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SPOORTHY STUDY CIRCLE Percentages Applications

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IntroductionPercent means for every 100. To get percentage

value of a ratio we multiply that ratio by 100.Percentages

help in comparing numerical data and information.

Some of the important percentages ratios which are helpfulin solving problems.

Let us take a number n,

100% of nis (100/100) n = n.

50% of nis (50/100) n = n/2=0.5n.

75% of nis (75/100) n = (3/4) n= 0.75n.

33.33% of nis (33.33/100) n = (1/3) n=

0.33n.

25% of nis (25/100) n = (1/4) n= 0.25n.

20% of nis (20/100) n = (1/5) n= 0.20n.

16.66% of nis (16.66/100) n = (1/6) n=

0.166n.

14.28% of nis (14.28/100) n = (1/7) n=

0.142n.12.5% of nis (12.5/100) n = (1/8) n=

0.125n.

11.11% of nis (11.11/100) n = (1/9) n=

0.1111n.

10% of nis (10/100) n = (1/10) n= 0.1n.

9.09% of nis (9.09/100) n = (1/11) n= 0.09n.

8.33% of nis (8.33/100) n = (1/12) n= 0.083n.

6.25% of nis (6.25/100) n = (1/16) n=

0.0625n.

250% of nis (250/100) n = (2.5) n

Suppose, we need to find 20% of 240, we take ratio directly,

20% of a number is (=1/5) of that number, so 20% of 240 =

(1/5) 240 = 48.

Find the following.

1.

25%360 + 33.33% 360 + 8.33% 360?

2.

16.66%60 + 12.5%96?

3.

66.66%120 + 62.5%240 + 37.5% 320?

4.

112.5%48 + 116.66%55?

5.

44%50?

6.

24%75 + 99.99%33.33?

Solution:

1. = (1/4)360 + (1/3)360 + (1/12)360 = 90 + 120 + 30 =

240

2.

= (1/6)60 + (1/8)96 = 10 + 12 = 22.

3.

66.66% = 50% + 16.66%, 62.5% = 50% + 12.5% and37.5% = 25% + 12.5% or 50% - 12.5%. 66.66%120 =

50%120 + 16.66%120 = 60 + 20 = 80. 62.5%240 =

50%240 + 12.5%240 = 120 + 30 = 150. 37.5%320 =

50%32012.5%320 = 160 40 = 120. 80 + 150 +120 = 350.

4.

100%48 + 12.5%48 + 100%55 +16.66%55 = 48 + 6 +

55 + 9.16 = 118.16

5.

x%y =y%x (Both are equal), 44%50 = 50%44 = 22

6.

= 75%24 + 33.33%99.99 = 18 + 33.33

When any value increases by 10%, it becomes 1.1

times of itself (Since 100% +10% =110% = (110/100) =

1.1), 20% 1.2 times of itself, 3.33% 1.33 times of

itself, 4% 1.04times of itself.

When any value decreases by 10%, it becomes 0.9

times of itself. (Since 100 10 = 90% = 90/100 =

0.9).20% 0.8times of itself, 35% 0.65times of itself

If 500 is increased by 50% then new value is 100%

+ 50% 150%500 = 1.5500 = 750

A value x is increased by 40%, it became 700,

value ofx? 1.4x =700, x = 500

If 500 is decreased by 55% then new value is 100%

55% 45%500 = 0.45 500 = 225. A value x is

decreased by 30%, it became 140, value of x? 0.7x =

140, x = 200.

So, if a value multiplied by greater than one, valuewill be increased, if it multiplied by less than one, value will

decreased.

1.23 373?

Solution: 123%373 123% = (100% +

20% + 3%) 373 = 100%373 + 2 10%373 + 3 1%373 =

373 + 2 37.3 + 3 3.73 = 373 + 74.6 + 11.19 = 458.79

'B s salary is 25% more than 'A s salary, by what

percentage 'A s salary less than 'B s salary?Solution: Assume As salary is 100.

0025 ?

25 25100( ) 125( ) 100( )A B A

25100

125 20

%, so answer is 20%.

A merchant gives successive discounts 20%, 10% and 20%

on an article, find the overall discount?

Solution: Assume initial price as 100 rupees.0 0 0

0 0 020 10 20

20 8 14.4100 80 72 57.6

100 57.6 =

0042.4

Price of a land passing through three hands rises on the

whole by 65%. If the first and second sellers earned 20%

and 25% profit respectively, find the profit earned by the

third seller. (S.S.C 2012) A 10% B 25% C 20% D 55%

Answer: Let initial price of a land is rs.100, at the end itbecame rs.165 (65% on whole).

First person earned 20%, he sold at 100 + 20% 100 = 120.

Second person gained 25%, he sold at 120 + 25% 120 = 120

+ (1/4)120 = 150

Third person gained rs.165150 = rs.15.

In percentage terms (15/150) 100 = 10%.


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Length and breadth of a rectangle are changed by +20% and

10% respectively, what is the percentage change in Area?

Solution: Length = l, breadth = b we know

that area ( A ) of rectangle is l b 0

020 1.2l l

and0

010 .9b b

A = 1.2 .9l b = 1.08A Overall change in area is +8% (Increment).

Method 2: Let initial area is 100.

Here, we can apply successive changes since area is product

of length and breadth.0 0

0 020 10

20 12100( ) 120 108( )

l b

A A

So answer is +8%

A Sells his goods 30% cheaper than B and 30% dearer

than 'C s , find by what percentage 'C s goods dearer than

'B s goods?

Solution: Let cost of 'B s goods is rs.100. Then, As cost

will be rs.70 (since 30% cheaper than B).

'A s Goods dearer than 'C s by 30%,So1.3 70 1.3 53.84C A C , so answer is 46.15 (100

53.84)

In an examination, A got 25% marks more than B, B got

10% less than C and C got 25% more than D. If D got 320

marks out of 500, the marks obtained by A were (S.S.C

2013) A 405 B 450 C 360 D 400

Solution: C marks = 125% D = (5/4) 320 = 400. B marks

= 90% C = 0.9 400 = 360. A marks = 125% B = (5/4)

360 = 450.

If the price of tea has gone up by 25%, then what should be

the percentage decrease in quantity consumed so that the

total expenditure on tea remains the same?

Solution: Let initial amount is rs.100.0

025 ?

25 25100 125 100

(Expenditure is

Constant here so you need to compromise with quantity of

tea) 125 should decrease to 100 to keep amount as same.

Answer is 25 100 20%125

Short-cut: ( ) 100100

RR

Method 2: Price Consumption = Expenditure, here

expenditure is constant.

Price and consumption are two parameters which are

inversely related, so a (1/x) increase in one parameter will

results in a (1/(x+1))decrease in another parameter. Price

gone up by 25% (=1/4), percentage decrease in quantity

becomes1 1

4 1 5

= 20%.

For 50% (=1/2) (1/3=) 33.33%.20% (=1/5) (1/6=) 16.66%, we can do problem 8, byusing above formula also.

And similarly,

If a (1/x)decrease in one parameter will results in a

(1/(x1))increase in other parameter.

An engineer is planning to decrease the breadth of a

rectangular room by 9.09% without increasing the area of

the room. By what percentage must he increase the length of

the room?

Solution: 9.09% (=1/11) decrease in breadth of the room

will results 1/ (11 1) (= 10%) increase in length of the

room.

Price of the rice is increased by 25% so that a person is able

to purchase of 10kg less of rice for 300 rupees, find initial

price of rice?

Solution: Let initial price of rice is x rupees.

Number of kgs =300

x, price of rice increased

by 25% so new price becomes 1.25x (or)5

4x

300

x-

54

300

x=10 x = 6 rupees.

Of the adult population in a certain city, 45% of men and

25% of women are married. Assuming that no man marries

more than one woman, and vice versa, the percentage of

total population of adults who are married, is

(S.S.C 2012) A 31.1% B 30% C 33.33% D 32.14%

Solution: We need ratio of total adults who are married to

total population. Ratio of men and women who are married

45%: 25% = 9: 5, ratio of men and women who are not

married (100% 45%): (100% 25%) = 55%: 75% = 11:

15 and and answer is

=9 5

1009 5 11 15

= 33.33%.

In a mixture of 60 liters of water and milk, milk contributes

75%. How much water should be added so that milk

becomes 20% of the mixture?

Solution: milk and water ratio is 75%: 25% = 3: 1, water

quantity = (3/4) 60 = 45 liters and milk quantity = 60 45

= 15.

15 20 1

45 80 4x

, 15 liters water should be added to

make milk content 20%.

ProfitThe amount paid in either purchasing or producing

an object is known as cost price and written as CP.

The amount got when it sold is called as the selling

price and written as SP.

If CP of an article is 100 rupees and SP of that

article is 120 rupees, then my profit is 120 100 = 20.

Profit in terms of percentage


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00 ?

20100( ) 120( )CP SP

20 100100

=20% (Getting rs.20 profit on

CP, so always profit percentage measured

With respect to cost price only)

If I say I got 25% profit by selling an article, it means SP is

25%more than CP so we can find SP as, SP=1.25CP (insimple terms CP is incremented by 25% )

If I say I got 80% profit by selling an article, it means SP is

80% more than CP so we can find SP as, SP =1.8 CP (in

simple terms CP is incremented by 80%)

Formulas,

Profit = SPCP

SP = Profit + CP

Profit percentage = (Profit/CP) 100.

Profit percentages always calculated on CP.

What is the SP, if CP of an article is rs.500, so that I should

get profit as 60%?

Solution: 1.6 1.6 500 800SP CP

What is the SP, if CPof an article is rs.800, so that I should

get profit as 30%? Answer: 104

Find profit percentage if CP and SP of an article is rs.500

and rs.600 respectively?

Solution:600 500 100

500 500

SP

CP

= 1 + 0.2 = 1.2,

1.2SP CP so profit is 20%.

If 1SP

CP , then we say we are getting profit.

If 1.65SP

CP

, profit is 65%, ifSP

CP

=1.05, Profit is 5%.

A person purchased a certain number of articles at 11

articles for rs.10 and sold them at 10 articles for rs.11. Find

the gain %? (S.S.C 2012) A 1 B 21 C 22 D 20

Solution: Selling price of one article is rs.11/10 and cost

price of one article is rs.10/11, SP/CP = (11/10)/ (10/11) =

121/100 = 1.21, so 21% profit.

LossIf CP of an article is rs.100 and SP of that article is

rs.80 then my loss is rs.20.

In terms of percentage,0

0?20 20100( ) 80( ) 100100CP SP =20%

If I say I got 25% loss by selling an article, it

means SP is 25%less than CP so we can find SP as,

(1 0.25) 0.75SP CP CP (Simple

terms CPis decremented by25%).

For 80% loss, (1 0.80) 0.20SP CP CP

For 55% loss, (1 0.55) 0.45SP CP CP

Formulas,

Loss = CP SP

SP = CP Loss

Loss % = (Loss/CP) 100.Loss percentage is always calculated on CP.

Problem: What is the SP if CP of an article is rs.500, so that

I should get loss as 60%?

Solution: (1 0.60) 0.40 500SP CP ,

200SP

Find loss percentage if CP and SP of an article are rs.500

and rs.300 respectively?

Solution:300 500 200

500 500

SP

CP

1 0.4 0.6 , 0.6SP CP So loss is 40%.

If 1SP

CP , then we can say its loss

If 0.65SP

CP

, loss is 35%, If 0.05SP

CP

, loss

is 95%.

Goods purchased for 240 rupees, one third of them sold at

10% profit, rest of them sold at 20% loss find overall

percentage of profit or loss?

Solution: Total money invested rs.240 so it is CP. One third

of them sold at10%.

Profit is = 1 240 803

1.1 80 88 .

Rest of them at 20% loss, loss is,

= 2 2403

=160

Krishna has 12 oranges with him, he sells n of them at a

profit of 10 % and remaining at a loss of 10%. He gains 5%

on the whole outlay. The value of n is (S.S.C2012) A 9 B

10 C 7 D 8

Solution: Loss always taken as negative value.

Method 1: CPof each orange isxrupees.

Then, 10%nx +(- 10%) (12-n)x = 5% 12x.

20%nx=15%12x,20nx=180x, n= 9.

Method 2: By using allegations

10% -10%

5%

5%-(-10%): 10%-5%

15% : 5%

So they are in 3: 1 ratio. n = (3/4)x 12 = 9.

A person calculates 20% profit on selling price find actual

profit percentage?

Solution: Assume SPis rs.100

Profit is 20% on SP, it becomes rs.20

We need CP to find actual profit


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CP=SPprofit 100 20 100 20 80

Actual profit is 20 10080

=25%.

The profit earned by a shopkeeper by selling an article at a

gain of 8% is rs.28 more than when he sells it at a loss of

8%. The cost price of the article is? (S.S.C 2012) A 175 B

165 C 170 D 190

Solution: (8 (8)) % of cost price = 28, 16%of CP = 28,

CP= 28/16 = 1.75, CPis 175.

By selling an article a person gains 60% profit, what is the

profit or loss if article sold at three fourth of selling price?

Solution: SP=1.6CP (since 60% profit)

Multiply by 34

on both sides, 3 1.24

SP CP

So profit is 20%.

A man bought 15kg of sugar for rs.540. He sold them at a

profit which is as much as the selling price of 3kgs. At what

price per kg he sold the sugar?

Solution: Letx is SPof sugar, profit is 3x.So, profit = SPCP, 3x = 15x540.

x = 540/12 = rs.45

In the above question, If he sold them at loss which is as

much as the selling price of 3kgs. At what price per kg he

sold the sugar?

Solution: Loss = 3x = 540 - 15x(= CP - SP)

x = 540/18 = rs.30

Marked Price (MP) or Listed PriceThe price at which an item or a product is intended

to be sold is called marked price. Actual price (= selling

price) that is, after offering a discount on marked price.

Assume marked price of an article is rs.100, after offering

20% discount, selling price of that article will be (100 -20)% 100 = 80%100 = 0.8 100 = rs.80.

Marked price of an article is rs.600, offers 20% discount on

it. Selling price will be 0.8 600 = rs.480.

Selling price of an article is rs.120 after offering a discount

of 20%, marked price will be SP= (100-20)%MP = 0.8MP,

MP= 120/0.8 = 150.

Cost price + % on CP = Marked price

Marked pricediscount = Selling price.

A merchant who marked his goods up by 50% subsequently

offered a discount of 20%. What is the percentage profit that

the merchant make after offering the discount?

Solution: Let CP is rs.100. Marked price will be 100 +50%100 = 150. Then, 20% discount allowed 20%150 = 30,

therefore selling price will be 150 30 = 120. So profit is

120/100 = 1.2, = 20% or (120100)/ (100) 100 = 20%.

Mahesh went to a shop to buy a TV. Shop owner said they

are two TV models, one was offering at 4/5 th of that price

and second model was offering at 7/8thof that price. Which

one he should buy if he decided to buy the TV on which he

was getting better percentage discount?

Solution: First model offering at 4/5thof that price it means

discount is, 1(4/5) = 1/5th= 20% of that price.

Second model offering at 7/8 th of that price, it means

discount is, 1 (7/8) = 1/8th = 12.5% of that price. So

Mahesh should buy 1stone since he is getting more discount

i.e. 20%.

A trader making 15% profit, after giving 8% discount on

marked price, what is the marked price?

Solution: Let CP = rs.100.8% discount on MP = 15% profit

= rs.115, so 92% onMP = 115,MP = (115/92) 100 = 5/4

100 = rs.125

MP= 125, 25% above CP.

Allegations in Percentages, Profit and Loss

Two vessels A and B contain milk and water are in the ratio

8: 5 and 5: 2 respectively. In what ratio they should mix it

so that milk will be 69(7/13) % in the new mixture?

Solution: Convert 69(7/13) % into fraction.

(900/13) % means, = 9/13. So milk quantity in the new

mixture is 9/13. Milk in first vessel is 8/13 and milk insecond vessel is 5/7.

8/13 5/7

9/13

2/(13x7) : 1/13

2 : 7

Two cycles together cost of rs.8000, by selling first one

merchant gains 10 % and on second one incurred loss of

10%, on overall incurred by 2.5%, find the cost price ofsecond cycle?

Solution: On 1stone gain 10% and on 2ndone gains -10%, so

by using allegations,

10% -10%

2.5%

12.5% : 7.5%

Ratio is 5: 3. Second one price is (3/8) 8000.

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