Percentage
Percent means for every hundred or per hundred. The numerator of “per hundred” is called the rate percent. Example 12/100 can be called as 12%, and 12% is the rate percent, and 12 is the rate. The other way to look at it is if some makes a profit of 20%, then one has gained 20/100 of the value invested.
Percent means for every hundred or per hundred. The numerator of “per hundred” is called the rate percent. Example 12/100 can be called as 12%, and 12% is the rate percent, and 12 is the rate. The other way to look at it is if some makes a profit of 20%, then one has gained 20/100 of the value invested.
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.
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=1.1005
-
To expressaas a percent: We have,a=ax 100%.bbb
-
Thus,1=1x 100%= 25%.44
Percentage
Increase/Decrease:
If
the price of a commodity increases by R%, then the reduction in
consumption so as not to increase the expenditure is:
|
|
R
|
x
100
|
%
|
|
(100
+ R)
|
If
the price of a commodity decreases by R%, then the increase in
consumption so as not to decrease the expenditure is:
|
|
R
|
x
100
|
%
|
|
(100
- R)
|
- Results on Population:
Let
the population of a town be P now and suppose it increases at the
rate of R% per annum, then:
-
1. Population after n years = P1 +Rn100
-
2. Population n years ago =P1 +Rn
100
- Results on Depreciation:
Let
the present value of a machine be P. Suppose it depreciates at the
rate of R% per annum. Then:
-
1. Value of the machine after n years = P1 -Rn100
-
2. Value of the machine n years ago =P1 -Rn
100
-
3. If A is R% more than B, then B is less than A byRx 100%.(100 + R)
-
4. If A is R% less than B, then B is more than A byRx 100%.(100 - R)
Basic
tips for faster calculations:
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.
Conversions:
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
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
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.
Important
relations in percentage
1. If the price of a commodity increases by r%, then percentage reduction in consumption, so as not to increase expenditure is
Example: If the cost of petrol increases by 40%, by what percent the person should reduce his consumption considering expenditure on petrol remains the same. Increase in price = 40%, by the formula, decrease in consumption is = = 28.57%
2. If the price of a commodity decreases by r%, then increase in consumption, so as not to decrease expenditure is
1. If the price of a commodity increases by r%, then percentage reduction in consumption, so as not to increase expenditure is
Example: If the cost of petrol increases by 40%, by what percent the person should reduce his consumption considering expenditure on petrol remains the same. Increase in price = 40%, by the formula, decrease in consumption is = = 28.57%
2. If the price of a commodity decreases by r%, then increase in consumption, so as not to decrease expenditure is
Example: If
the cost of petrol decreases by 10 %, by what percent can a person
increase his consumption considering expenditure on petrol remains
the same?
Decrease in price = 10%, by the formula, increase in consumption is = 11.11%
3. If A’s income is r% more than B’s then B’s income is % less than A’s.
Example: If A’s income is 20% more than B’s, then what percent is B’s income lesser than A?
A’s income more than B = 20%, by the formula, B’s income is = × 100 = 16.66 % less of A
4. If A’s income is r % less than B’s then B’s income is % more than A’s
Example: If A’s income is 20 % less than B’s, then what percent is B’s income more than A?
A’s income less than B = 20%, by the formula, B’s income is = × 100 = 25% more of A
5. If the present population of a town is p and let there be an increase of X % per annum. Then:
(i) Population after n years =
(ii) Population n years ago =
This is the compound interest formula, which we will study in detail later. If the decrease or depreciation is r%, then population or value of a machine (after depreciation) after n years
CALCULATIONS IN PERCENTAGES
Let’s start with a number A (= 1A)
1. A increased by 10% would become A + 0.1A = 1.1A
2. A decreased by 10% would become A – 0.1A = 0.9 A
3. A increased by 200% would become A + 2A = 3A
4. A decreased by 50 % would become = 0.5A
5. Use decimal fractions while adding and subtracting and normal fractions while multiplying.
Decrease in price = 10%, by the formula, increase in consumption is = 11.11%
3. If A’s income is r% more than B’s then B’s income is % less than A’s.
Example: If A’s income is 20% more than B’s, then what percent is B’s income lesser than A?
A’s income more than B = 20%, by the formula, B’s income is = × 100 = 16.66 % less of A
4. If A’s income is r % less than B’s then B’s income is % more than A’s
Example: If A’s income is 20 % less than B’s, then what percent is B’s income more than A?
A’s income less than B = 20%, by the formula, B’s income is = × 100 = 25% more of A
5. If the present population of a town is p and let there be an increase of X % per annum. Then:
(i) Population after n years =
(ii) Population n years ago =
This is the compound interest formula, which we will study in detail later. If the decrease or depreciation is r%, then population or value of a machine (after depreciation) after n years
CALCULATIONS IN PERCENTAGES
Let’s start with a number A (= 1A)
1. A increased by 10% would become A + 0.1A = 1.1A
2. A decreased by 10% would become A – 0.1A = 0.9 A
3. A increased by 200% would become A + 2A = 3A
4. A decreased by 50 % would become = 0.5A
5. Use decimal fractions while adding and subtracting and normal fractions while multiplying.
SOLVED
EXAMPLES
Q1. Express as percentages: .
Ans. To convert to percent, multiply each by 100
Q2. On my sister’s 15th birthday, she was 159 cm in height, having grown 6% since the year before. How tall was she the previous year?Ans. Height this year = 159 cm,
Q1. Express as percentages: .
Ans. To convert to percent, multiply each by 100
Q2. On my sister’s 15th birthday, she was 159 cm in height, having grown 6% since the year before. How tall was she the previous year?Ans. Height this year = 159 cm,
growth
= 6 %,
let
last year height be A
Now A = 159
Now A = 159
A
= = 150
Last year height = 150
Last year height = 150
Q3.
Arun spent 25 % of his pocket money, and has Rs 125 left. How much
had he at first?Ans. Pocket
money spent = 25%,
left
= 75%,
Let
original pocket money be A
Therefore A = 125
Therefore A = 125
A
= 125 × (By now students should be able to do this in single
step), A = 166.66
Q4. If the cost of electricity increases by 30%, by what percent one should reduce his spend in order that spent on electricity stays the same? Ans. As per the formula, × 100 The reduced percentage spent = × 100 = 23.07%
Q5. In an election, Congress secured 10% of the total votes more than BJP (consider only two parties in the election and everyone voting). If BJP got 126000 votes, by how many votes did it lose the election? Ans. Let congress secured X % of the total votes, therefore BJP had secured (X – 10) % of votes, being a two party election:
X + X – 10 = 100
2X = 110
X = 55
Therefore Congress has 55% of vote and BJP has 45%, since BJP got 126000 votes
of total votes = 126000
Total votes = 280000
Congress votes = 55/100 × 280000 = 154000
Difference = 28000, which is the victory margin.
Q6. If the population is15,00,000 and the expected birth rate is 50%, while the expected death rate is 31%, What will be the net change in the in the population at the end of the one year.Ans. The current population is 1500000
Number of births will be 50/100 × 1500000 = 750000
Number of deaths was 31/100 × 1500000 = 465000
Net change = 750000 – 465000 = 285000
Q4. If the cost of electricity increases by 30%, by what percent one should reduce his spend in order that spent on electricity stays the same? Ans. As per the formula, × 100 The reduced percentage spent = × 100 = 23.07%
Q5. In an election, Congress secured 10% of the total votes more than BJP (consider only two parties in the election and everyone voting). If BJP got 126000 votes, by how many votes did it lose the election? Ans. Let congress secured X % of the total votes, therefore BJP had secured (X – 10) % of votes, being a two party election:
X + X – 10 = 100
2X = 110
X = 55
Therefore Congress has 55% of vote and BJP has 45%, since BJP got 126000 votes
of total votes = 126000
Total votes = 280000
Congress votes = 55/100 × 280000 = 154000
Difference = 28000, which is the victory margin.
Q6. If the population is15,00,000 and the expected birth rate is 50%, while the expected death rate is 31%, What will be the net change in the in the population at the end of the one year.Ans. The current population is 1500000
Number of births will be 50/100 × 1500000 = 750000
Number of deaths was 31/100 × 1500000 = 465000
Net change = 750000 – 465000 = 285000
Q7. What is the % change in the area of a square (which will become rectangle) if its length side is increased by 10% and its width side is decreased by 10%?
Ans.
In
these types of problems, assume a percent (100) base and then move
forward.
Let the side of square be 100
So length side will become
1.1(10% increase) × 100 = 110
And the width side will become
0.9(10% decrease) × 100 = 90
Old Area = 100 × 100 = 10000
New Area = 110 × 90 = 9900
Difference = 10000 – 9900 = 100
Difference percent = 100/10000 × 100
= 1% decrease in the area
Q8. Ram obtains 40 % of the marks in a paper of 200 marks. Shyam is ahead of Ram by 25 % of Ram’s marks, while Bhuvan is ahead of Shyam by one ninth of his own marks. How many marks does Bhuvan get?Ans. Ram’s marks = 40/100 × 200 = 80
Shyam’s Marks = 1.25 (25% ahead) × 80 = 100
Let Bhuvan’s Marks be A, therefore
A = 100 (Shyam’s marks) + 1/9A
8/9 A = 100, A = 112.5
Let the side of square be 100
So length side will become
1.1(10% increase) × 100 = 110
And the width side will become
0.9(10% decrease) × 100 = 90
Old Area = 100 × 100 = 10000
New Area = 110 × 90 = 9900
Difference = 10000 – 9900 = 100
Difference percent = 100/10000 × 100
= 1% decrease in the area
Q8. Ram obtains 40 % of the marks in a paper of 200 marks. Shyam is ahead of Ram by 25 % of Ram’s marks, while Bhuvan is ahead of Shyam by one ninth of his own marks. How many marks does Bhuvan get?Ans. Ram’s marks = 40/100 × 200 = 80
Shyam’s Marks = 1.25 (25% ahead) × 80 = 100
Let Bhuvan’s Marks be A, therefore
A = 100 (Shyam’s marks) + 1/9A
8/9 A = 100, A = 112.5
Practice
1 :
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?
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?
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8.
A student multiplied a number by |
3 |
instead
of |
5 |
. |
|
5 |
3 |
What
is the percentage error in the calculation?
|
9.
In an election between two candidates, one got 55% of the total valid
votes, 20% of the total votes were invalid. If the total number of
votes was 7500, the number of valid votes that the other candidate
got, was:
10.
Three candidates contested an election and received 1136, 7636 and
11628 votes respectively. What percentage of the total votes did
the winning candidate get?
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12.
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.
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13.
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:
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Practice
2:
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 salt 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.
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 they
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%
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