Profit & Loss
In this chapter the concepts of Profit &
loss, Simple Interest & Compound Interest and Stocks & shares will
be covered in detail. You are advised to go through the topics of percentages and averages thoroughly before you take up these topics. Few examples are given at the beginning of this chapter which will help you to recollect concepts, which were taught in
your classes. Also, in the following
tables, few standard results are given
which you must learn by heart, that will help you in exercises to come.
For fixed
total expenditure
Price goes up by .... %
|
Consumption comes down by ....%
|
20
|
16.66
|
25
|
20
|
33.33
|
25
|
50
|
33.33
|
100
|
50
|
For fixed total expenditure
Price comes down by .... %
|
Consumption goes up by ....%
|
 20 |
25
|
25
|
33.33
|
33.33
|
50
|
50
|
100
|
75
|
300
|
El. The decrease in the price of petrol in the market by
25% led a man to increase his
consumption of petrol by so much that his total expenditure on petrol did not
change. What was the
percentage increase?
Sol. Let E
be the expenditure, P be the original price of petrol and Co be the original consumption.
Then E=P x
CO ... (1)
The new price of petrol is 25% lower than
original value, hence new price = 0.75 x P.
The expenditure should remain the same and so let us assume that the new
consumption is Cn, such that E = 0.75P x Cn ...
(2)
Equating (1) and (2), we get P x Co = 0.75P x Cn.
Thus,
Cn = C0/0.75 =
1.33C0.
Thus,
he should increase his consumption by 33%.
Short-cut: Applying
formula given above, the percentage increase in consumption = 100 x 25 = 2500 = 33%
100 – 25 75
Percentage error
= The error x
100 %
True value
Successive change in percentage
If a number A is
increased successively by X% followed by Y% and then by Z%, then the final
value of A will be
A(1 + X/100 )(1 + Y/100) (1 +
Z/100)
In a
similar way, at any point or stage, if the value is decreased by any percentage, then
we can replace the same by a negative sign. The same formula can be used for two or more successive changes. The final value of A in
this case will be
A(1 - X/100 )(1 - Y/100) (1 - Z/100)
Percentage change and
effect on products
Let the
expenditure on a commodity (E) = price (P) x consumption (C). If price and consumption each are increased by 20% and 25% respectively, then the
total increase in expenditure will be New E = 1.2P
x 1.25C = 1.5PC = 1.5E = 50% increase.
The net percentage
change when two variables are increased /
decreased by given percentages, say a% and
b% will be
a + b + ab/100
For the same data given above, applying
the formula, we get the net percentage change in
expenditure
=20+25+20x25 =20+25+5= 50%.
100
In case a given value decreases by any
percentage, we will use a negative sign before that.
E2. If the length of a rectangle is decreased by 40% and the breadth is increased by 30%, then find the
percentage change in the area of the rectangle.
Sol. Area of rectangle = length x breadth
Here, both length and breadth are changed. So, using the
formula,
net percentage change in area = — 40 + 30 + (-40)(30)
100
= -22, (-)ve sign signifies decrease.
Hence the area of
the rectangle decreases by 22%.
The
above formula can be used to find out the following:
(1)
Percentage effect on expenditure.
(2)
Percentage effect on area of
rectangle/square.
Profit and Loss
Suppose a shopkeeper buys an article
from a manufacturer. The price at which he buys the article is called the cost price
of the article. We write C.P. for cost price.
The shopkeeper sells the
article at a price which is generally more than its cost price. The amount for
which he sells the article is known
as the selling price. We write S.P. for selling price.
The excess of
the selling price over the cost price of an article is called the
profit or the gain. So,
Gain or Profit = Selling Price — Cost Price
Sometimes, the shopkeeper has to sell
the article for a price which is less than its cost price. In this case, the excess
of the cost price over the selling price is called the loss. So,
Loss = Cost price — Selling price
Gain or Profit = Selling Price — Cost Price
Profit or Gain = S.P. — C.P.
Loss = C.P. — S.P
Profit and loss are always calculated with C.P. as the base.
Marked Price (M.P.): The price at which the article is
marked. If the article is sold at this price,
then the selling price (S.P.) is
equal to marked price (M.P.). But generally some discounts might be available on the marked price, then marked price
less discount will be equal to the selling price. Item at selling price
Item at selling price
= CP + Y
Add
gain (+Y)
Item at a cost price = CP
Less
Loss (—X)
Item
at selling price =
CP — X
Gain% = Gain x 100% = Y x
100%
C.P C.P
Loss% = Loss/CP x 100% = X/CP x 100%
SP = CP x 100+Gain% or CP x 100—Loss%
100 100
CP = SP x
100 or SP
x 100
100+Gain% 100—Loss%
If C.P. of both the items is same and the
percentage loss and gain are equal, then net
loss or profit is zero.
E3. Two shirts
were having a cost price of Rs.200 each. One was sold at a profit of 15% and
the other was sold at a loss of 15%. Find the net profit or loss.
(100 +
15)
Sol. SPI
=200x 100 = Rs.230 .
(100-15)
SP2 = 200 x 100 = Rs.170.
Total S.P. received = S.P1 + S.P2 = 230 + 170 =
Rs.400 Total C.P. = 200 + 200 = 400.
Net result: No profit, No Loss.
Important
If two
items are SOLD, each at rupees S, one at a gain of X % and other at a
loss of X%, then the net result is always a loss.
Item having CP1
Sold at a loss of X%SP Net Result:
(Same for both
the items) Loss
Sold
at a profit of X%
Item having CP2
E4. Two articles were sold at Rs.100 each.
After selling it was realized that on one, a profit of 10% was made, and on the
other, a loss of 10% was made. What is the net result?
Sol.
|
Item 1
|
Item 2
|
Selling
Price
|
Rs.100
|
Rs.100
|
Profit
%
|
10
|
-
|
Loss %
|
—
|
10
|
Cost
Price
|
100/1.1
|
100/0.9
|
= 90.90
|
=
111.11
|
Total
S.P. received = 100 + 100 = Rs.200.
Total
C.P. = 90.90 + 111.11 = Rs.202.01.
Loss% = 200-202.01 x 100 = -2.01
x 100 = -1%.
202.01 202.01
the same calculation can be done
by a very simple formula
Loss % = X2 / 100 and
Value of loss = 2X2S
1002 — X2
Where
X is the percentage profit and loss made on each of the items and S is the
common selling price received on both.
In case of discounts being offered, the price
on which the discount was offered is known as the marked price and the price that is finally received is known as
the selling price.
E5.
A merchant
purchases an item for Rs.500. He marks the item at a price of Rs.700 but
allows a discount of 10% on cash payment.
What is the total profit in terms of amount and percentage made by the
merchant?
Sol. C.P. =
Rs.500, M.P. = Rs.700.
Hence S.P. = 700 (1 - 10/100) =
Rs.630.
Thus, profit = Rs.630 — Rs.500 =
Rs.130. Profit % = (130/500) x 100 = 26%.
If
a trader uses a false scale for selling his goods, then the overall gain made by
him in this process will be
Gain
% = ( error x 100
)
True value - error
E6.
A milkman claims to sell milk at the cost price
but uses a measure of 800 ml instead of a litre. Find the net profit
made by him.
Sol. Using the
formula given above 200
Gain% =
200 x 100 = 200 / 800 x 100 = 25.
[1000 -200]
If a tradesman defrauds (by means of a false balance or
otherwise) to the tune of X% in buying and also defrauds to the tune of X% in
selling, his overall percentage gain will be
(100 + common gain %)2 - 100 %
100
E7.
A trader
defrauds the seller by 10% when he purchases goods from him, and
while selling the same to a customer, he defrauds once again by 10%.Find the
net gain made by the trader.
Sol. The
required answer is
(100+10) - 100
% = 12100 – 100 = 121
- 100 = 21%
100 100
If a tradesman defrauds (by means of a false balance
or otherwise) to the tune of X% in buying and also defrauds to the tune of Y%
in selling, his overall percentage gain will be
(100 + X%)(100+Y%) - 100
0/o
100
E8. A milkman defrauds by means of a false measure to the tune of 20% in buying and also defrauds to
the tune of 25% in selling. Find his overall % gain.
Sol. The
milkman defrauds 20% in buying and also defrauds 25% in selling, so his
overall % gain will be
(100 + 20%)(100 + 25%) —100 % = 50%.
100
If selling price of X articles is equal to the cost price of
Y articles, then the net profit percentage is given by
= Y—X x100
X
E9.
The cost
price of 20 pens is equal to the selling price of 25 pens. What is the net loss
percentage?
Sol.
Here, cost price of 20 pens = selling price of 25 pens.
So the net loss
percentage = 20 - 25 x 100 = —20%. Here minus sign indicates the loss.
25
E10.
Bhuvan,
a fruit seller bought bananas at the rate of Rs.5 a dozen. He sold 2 bananas
for Re.1. Find his net profit percentage
Sol. Here, cost price of a
dozen = Rs.5. Also, selling price of 10 bananas = Rs.5.
Since, selling price of 10 bananas is
equal to the cost price of 12 bananas, so the net profit percentage
12-10 x100=20%
10.
In case of the partnership business (where more
than one person is involved in the
business), if the period of investment is the same for each partner, then
the profit or loss is divided in the ratio of their investments.
If X and Y are partners in a business, then
Investment of X = Profit
of X or Investment
of X = Loss
of X
Investment of Y Profit of Y Investment of Y Loss of Y
E11.A and B together invested Rs.12000 in
a business. At the end of the year, out of a total profit of Rs.1800, A's share was Rs.750. What was the investment
of A?
Sol.
Since profits are shared in the ratio of
their investments
A's
investment = Profit
share of A
B's
investment Profit share
of B
(Money invested by A and B for the
same period)
= 750 = 750
= 5
1800 — 750 1050 7
Investment of A = 5/5+7 x 12000 = Rs.5000
E12.A and B together
started a business by investing Rs.5000 and
Rs.7000 respectively. At the end of the year, the total loss was Rs.1800. What
is the share of A and B in loss?
Sol. Ratio
of investment of A and B is 5000 : 7000 = 5 : 7.
Share of A in loss =
1800 x 5 = Rs.750 and the share
5+7
of B in loss =1800 x 5+7 = Rs.1050 .
E13.In a business A,
B, and C invested Rs.380, Rs.400 and Rs.420 respectively. Divide a net profit of Rs.180 among the partners.
Sol. A's profit : B's profit :
C's profit
= A's investment : B's investment : C's
investment = 380 : 400 : 420 = 19 : 20 : 21.
Profit share of A =
19/60x 180 = Rs.57. Profit
share of B = 20/60 x 180 = Rs.60.
Profit share of C =21/ 60 x 180 = Rs.63.
E14.A started a
business with a capital of Rs.10000. Four months later, B joined him with a capital of Rs.5000. What is the
share of A in a total profit of Rs.2000 at
the end of the
year?
Sol. Profit of A = Amount x No. of months
Profit of B
Amount x No. of months
= 10000x12 =
3
5000 x 8
profit share of A =
3 x 2000
= Rs.1500
3 + 1.
If more than two
persons invest money in a business, then MEI of A : MEI of B : MEI of C
= Profit for A :
Profit for B : Profit for C.
E15.A, B and C enter
into a partnership. A contributes Rs.320 for
4 months, B contributes Rs.510 for 3 months and C contributes Rs.270 for 5 months. If the total profit is Rs.208, then
find the profit share of each of the partners.
Sol. As profit : B's
profit : C's profit
=
Rs.320 x 4 : 510 x 3 : 270 x 5
= 1280 : 1530 : 1350
= 128: 153 : 135
Profit of A = 128
x 208
128+153+135
=128
x
208 = Rs 64
416
Profit of B = 153 x 208
128+153+135
= 153x
208 = Rs 76.50
416
Profit of C = 135 x 208
128+153+135
=
135 x 208 = Rs.67.50.
416
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