect of 10% increase and 5% decrease is N%
Now look at the following scenario:
% %
+ % OR + ? % + Hence we can write:
%5.4100100
)5(10)5(10
100
)5(1010)5(100100
100
)5(1010010)5(100100100100
100
)5(100
100
10100
100
100`
100
95
100
110
100
100
xyyxN
N
N
SN
S
SN
S
Hence we can conclude that; if we are given two successive
percentages x% and y%, its net effect will be given by: %100
xyyxN
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By putting the values of x and y with sign i.e.; for percentage increase
put the values with positive sign and for the percentage decrease put
the values with negative sign.
Now we would be able to apply in the following questions withoutgoing into the details:
Q.1: A man increases the price of his article by 17%. After that
he allows the discount of 17% on the increased price to his
customers. Find his net profit or loss percentage in the deal.
Q.2: The population of a town decreases by 12% during the year
1988. If the percentage increase in the population of the town
in the beginning of 1990 is observed to be 1.2%; find at what %
it is increased during 1989?
Question 3: Find the net discount of two successive discounts
of 10% and 20%?
Short Trick:
Sol: % = =.%.%
Short Trick:
Sol: . = = %
Short Trick:
Sol: % = =%%
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Percentage Change in the Product of Two or More Quantities:
Let us discuss it with the help of the following example:
Q: In the product, = , a is increased by 10% while b isincreased by 20%. The new product so formed would be whatpercent more than the initial product?
Sol: By conventional Method:
% =
%
% = % % = %
% = %OR (In Short Way):% = =%Q: In the product; = ; a is increased by 10% whileb is increased by 20% and c is decreased by 10%. The new
product so formed would be what percent more than the initial
product?
Sol: % = %; %; %% = %; % %
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% = % %% = % %%=.%Hence we could conclude that we can also use the net effect method
for finding the percentage change in the product of two or more
quantities.
We will extensively use it for finding the percentage change in AREA
(Product of Two Quantities) of 2D figures and the percentage change
in the VOLUME (Product of Three Quantities) of 3D figures. We will
we discuss the questions related to percentage change in AREA of 2Dfigures (Volume questions are asked in SSC and equivalent exams; so
we will skip its discussion here)
Q.1: The length of a rectangle is increased by 10% while its breadth the
reduced by 5%. The area of the new rectangle would be what
percentage more than the area of the original area of the rectangle?
Sol: % = = . %Q.2: The adjacent sides of a square are increased by 10% and 20%respectively. The area of the rectangle so formed would be what
percent more than the area of the square?
Sol: % = =%Q.3: Each side of a square is decreased by 30%. Find the percentage
change in the area of the square?
Sol: % = =%Q.4: The length of a rectangle is increased by 10%. By what percentage
its breadth should be reduced in order to
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1.Increase the area by 4.5%.
2.Maintain the same area
Sol.1:
. % =
= %Sol.2: % = = %Q.5: The radius of a circle is increased by 2 cm; from 5 cm to 7 cm. Find
the percentage change in the area of the circle.
Sol: %= = % % = =%Q.6: The base and height of a triangle is increased by 10% and 20%;
then the area of the triangle increases to 529 meter square. Find the
original area of the triangle.
Sol: % = =% = % % = Q.7: The perimeter of a square is increased by 10%. Find the percentage
change in the area of the square.
Sol: % % = % % = =%
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Q.8: The circumference of a circle is decreased by 20%. Find the
percentage change in the area of the circle.
Sol:
% % = %
% = =%Q.9: The perimeter of a rectangle is increased by 30%. Find the
percentage change in the area of the rectangle.
Sol: %( ) %
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