Module 1:

Mathematics

PartA :

PERCENTAGE Introduction:The word

'per cent' means per

hundred. Thus, 19 per cent

means, 19 parts out of 100

parts. This can also be

written as 19/100,

therefore, per cent is a

fraction whose denominator is 100.

Example 1 :Express 3/4 in rate per cent

Required rate per cent = 3/4 x 100% = 75%

Example 2: 8% can be converted to a fraction as 8/100

Example 3: To find out 25% of 500

Solution 3: Required value=25% of 500=(25/100)*500=125

Let us consider a number N,

25% of N = (25/100)*N = N/4 (value)

Example 4: 9% of

what number is 36

Solution 4: the

required number

(base number) =

36/9%

(36/9) * 100 = 400

Example 5: If 30%

of a number is 48

then what is 70% of

the number?

Solution 5: Here,

unitary method can

be used to save the

time. 30%

30%--> 48

1%--->48/30

70%=(48/30)*70=112

Hence, the required value is 112

Example 6: If 40% of the number exceeds 25% of it by 54.

Find the number.

Solution 6: Using the formula

any value/its rate % of number = number (i.e. base number)

54/(40-25)%=number

number=(54/(40-25))*100=360

required number=360

Example 7: To find '30 is what per cent of 150' or 'what

percentage of 150 is 30 ?

Solution 7: We find here that 150 is the basis of comparison

and hence 150 will be in the denominator. The required

percentage = (30 / 150)* 100 = 20%.

CONVERTING A PERCENTAGE INTO DECIMALS

67% may be

converted into

decimals as 0.67,

because 67/100 =

0.67

8% can be written

as 0.08

253% can be

written as 2.53

0.25% may be

converted into decimals as 0.0025

CONVERTING A

DECIMAL INTO A

PERCENTAGE

0.45 may be expressed as

45%

0.032 is equivalent to

3.2%,

1.7 is equivalent to 170%

EFFECT OF

PERCENTAGE CHANGE

ON ANY NUMBER

If any number is increased by X%, then new number =

original number x (100+ X)/ 100

or = original number x (1 + decimal equivalent of X %).

Similarly, if any number (quantity) is decreased by X %, then

new number (quantity) = original number x (100- X)/ 100

or = original number x (1 - decimal equivalent of X%).

Example 8: The present salary of A is Rs 3,000. This will be

increased by 15% in the next year. What will be the

increased salary of A?

Solution8:The increased salary = 3000 (1 + 0.15) or 3000 x

((100 + 15)/ 100)

= 3000 x 1.15 = Rs 3450.

If a number is changed by x%, then it is changed again by

y%, then net percentage change = x+y+(xy/100)

If x or y indicates decrease in percentage, then put a -ve

sign before x or y, otherwise positive sign remains.

Example 9: If a number is increased by 12% and then

decreased by 18% then find the net percentage change in

number

Solution 9: Using the formula net % change = x + y + xy/100

x=12 y=-18 net % change in area =12-18+((12)* -(18))/100 = -

8.16

Application of the Formula: net % change in product = x +

y +xy/100

Example 10: If the length of rectangle increases by 30%

and the breadth decreases by 12%, then find the % change in

the area of the rectangle.

Solution 10: Since, length x breadth = area we get net %

change in product = x + y + xy/100 where x = 30, y = -12 net

% change in area = 30 - 12 + (30*-12)/100=18-3.6=+14.4 It

implies that there is 14.4% increase in the area of the

rectangle.

To Keep the Product of Two Variable Quantities Fixed

Put net % change in product = 0, x+y+xy/100=0

y = -(x/(100+x))*100, -ve sign shows decrease,

from the above derivation, we thus find that

if one A increases by x%, then B decreases by

(x/(100+x))*100% and if A

decreases by x% then

putting (-) x, in place of x,

we find that the other

quantity B increases by

((x/(100-x))/100%.

Example 11: If the price

of coffee is increased by

10%, then by how much

percentage must a house

wife reduce her

consumption, to have no

extra expenditure?

Solution 11: Since price x consumption = expenditure and

expenditure has to be kept fixed so, when the price

increases by 10%, the %reduction in consumption

=(10/(100+10))*100=9 1/11%

RATE CHANGE AND CHANGE IN QUANTITY

AVAILABLE FOR FIXED EXPENDITURE

Example 12: A reduction

of 25% in the price of

sugar enables the person to

get 10 kg more on a

purchase for Rs 600. Find

the reduced rate of sugar.

Solution 12: Let the

original rate = Rs X per kg.

Since, there is a rate

reduction of 25%, so,

New rate (or reduced rate)

= (1 - 0.25) X

= 0.75 X = 3X/4

Expenditure = Rs 600.

(Expenditure/X)+change in quantity available =

Expenditure/New rate

(600/X+10)=600/(3X/4)

(600/x)(4/3-1)=100

x=20

therefore reduced rate =3x/4=3/4*20=Rs 15/kg.

% EXCESS OR % SHORTNESS

If A exceeds B by x%, B is less than A by x/(100+x)* 100%.

Similarly, if A is less than B by x%, then

B is more than A by x/(100-x) *100%

Example 13: If the income of Santa is more than that of

Banta by 25%, then by how much percentage Banta's income

is less than that of Santa?

Solution: 13

Banta=25/(100+25)*100%

Therefore income of Banta is 20%

less than of Santa.

Evaluate your grasp:

1. A number 'A' exceeds 'B' by

25%. By what per cent is 'B' short

of 'A'? 2. The daily wage is increased by 15 %, and a person now

gets Rs 23 per day. What was his daily wage before the

increase?

3. The ratio of number of boys and girls in a school is 3 : 2 if

20% of the boys and 25% of the girls are holding

scholarship, find the % of school students who hold

scholarship.

4. A reduction of Rs 2 per kg enables

a man to purchase 4 kg more sugar for

Rs 16. Find the original price of sugar.

5. From a man's salary, 10% is

deducted on tax, 20% of the rest is

spent on education, and 25% of the

rest is spent on food. After all these

expenditures, he is left with Rs 2,700.

Find his salary.

6. Increase 200 by 60%

7. Decrease 200 by 40 %

8. The weight of a sand bag is 40 kg. In a hurry, it was

weighed as 40.8 kg. Find the error percentage.

9. If X is 20% less than Y,then find: x/y = ?

10. If 2 1/2%( 2 and a half percent) of the weight of a

table is 0.2 kg, then what will be 120% of it?

11. Find 0.02% of 6500

12. 7/8 is what % of 144?

13. If the price of 1 Kg of cornflakes is increased by 25%,

the increase is Rs 10. Find the new price of the cornflakes

per kg?

14. If A is more than B by 10%, then find A/B.

15. 12 is 25% of 20% of what?

16. Express 7x/y in terms of percentage.

Answers:

Question

No.

Correct

Answer:

Question

No.

Correct

Answer:

1 20 9 4/5

2 20 10 9.6 kg

3 22% 11 13/10

4

Rs 4 per

kg. 12 0.607%

5 Rs 5000 13 Rs 50

6 320 14 11/10

7 120 15 240

8 2% 16 700x/y

Part B: Profit, Loss and Discount Solved Examples:

Example1: By selling an article for Rs 450, a man loses 25%.

At what price should he sell in order to gain 25%?

Solution1: S1/(100 + x1)=S2/(100 + x2)

(S1and S2 are two selling prices)

450/(100 + ( - 25)) = S2/(100 + 25) .: S2 = 750.

[(-)ve sign indicates loss]

He sells the article at Rs 750.

Example2: The cost price of 25 articles is equal to the

selling price of 20 articles. Find the gain %.

Solution2: As per question, 25 * CP = 20 * SP

SP/CP = 25/20

% gain = (SP/CP - 1) * 100

(25/20 - 1) * 100 = 25 %

There is 25% gain in the transaction.

Example3: A person sells 36

oranges per rupee and suffers a

loss of 4%. Find how many

oranges per rupee to be sold to

have a gain of 8%?

Solution3: Always find the unit

price, i.e. for one orange. Here,

Sale price per orange = Rs

1/36=S1

S1/(100+x1)=S2/(100+x2)

(1/36)/(100+(-4)) =S2/(100+8)

.:S2=1/32

He sells 32 oranges per rupee.

Example4: A shopkeeper purchases 10 kg of rice at Rs 600

and sells at a loss as much the selling price of 2 kg of rice.

Find the sale rate of rice/kg.

Solution4: Let Selling price be Rs x /kg. Loss = C.P. ­S.P.

2*x=600-10*x .: x = Rs 50 per kg.

(Since loss of 2 kg of SP. of rice) Hence the Selling price of

rice is Rs 50 per kg.

Example5: By selling a horse for Rs 455, a man loses 9%. If

he sells it for Rs 555, what would be his gain or loss per

cent?

Solution5: S1/(100 + X1)=S2/(100 + X2)

455/(100+(-9))=555/(100+X2) .:X2=+11%.

[( + )ve sign indicates it is gain.] The man has a gain of 11 %.

Example6: If a merchant estimates his profit as 20% of the

selling price, what is his real profit per cent?

Solution6: Real profit is that which is calculated on CP

Profit %=% profit on SP/(100-% profit on SP)*100

=20/(100-20)*100=25%

NB: Real % profit is always more than the % profit on S.P.

Example7: How much per cent above the cost price should a

shopkeeper mark his goods so as to earn a profit of 26%

after allowing a discount of 10% on the marked price?

Solution7: Marked price* (1 - % discount) = Cost price (1+ %

gain)

[M = Marked price, C = Cost price]

M *(100 - d) = C* (100 + g)

M *(100 - 10) = C *(100 + 26)

M =126/ 90 C = 1.4 C = (1 + 0.4) C

i.e. M is + 0.4 or 40% above C

Marked price is 40% above the cost price.

Example8: A vendor sells 10

apples for a pound gaining

thereby 40%. How many apples

did he buy for a pound?

Solution8: Always the unit price

is to be put. i.e. Sale price for 1

apple = 1/10

% x = (SP/CP - 1) * 100.

40 = [1/10C- 1] * 100

1/10C = 40/100 + 1 = 14/10

C = 1/14. So, he bought 14 apples

per pound.

Example9: A man sold two watches for Rs 1000 each. On one

he gains 25% and on the other, 20% loss. Find how much %

does he gain or lose in the whole transaction?

Solution9: Here, S1= S2, Overall % gain or loss

={1-[2*(100+X1)* (100+X2)]/[(100+X1) +(100+X2)]} * 100 %

={1-[2*(100+25)*(100-20)]/[(100 + 25) + (100 - 20) ]}*100%

=[1-(2 * 125 * 80)/205] * 100 =100/41 %

.:The man had 2 (18/41) % loss in the whole transaction.

Example10: A cloth merchant says that due to slump in the

market, he sells the cloth at 10% loss; but he uses a false

metre-scale and actually gains 15%. Find the actual length of

the scale.

Solution10: Here cost price is not

equal to selling price because he

sells the cloth at 10% loss.

(100 + G) / (100+x) = True Scale/

False Scale

Here, overall gain G = 15% and loss

x = -10%

[(-)ve sign for loss.] Let false scale length = x cm

(100+15)/( 100-10) =100/x

x = 90/115 * 100 = 78.25 cm.

.: Actual length of scale is 78.25 cm instead of 1 metre.

Example11: A man sells a book at a profit of 20%. If he had

bought it at 20% less and sold it for Rs 18 less, he would

have gained 25%. Find the cost price of the book.

Solution 11: Assume the cost price of the book = Rs 100

It sells at 20 % profit. .: SP =100 * 1.2 = 120

If he bought it at 20% less, i.e

CP= Rs 80 and sells at 25 % profit then, SP=80 * 1.25 = 100

So, S1 - S2 = Rs 20 when cost price is . Rs 100

but S1-S2 =Rs 18, so, the cost price is Rs 90

Hence, the cost price of the book is Rs 90.

Example12: What profit

percent is made by a

farmer selling eggs at a

certain price if by selling

at 3/4 of that price

there may be a loss of

10%?

Solution 12: 3/4 (SP) =

90 % of CP .: S.P. = 120%

of CP

.:The eggs been sold at a

profit of (120 - 100), i.e

20%.

Example13: By selling 66 metres of cloth, I gain the selling

price of 22 metres. Find the gain percent.

Solution13: Here Gain = sell price of 22 metres = x (say)

% gain = x/(N-x)*100 %, where x=22 and N = 66

% gain =22/( 66 - 22) * 100 % = 50 %.

Practice Questions:

1. A man sold his book for Rs 891, thereby gaining 1/10 of its

cost price. The cost price is:

2. A shopkeeper earns a profit

of 12% on selling a book at

10% discount on the printed

price. The ratio of the cost

price and printed price is:

3. A shopkeeper increased the

price of a product by 50%

from its cost but had to sell at

a 50% discount. The

shopkeepers loss was:

4. A shopkeeper purchased a

electric heater marked at Rs 200 at successive discounts of

10% and 15% respectively. He spent Rs 7 on packaging and

sold it at Rs 200. Find his gain %.

5. The cost price of 20 pencils is equal to the selling price of

25 pencils. The loss % in the transaction is:

Ans Key:

1. Rs 810 2. 45:56

3. 25% 4. 25

5. 20