Question

Avi took a loan of 50,000 from the bank at a rate of 15%. Find the compound interest after 2 years when the interest is: (a) Compounded annually. (b) Compounded half yearly. ​

Answer 1

$\large\underline{\sf{Solution-a}}$

Compounded annually

Principal, P = 50000

Rate of interest, r = 15 % per annum compounded annually

Time, n = 2 years

We know,

Amount received on a certain sum of money of Rs p invested at the rate of r % per annum compounded annually for n years is given by

$\boxed{\tt{ \: \: Amount \: = \: p \: {\bigg[1 + \dfrac{r}{100} \bigg]}^{n} \: }}$

So, on substituting the values, we get

$\rm \: \: \: Amount \: = \: 50000 \: {\bigg[1 + \dfrac{15}{100} \bigg]}^{2} \:$

$\rm \: \: \: Amount \: = \: 50000 \: {\bigg[1 + \dfrac{3}{20} \bigg]}^{2} \:$

$\rm \: \: \: Amount \: = \: 50000 \: {\bigg[\dfrac{20 + 3}{20} \bigg]}^{2} \:$

$\rm \: \: \: Amount \: = \: 50000 \: {\bigg[\dfrac{23}{20} \bigg]}^{2} \:$

$\rm\implies \:Amount \: = \: 66125$

So,

Compound interest is given by

$\rm \: Compound\:interest \: = \: Amount \: - \: Principal$

$\rm \: Compound\:interest = 66125 - 50000$

$\rm\implies \:Compound\:interest \: = \: 16125$

$\large\underline{\sf{Solution-b}}$

When compounded half yearly

Principal, P = 50000

Rate of interest, r = 15 % per annum compounded half yearly

Time, n = 2 years

We know,

Amount received on a certain sum of money of Rs p invested at the rate of r % per annum compounded half yearly for n years is given by

$\boxed{\tt{ \: \: Amount \: = \: p \: {\bigg[1 + \dfrac{r}{200} \bigg]}^{2n} \: }}$

So, on substituting the values, we get

$\rm \: \: \: Amount \: = \: 50000 \: {\bigg[1 + \dfrac{15}{200} \bigg]}^{4} \:$

$\rm \: \: \: Amount \: = \: 50000 \: {\bigg[1 + \dfrac{3}{40} \bigg]}^{4} \:$

$\rm \: \: \: Amount \: = \: 50000 \: {\bigg[\dfrac{40 + 3}{40} \bigg]}^{4} \:$

$\rm \: \: \: Amount \: = \: 50000 \: {\bigg[\dfrac{43}{40} \bigg]}^{4} \:$

$\rm\implies \:Amount \: = \: 66773.46$

So,

Compound interest is given by

$\rm \: Compound\:interest \: = \: Amount \: - \: Principal$

$\rm \: Compound\:interest = 66773.46 - 50000$

$\rm\implies \:Compound\:interest \: = \: 16773.45$

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ADDITIONAL INFORMATION:-

1. Amount received on a certain sum of money of Rs p invested at the rate of r % per annum compounded quarterly for n years is given by

$\boxed{\tt{ \: \: Amount \: = \: p \: {\bigg[1 + \dfrac{r}{400} \bigg]}^{4n} \: }}$

2. Amount received on a certain sum of money of Rs p invested at the rate of r % per annum compounded monthly for n years is given by

$\boxed{\tt{ \: \: Amount \: = \: p \: {\bigg[1 + \dfrac{r}{1200} \bigg]}^{12n} \: }}$

Answer 2

Given

• Principal = ₹50,000.
• Rate = 15%.
• Time = 2years.

To Find

• (a) Compounded annually.
• (b) Compounded half-yearly.

Formula to be used

• I = P×R×T/100.
• A = P + I.

Solution

This problem is solved by simple interest method.

(a) Compounded annually

Principal = ₹50,000.

Rate = 15%.

Time = 2years.

For the first year

P = ₹50,000.

R = 15%.

T = 1year.

I = P×R×T/100

⇢ I = 50000×15×1/100

⇢ I = 500 × 15

⇢ I = ₹7,500.

Amount = P + I

⇢ Amount = 50000 + 7500

⇢ Amount = ₹57,500.

For the second year

Amount of first year = Principal of second year.

P = ₹57,500.

R = 15%.

T = 1year.

I = P×R×T/100

⇢ I = 57500×15×1/100

⇢ I = 575 × 15

⇢ I = ₹8,625.

Final Amount = P + I

⇢ Final Amount = 57500 + 8625

⇢ Final Amount = ₹66,125.

C.I. = Final Amount - Original Principal

⇢ C.I. = 66125 - 50000

⇢ C.I. = ₹16,125.

(b) Compounded half-yearly

When interest is compounded half-yearly then we have multiply the time by 2 and divide the rate by 2.

P = ₹50,000.

R = 15% = 15/2 = 7.5%.

T = 2years = 4years.

For the first year

I = P×R×T/100

⇢ I = 50000×7.5×1/100

⇢ I = 500 × 7.5

⇢ I = ₹3,750.

A = P + I

⇢ A = 50000 + 3750

⇢ A = ₹53,750.

For the second year

P = ₹53,750.

R = 7.5%.

T = 1year.

I = P×R×T/100

⇢ I = 53750×7.5×1/100

⇢ I = ₹4,031.25.

A = P + I

⇢ A = 53750 + 4031.25

⇢ A = ₹57,781.25.

For the third year

P = ₹57,781.25.

R = 7.5%.

T = 1year.

I = P×R×T/100

⇢ I = 57781.25×7.5×1/100

⇢ I = ₹4,333.59

A = P + I

⇢ A = 57781.25 + 4333.59

⇢ A = ₹62,114.84.

For the fourth year

P = ₹62,114.84.

R = 7.5%.

T = 1year.

I = P×R×T/100

⇢ I = 62114.84×7.5×1/100

⇢ I = ₹4,658.61.

Amount = P + I

⇢ Amount = 62114.84 + 4658.61

⇢ Final Amount = ₹66,773.45.

C.I. = Final Amount - Original Principal

⇢ C.I. = 66773.45 - 50000

⇢ C.I. = ₹16,773.45.

Final Answer

• (a) Compounded Annually = ₹16,125.
• (b) Compounded half-yearly = ₹16,773.45.

Used Abbreviations

P = Principal.

A = Amount.

R = Rate.

T = Time.

I = Simple Interest.

C.I. = Compound Interest.

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