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## Note on Compound Interest

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Suppose, Deepak borrows Rs. 1000 at 10% interest from Luna. The simple interest on this sum at the end of one year will be $$Rs. \frac {1000 \times 1 \times 10} {100}$$ = Rs. 100. If Deepak pays this interest to Luna, Luna can get back Rs. 1100. In case, Deepak pays this interest to Luna then Luna has right to charge interest on Rs. 1100 for next year. The compound interest is infact a simple interest computed on the previous simple amount. When Deepak calculates the simple interest for 2 years then,

$$I = \frac {1000\times 2\times 10} {100}$$ = Rs. 200 but when he calculates the compound interest for 2 years then compound interest for 2 years = simple interest for second year
$$=\frac{1000 \times 1 \times 10} {100} + \frac{1100\times 1 \times 10} {100}$$ (Principal for 2nd year = Rs. 1000 + interest of 1st year)

=100 + 110

= Rs. 210

In this process we get Rs. 10 profit by the way of compound interest.

The following points should be remembered before calculating the compound interest.

1. The compound interest for every succeeding year is always greater than the compound interest for the previous year.
2. The amount of the previous year becomes the principal for the coming year.
3. The final amount is equal to the sum of the original principal and the interest for all the years.
4. The compound interest for the entire period is the sum of the interest for all the years that is the difference between the final amount and the original principal.

The installment is the regular interval of time in which the compound interest is calculated. The payment might be yearly, half-yearly, quarterly, monthly, daily etc. Here we use only yearly and half-yearly installments.

#### Derivation of yearly compound interest

 Year Principal Time Rate Interest Amount 1st P 1 year R% $$\frac {PR}{100}$$ $$P +\frac {PR}{100} = P((1 + \frac {R} {100})$$ 2nd $$P (1 + \frac {R}{100})$$ 1year R% $$P\times(1 + \frac {R} {100})\times \frac {R}{100}$$ $$P\times(1 + \frac {R} {100}) + P\times(1 + \frac {R} {100})\times \frac {R}{100}$$ $$= P\times(1 + \frac {R} {100})\times (1 + \frac {R} {100})$$ $$= P\times(1 + \frac {R} {100})^2$$ 3rd $$P (1 + \frac {R}{100})^2$$ 1year R% $$P\times(1 + \frac {R} {100})^2\times \frac {R}{100}$$ $$P\times(1 + \frac {R} {100})^2 + P\times(1 + \frac {R} {100})^2 \times\frac {R}{100}$$ $$=P\times(1 + \frac {R} {100})^2 \times(1 + \frac {R} {100})$$ $$=P\times(1 + \frac {R} {100})^3$$ 4th $$P\times(1 + \frac {R} {100})^{T-1}$$ 1year R% $$P\times(1 + \frac {R} {100})^{T-1} \times \frac {R} {100}$$ $$P\times(1 + \frac {R} {100})^{T + 1-1}$$ $$=P\times(1 + \frac {R} {100})^T$$

So, the yearly compound amount for T years at R% p.a. = $$P\times \left (1 + \frac {R} {100}\right)^T$$

Compound interest for T years = Compound amount for T years - original principal

$$= P \times \left(1 + \frac {R} {100}\right)^T - P$$

$$=P \left \lbrace \left (1 + \frac {R} {100} \right )^T - 1 \right \rbrace$$

#### Half-yearly compound interest

In case of half-yearly compoundinterest, time will be double and rate will be halved.

Since yearly compound interest = $$P \left (1 + \frac {R} {100} \right ) ^ T - P$$

$$= P \times \left(1 + \frac {R} {100}\right)^2T - P$$

$$=P \left \lbrace \left (1 + \frac {R} {100} \right )^2T - 1 \right \rbrace$$

a. The amount of the previous year becomes the principal for the coming year.

b. The compound interest for every succeeding year is always greater than compound interest for the previous year.

c. The final amount is equal to the sum of the original principal and the interest for all the years.

d. The compound interest for the entire period is the sum of the interest for all the year that is a difference between the final amount and the original principal.

.

### Very Short Questions

Solution:

Principal(P) = Rs 5000,
Rate (R) = 12%
Time (T) = 2 years
Compound interest (C.I) = ?
Now,

\begin{align*} C.I &= P \left[ \left( 1 + \frac{R}{100}\right)^T - 1\right]\\ &= 5000 \left[ \left( 1 + \frac{12}{100}\right)^2 - 1\right]\\ &= 5000[(1.12)^2 -1]\\ &= 5000[1.2544 - 1] \\ &= 5000 \times 0.2544\\ &= Rs \: 1272\: \: _{Ans.}\end{align*}

Solution:

Principal (P) = Rs 700
Compound interest (A) = Rs 847
Time (T) = 2 years
Rate of interest (R) = ?

\begin{align*} Compound \: amount \: (CA) &= P \left(1 + \frac{R}{100}\right)^T \\ or, 847 &= 700\left(1 + \frac{R}{100}\right)^2 \\ or, \frac{847}{700} &= \left(1 + \frac{R}{100}\right)^2 \\ or, 1.21 &= \left(1 + \frac{R}{100}\right)^2\\ or,(1.1)^2 &= \left(1 + \frac{R}{100}\right)^2 \\ or, 1.1 &= 1 + \frac{R}{100}\\ or, \frac{R}{100} &= 1.1 - 1\\ or, R &= 0.1 \times 100 \\ \therefore R &= 10\% \:\:\:_{Ans.}\end{align*}

Solution:

Principal(P) = Rs 50,000
Time (T) = 2 years
Rate (R) = 10%
Compound interest (C.I) = ?

\begin{align*} C.I &= P \left[ \left( 1 + \frac{R}{100}\right)^2T - 1\right]\\ &= 50,000\left[ \left( 1 + \frac{10}{100}\right)^{2 \times 2} - 1\right] \\ &= 50,000 [(1 + 0.05)^4 - 1]\\ &= 50,000 [1.2155 -1] \\ &= 50,000 \times 0.2155 \\ &= Rs \: 10775 \:\: \: _{Ans.} \end{align*}

Solution:

Principle (P) = Rs 12,000
Time (T) = 1 year
Month(M)=6 month
Rate (R) = 10%
Compound interest (C.I) = ?

\begin{align*} C.I &= P \left[ \left( 1 + \frac{R}{100}\right)^T \left( 1 + \frac{MR}{1200}\right)- 1\right]\\ &= 12,000 \left[ \left( 1 + \frac{10}{100}\right)^1 \left( 1 + \frac{6 \times 10}{1200}\right) - 1 \right] \\ &= 12,000 [(1.1) (1.05) - 1]\\ &= 12,000\: [1.155 - 1] \\ &= 12,000\times 0.155 \\&= Rs 930\\ \end{align*} $$\therefore Compound \: interest = Rs \: 930 \:\: _{Ans.}$$

Solution:

Principle (P) = Rs 25,000
Time(T) = 2 years
Rate (R1) = 4%
Rate (R2) = 5%
Compound Amount (CA) = ?

\begin{align*} Compound \: amount \: (CA) &= P \left(1 + \frac{R_1}{100}\right) \left(1 + \frac{R_2}{100}\right)
\\ &= 25000\left(1 + \frac{4}{100}\right) \left(1 + \frac{5}{100}\right) \\ &= 2500 \times 1.04 \times 1.05 \\ &= Rs \: 27300 \:\:\: _{Ans.} \end{align*}

Solution:

Compound Amount (CA) = Rs 5191.68
Time (T) = 1 year
Rate (R) = 8%
Principal (P) = ?

\begin{align*} Compound \: amount \: (CA) &= P \left(1 + \frac{R}{200}\right)^{2T} \\ or, 5191..68 &= P \left(1 + \frac{8}{200}\right)^2\\ or, 5191.68 &= P(1.04)^2\\ or, P &= \frac{5191.68}{1.0816}\\ &= \: Rs \: 4800 \end{align*}

$$\therefore$$ The principal (P) = Rs \: 4800 $$_{Ans.}$$

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• ### Santosh borrowed Rs130000 from Suresh at the rate of 21% per annum. Find the simple interest at the end of 2 years and interest compounded yearly.

Rs 54600,Rs 60333

Rs 40000, Rs45000

Rs  46000, Rs 48222

Rs 60500, Rs 70000

Rs 21745

Rs 22678

Rs 20000

Rs 24451.87

• ### A man deposited Rs 2000  in the fixed deposit account of the bank for two years at the rate of 12% per annum. The interest is compounded  is semi-annually.How much will be the amount and the compound interest at the end of 2 years?

Rs 2289,489.3

Rs 2124.44,Rs 505.65

Rs 2524.95, Rs 524.95

Rs 2222.22, Rs 421.67

Rs 75

Rs 60

Rs 65

RS 70

Rs 305

Rs 205

Rs 405

Rs 505

Rs 4660

Rs 48000

Rs 45000

Rs 5000

Rs 42000

Rs 38000

Rs35000

Rs 4000

Rs 19

Rs 18

Rs 20

Rs 15

Rs 13000

Rs 11500

Rs 13500

Rs 12000

Rs 14000

Rs 20000

Rs 16000

Rs 12000

Rs 275

Rs 150

Rs 200

Rs 300

5 years

2 years

6 years

1 years

3years

10 years

6 years

2 years

20%

17%

10%

15%

Rs 2000

Rs 1500

Rs 1200

Rs 2500

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##### PRATIK ACHARYA

if SI=1000 and CI=2050 and T=2years find principal and rate