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