Mathematics of Finance
 Mark up
There are two types of mark-up:
I. Dependent upon cost
𝑆𝑎𝑙𝑒 𝑃𝑟𝑖𝑐𝑒 = 𝐶𝑜𝑠𝑡 + 𝑀𝑎𝑟𝑘𝑢𝑝
𝑆𝑎𝑙𝑒 𝑃𝑟𝑖𝑐𝑒 = 𝐶𝑜𝑠𝑡 + 𝐶𝑜𝑠𝑡 𝑟𝑎𝑡𝑖𝑜
𝑺 = 𝑪(𝟏 + 𝒓)
II. Dependent upon sale
𝐶𝑜𝑠𝑡 = 𝑆𝑎𝑙𝑒 𝑃𝑟𝑖𝑐𝑒 − 𝑀𝑎𝑟𝑘𝑢𝑝
𝐶𝑜𝑠𝑡 = 𝑆𝑎𝑙𝑒 𝑃𝑟𝑖𝑐𝑒 − 𝑆𝑎𝑙𝑒 𝑟𝑎𝑡𝑖𝑜
𝑪 = 𝑺(𝟏 − 𝒓)
1. Find the selling price of a piece of furniture whose
cost of production is Rs. 1400 & percentage mark-up is
35% on cost. What is the amount of Profit per unit?
Mark-up =35%
Cost = Rs. 1400
Selling Price ?
𝑺 = 𝑪(𝟏 + 𝒓)
𝑆 = 1400(1 + 0.35)
𝑆 = 1400(1.35)
𝑺𝒆𝒍𝒍𝒊𝒏𝒈 𝑷𝒓𝒊𝒄𝒆 = 𝟏𝟖𝟗𝟎
Now we calculate profit amount: We know that
Profit = Selling Price – Cost
P = 1890 – 1400
P = 490
So, Amount of Profit is Rs. 490
2. In the year 1990, Product A was sold for Rs. 300 per unit
making a gross profit of 20% on sales. The total production
cost was made up of 25% of direct material, 40% of direct
labour & 35% of factory overhead. Due to general rises in
prices in 1995, the selling price of the product increased by
15%. The total production cost has also increased resulting
in the increase of direct material, direct labour & factory
overhead cost by 10%, 15% & 12% respectively. What will
be gross profit per unit in 1995?
In 1990,
Profit = 20% of sale & 𝑆𝑎𝑙𝑒 𝑃𝑟𝑖𝑐𝑒 = 300
𝑪 = 𝑺(𝟏 − 𝒓) ⟹ 𝑪 = 300(1 − 0.20) ⟹ 𝑪 = 300(0.8)
𝑪𝒐𝒔𝒕 = 𝟐𝟒𝟎
Cost of Material =
Cost of Labour =
Cost of F.O.H =
In 1995,
25
x 240
100
40
x 240
100
35
x 240
100
⟹ 60
⟹ 96
⟹ 84
𝑆𝑎𝑙𝑒 𝑃𝑟𝑖𝑐𝑒 = 300 + 15% 𝑜𝑓 300 = 𝟑𝟒𝟓
10
x 60 ⟹ 66
100
15
96 +
x 96 ⟹ 110.4
100
12
84 +
x 84 ⟹ 94.08
100
𝑻𝒐𝒕𝒂𝒍 𝑪𝒐𝒔𝒕 = 𝟐𝟕𝟎. 𝟒𝟖
Cost of Material = 60 +
Cost of Labour =
Cost of F.O.H =
Gross Profit = Selling Price – Cost ⟹ P = 345 – 270.5
Gross Profit = 74.5
3. The manufacturing cost of a refrigerator was Rs. 5000
composed of 30% of direct material, 25% of direct
labour & 45% of factory overhead costs. Due to
improved method of manufacturing, the cost of direct
material has decreased by 8% but direct labour & factory
overhead costs have increased by 4% & 2% respectively.
What will now be the cost of production of the refrigerator?
Old manufacturing method:
30
x 5000
100
25
x 5000
100
45
x 5000
100
Cost of Material =
Cost of Labour =
Cost of F.O.H =
⟹ 1500
⟹ 1250
⟹ 2250
𝑻𝒐𝒕𝒂𝒍 𝑪𝒐𝒔𝒕 = 𝟓𝟎𝟎𝟎
New manufacturing method:
8
x 1500 ⟹ 1380
100
4
1250 +
x 1250 ⟹ 1300
100
2
2250 +
x 2250 ⟹ 2295
100
𝑻𝒐𝒕𝒂𝒍 𝑪𝒐𝒔𝒕 = 𝟒𝟗𝟕𝟓
Cost of Material = 1500 −
Cost of Labour =
Cost of F.O.H =
4. Selling price of a box is Rs. 1250. If the mark-up on the
selling price 25% & the cost price is Rs. 937. What
should be mark-up on sales?
𝑪 = 𝑺(𝟏 − 𝒓)
𝐶 = 1250(1 − 0.25)
𝐶 = 937.5
 Simple Interest:
𝑰 = 𝑷𝒓𝒕
𝐻𝑒𝑟𝑒:
𝑰 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑆𝑖𝑚𝑝𝑙𝑒 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝑷 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑃𝑟𝑒𝑠𝑒𝑛𝑡/𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑚𝑜𝑢𝑛𝑡
𝒓 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑒𝑟 𝑎𝑛𝑢𝑚
𝒕 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑡𝑖𝑚𝑒 𝑖𝑛 𝑦𝑒𝑎𝑟𝑠
Formula of Amount in simple Interest:
𝐴𝑚𝑜𝑢𝑛𝑡 = 𝑃 + 𝑃𝑟𝑡
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒓𝒕)
5. How much money would Mr. A receive after 3 years if
he invests Rs. 12000 today at 6% simple interest?
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒓𝒕)
𝐴 = 12000(1 + (0.06 x 3))
𝐴 = 12000(1 + 0.18)
𝐴 = 12000(1.18)
𝑨 = 𝟏𝟒, 𝟏𝟔𝟎
So, Mr. A receive Rs. 14,160 after 3 years
6: Mr. A invested two sum of money on simple interest.
The first was Rs. 2200 invested at 8% per anum and the
second was Rs. 1960 at 10%. The two sum were allowed
to grow till both amounted to the same figure. In how
many years did this happen & what was the amount?
Suppose both
𝑨𝟏 = 𝑷𝟏 (𝟏 + 𝒓𝒕)
𝑨𝟐 = 𝑷𝟐 (𝟏 + 𝒓𝒕)
𝐴1 = 2200(1 + (0.08 x 𝑡))
𝐴2 = 1960(1 + (0.1 x 𝑡))
𝑨𝟏 = 𝑨𝟐
2200(1 + 0.08𝑡) = 1960(1 + 0.1𝑡)
2200 + 176𝑡 = 1960 + 196𝑡
2200 − 1960 = 196𝑡 − 176𝑡
240 = 20𝑡
𝒕=
𝟐𝟒𝟎
= 𝟏𝟐
𝟐𝟎
𝑨𝟏 = 𝑷𝟏 (𝟏 + 𝒓𝒕)
𝑨𝟐 = 𝑷𝟐 (𝟏 + 𝒓𝒕)
𝐴1 = 2200(1 + (0.08 x 12))
𝐴2 = 1960(1 + (0.1 x 12))
𝐴1 = 2200(1 + 0.96) ⟹ 4312
𝐴2 = 1960(1 + 1.2) ⟹ 4312
So, In 12 years the sum of both amounts will be same Rs.
4312/=
7: Mr. A opened two accounts to grow P by simple
interest. The first amount with a principle of Rs. 2500
grew at 8% per anum and the second with a principle of
Rs. 4400 at 10%. Till such time that the second amount
becomes double of the first amount. In how many years
did this happen & what were the two amounts?
Suppose both
𝑨𝟏 = 𝑷𝟏 (𝟏 + 𝒓𝒕)
𝑨𝟐 = 𝑷𝟐 (𝟏 + 𝒓𝒕)
𝐴1 = 2500(1 + (0.08 x 𝑡))
𝐴2 = 4400(1 + (0.1 x 𝑡))
𝑨𝟐 = 𝟐𝑨𝟏
4400(1 + 0.1𝑡) = 2 (2500(1 + (0.08 𝑡)))
4400 + 440𝑡 = 2(2500 + 200𝑡) ⟹ 4400 + 440𝑡 = 5000 + 400𝑡
440𝑡 − 400𝑡 = 5000 − 4400
40𝑡 = 600
𝒕=
𝟔𝟎𝟎
= 𝟏𝟓
𝟒𝟎
𝑨𝟏 = 𝑷𝟏 (𝟏 + 𝒓𝒕)
𝑨𝟐 = 𝑷𝟐 (𝟏 + 𝒓𝒕)
𝐴1 = 2500(1 + (0.08 x 𝑡))
𝐴2 = 4400(1 + (0.1 x 𝑡))
𝐴1 = 2500(1 + 1.2) ⟹ 5500
𝐴2 = 4400(1 + 1.5) ⟹ 11000
So, In 15 years the second amount will be double of first
amount
8: A debt of Rs. 2000 is due in 6 months if rate of interest
is 6%, what is the value of debt if it is paid after two
months?
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒓𝒕)
1
2000 = 𝑃 (1 + (0.06 x ))
3
2000 = 𝑃(1 + (0.06 x 0.334))
2000 = 𝑃(1 + (0.02)
2000 = 𝑃(1.02)
𝑃=
2000
= 1960.78
1.02
9: A debt of Rs. 2000 is due in 8 months if rate of interest
is 6%, what is the value of debt if it is paid after 3
months?
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒓𝒕)
2000 = 𝑃 (1 + (0.06 x
5
))
12
2000 = 𝑃(1 + (0.06 x 0.4167))
2000 = 𝑃(1 + (0.025)
2000 = 𝑃(1.025)
𝑃=
2000
= 1951.2
1.025
10: Mrs. A plans to purchase a T.V set. She has been
offered the option of paying Rs. 2000 down payment &
Rs. 3000 after 4 month OR of paying Rs. 3000 down
payment & Rs. 2000 after 5 months. If rate of interest is
9%, which option would be better for her?
1st Option:
𝐷𝑜𝑤𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 = 2000
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒓𝒕)
3000 = 𝑃 (1 + (0.09 x
4
))
12
3000 = 𝑃(1 + (0.09 x 0.334)) ⟹ 3000 = 𝑃(1 + (0.03)
3000 = 𝑃(1.03) ⟹ 𝑃 =
3000
1.03
= 2912
𝐓𝐨𝐭𝐚𝐥 𝐀𝐦𝐨𝐮𝐧𝐭 = 2000 + 2912 = 𝟒𝟗𝟏𝟐
2nd Option:
𝐷𝑜𝑤𝑛 𝑝𝑎𝑦𝑚𝑒𝑛𝑡 = 3000
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒓𝒕)
2000 = 𝑃 (1 + (0.09 x
5
))
12
2000 = 𝑃(1 + (0.09 x 0.4167)) ⟹ 2000 = 𝑃(1 + (0.0375)
2000 = 𝑃(1.0375) ⟹ 𝑃 =
2000
1.3753
= 1927
𝐓𝐨𝐭𝐚𝐥 𝐀𝐦𝐨𝐮𝐧𝐭 = 3000 + 1297 = 𝟒𝟗𝟐𝟕
𝐅𝐢𝐬𝐭 𝐎𝐩𝐭𝐢𝐨𝐧 𝐢𝐬 𝐛𝐞𝐬𝐭
 Compound Interest:
𝑨 = 𝑷(𝟏 + 𝒊)𝒏
𝐻𝑒𝑟𝑒:
𝑨 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝐴𝑚𝑜𝑢𝑛𝑡 (𝐼𝑛𝑐𝑙𝑢𝑑𝑒 𝑐𝑜𝑚𝑝𝑢𝑛𝑑 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡)
𝑷 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑃𝑟𝑒𝑠𝑒𝑛𝑡/𝑃𝑟𝑖𝑛𝑐𝑖𝑝𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑎𝑚𝑜𝑢𝑛𝑡
𝒊 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑝𝑒𝑟 𝑃𝑒𝑟𝑖𝑜𝑑
𝒏 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑜𝑓 𝑝𝑒𝑟𝑖𝑜𝑑
Formula of Amount in compound interest:
𝐴𝑚𝑜𝑢𝑛𝑡 = (𝑃 + 𝑃𝑖)𝑛
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
11. A man invested Rs. 5000 today at 6% compounded
semi-annually interest. What would he receive after 8½
years?
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
0.06 17
𝐴 = 5000 (1 +
)
2
𝐴 = 5000(1.03)17
𝐴 = 5000(1.65)
𝑨 = 𝟖, 𝟐𝟓𝟎
So, He receive Rs. 8,250 after 8½ years
12. Mr. A invested Rs. 1000 for which he will receive
interest at 6% compounded semi-annually for the first 5
years & 8% compounded quarterly for the next 4 years.
What compound amount would be received at the end of
the ninth year?
For first 5 years:
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
0.06 10
𝐴 = 1000 (1 +
)
2
𝐴 = 1000(1.03)10
𝐴 = 1000(1.34)
𝑨 = 𝟏𝟑𝟒𝟑. 𝟗
For next 4 years:
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
0.08 16
𝐴 = 1343.9 (1 +
)
4
𝐴 = 1343.9(1.02)16
𝐴 = 1343.9(1.37)
𝑨 = 𝟏𝟑𝟒𝟑. 𝟗
13. Find the compound amount and compound interest
when Rs. 4500 invested for 3 years and 2 months at 6%
compound semi-annually?
For first 3 years:
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
0.06 6
𝐴 = 4500 (1 +
)
2
𝐴 = 4500(1.03)6
𝐴 = 4500(1.194)
𝑨 = 𝟓𝟑𝟕𝟑. 𝟐𝟑
For next 2 months:
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒓𝒕)
𝐴 = 5373.2 (1 + (0.06 x
2
))
12
𝐴 = 5373.2(1 + (0.06 x 0.1667))
𝐴 = 5373.2(1 + (0.01)
𝐴 = 5373.2(1.01)
𝐴 = 5426.96
14. Find the present value of Rs. 2479.27 due at the end
of 8½ years if money is worth 6% compounded semiannually?
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
0.06 7
2479.27 = 𝑃 (1 +
)
2
2479.27 = 𝑃(1.03)7
2479.27 = 𝑃(1.6528)
𝑃=
2479.27
1.6528
𝑷 = 𝟏𝟓𝟎𝟎
15. If Rs. 5000 accumulate to Rs. 9739.5 in 17 years,
what is the rate of interest compounded annually?
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
9739.50 = 5000(1 + 𝑖 )17
9739.50
= (1 + 𝑖)17
5000
1.9479 = (1 + 𝑖)17
1
17
(1.9479)
=1+𝑖
1.04 = 1 + 𝑖
1.04 − 1 = 𝑖
0.04 = 𝑖
𝒊 = 𝟒%
16. How long will it take Rs. 3000 to accumulate to the
amount of Rs. 4958.54 at 6% compounded semiannually?
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
0.06 𝑛
4958.54 = 3000 (1 +
)
2
4958.54
= (1 + 0.03)𝑛
3000
1.6528 = (1.03)𝑛
Take log both sides
log 1.6528 = log1.03 𝑛
log 1.6528 = 𝑛 log 1.03
log 1.6528
𝑛=
log 1.03
𝑛=
0.1282
0.128
𝑛 = 17 𝑝𝑒𝑟𝑖𝑜𝑑𝑠 𝑎𝑛𝑑 8.5 𝑦𝑒𝑎𝑟𝑠
17. If Rs. 900 will accumulate to Rs. 1500 in 10 years,
what is the rate of interest compounded quarterly?
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
𝑖 40
1500 = 900 (1 + )
4
1500
𝑖
= (1 + )40
900
4
𝑖 40
1.667 = (1 + )
4
1
40
(1.667)
𝑖
=1+
4
1.0128 = 1 +
𝑖
4
1.0128 − 1 =
𝑖
4
0.0128 =
𝑖
4
4(0.0128) = 𝑖
0.0512 = 𝑖
𝒊 = 𝟓. 𝟏𝟐%
18. Mr. A invested Rs. 6000 in a business which is
expected to earn a profit of 12% per anum on the
balance of amount invested at the end of each year. If the
profit earned is invested in the business each year, what
would be the value of his accumulated invested at the
end of the 8 years?
𝑨𝒎𝒐𝒖𝒏𝒕 = 𝑷(𝟏 + 𝒊)𝒏
𝐴 = 6000(1 + 0.12)8
𝐴 = 6000(1.12)8
𝐴 = 6000(2.4759)
𝑨 = 𝟏𝟒𝟖𝟓𝟓. 𝟒
So, He receive Rs. 14,855.4 after 8 years