Percentage
Percentage
Percent mean that many hundredths.
To express x % as a fraction :
x % = x/100
Thus 20 % = 20/100 = 1/5
To express a/b as a percent :
a/b = [(a X 100 )/b ] %
Thus 1/5 = [( 1 X 100 )/ 5] %
= 20 %
Example :-
1) Express as a fraction 64 %.
Sol. 64 % = 64/100
= 16/25
2) Express as a decimal 34 %
Sol. 34 % = 34/100
= 0.34
3) Express as a rate percent 3/7
Sol. 3/7 = [(3 X 100 )/ 7 ]
6
= 42-------%
7
Rule I.
If the price of a commodity increases by r%,
then reduction in consumption , so as not to increase the expenditure , is :
or
If A s income is r % more than that of B , then :
B s income is less than that of A by :
We use same formula in above both condition
r X 100
----------------- %
100 + r
Example :-
1) If the price of sugar is increased by 15 %, find by how much percent a householder must reduce her consumption of sugar so as
not to increase the expenditure .
Sol.
Reduction in consumption
15 X 100
= ------------------ %
100 + 15
1500
= ---------- %
115
1
= 13------ %
23
2) If A s monthly income is 40 % more than that of B , how much percent is B s income less than that of A ?
Sol.
40 X 100
= --------------- %
100 + 40
4
= 28------ %
7
Rule II.
If the price of a commodity decreases by r % , then increase in consumption , so as not to decrease the expenditure , is :
or
If A s income is r % less than that of B , then :
B s income is more than that of A by :
We use same formula in above both condition
r X 100
---------------- %
100 - r
Example :-
1) If A s height is 25 % less than that of B , how much percent is B s height more than of A ?
Sol.
25 X 100
= ------------------- %
100 - 25
2500
= ----------- %
75
1
= 33 ------ %
3
Rule III.
Increase in population :
let population be P
rate be R
year n
i) Population after n years
= P [ 1 + R/100 ]^n
ii) Population n year ago
= P / [ 1 + R/100 ]^nExample :-
The population of a village is 176400. If it increases annually at 5 % , what will be its population 2 years hence ? What was it 2 years ago ?
Sol.
Population after 2 years
= 176400 [ 1 + 5/100 ]^2
= 176400 X [ 21/20 ]^2
= 194481
Population 2 years ago
= 176400/ [ 1 + 5/100 ]^2
= 176400 X [ 20 / 21 ]^2
Rule IV.
Depreciation
Let present value of machine be p
rate R
year n
i) Value of machine after n years
= p[ 1 - R/ 100]^n
ii) Value of machine n years ago
= p/[ 1 - R/100 ]^n
Example :-
The value of bus depreciates at the rate of 10 % per annum . If its present value is Rs. 81000 , what will be its worth after 2 years ?What was the value of the bus 2 years ago ?
Sol.
Value of the bus after 2 years
= 81000 [ 1 - 10/ 100 ]^2
= 81000 [ 9/10 ]^2
= 65610.
Value of bus 2 years ago
= 81000 / [ 1 - 10 / 100 ]^2
= 81000 / [ 10/9 ]^2
= 100000
Rule V.
If population p of the city increases first year R1 % , second year R2 % , third year R3 %
then after three year
=p[1+R1/100 ] [1+ R2/100 ] [1 + R3/100 ]
Example :-
If population of the village 25000 .It increases first year 5 % , second year 25 % ,
third year 20 % . What is its population after three year .
Sol.
= 25000 [1 + 5/100] [1+ 25/100] [1+ 20/100]
= 25000 [21/20] [5/4] [6/5]
= 7875/2
= 3937
Rule VI.
If a man spend Rs x % on food , Rs y % on transport , Rs z % on education and save Rs r than monthly income of a man
100 100 100
= -----------X------------X-----------X r
100-x 100-y 100-z
Example :-
Sonu spend 50 % on food , 50% on transport , 50% on education and he save Rs 5000 . What was sonu's monthly income ?
Sol.
Monthly income
100 100 100
= ------------X----------------X--------------X 5000
100 -50 100 - 50 100 -50
100 100 100
= --------------X---------------X----------------X 5000
50 50 50
= 40000 .
Rule VII
If a man spend x% on food , y% on transport , 2% on education and he save Rs r . What was the monthly income .
100
= ---------------------------- X r
100 -( x + y + z )
Example :-
Shubh spend 15 % on food , 15 % on transport , 20 % on education and he save Rs 5000 . What was the monthly income .
Sol.
Monthly income
100
= -------------------------------------X 5000
100 - ( 15 + 15 + 20 )
100
= ---------------------X 5000
50
= 100 X 100
= 10000
Rule VIII
A person spend 1/x on food , a/y on transport , 1/z on cloth and save Rs r What was the monthly income.
x y z
= ----------------X---------------X---------------X r
x - 1 y - a z - 1
Example :-
Vinod spend 1/5 part on food , 2/3 on scooter
,1/5 on cloth and he save Rs 4000. what was the monthly income.
Sol.
5 3 5
= ------------------X---------------X--------------X 4000
5-1 3-2 5-1
5 3 5
= -----------------X---------------X--------------X 4000
4 1 4
=5X3X5X250
=31250
Exercise
1) The fraction equivalent to 2/5 % is .
a) 1/40 b)1/125 c)1/250 d)1/500
2) 0.025 in terms of rate percent is ,
a)25% b)2.5% c)0.25% d)37.5%
3)What percent is 3% of 5% ?
a) 60% b)1.5% c) .15% d).015%
Answer
1) c 2) b 3) a
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