Question

What is be the compound interest (in Rs.) accrued on an amount of Rs. 15000 at the rate of 20 per cent annum in two years, if the interest is compounded half-yearly?

Note- Full Explanation needed
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Answer 1

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Answer 2

Answer:

₹6961.5 is be the compound interest accrued if the interest is compounded half-yearly.

Step-by-step explanation:

Given:

• Principal = 15000

• Time = 20 years

• Rate of interest (half-yearly) = 20/2= 10%

To Find Out:

The compound interest accrued if the interest is compounded half- yearly.

Explanation:

Rate of interest (half-yearly)

= 20/2=10%

Now,

Principal = 15000

Time - 2 = 4 half years

By the net% effect we would calculate the effective compound rate of interest for 4 half years = 46.41% (Refer to sub-details)

Therefore,

$\rm{CI = 46.41\% \: of \: 15000} \\ \\ CI = \dfrac{46.41 \times 15000}{100} \\ \\ \: \: \: = { \blue{ 6961.5}}$

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Sub - Details:

Calculation of effective compound rate of interest for 4 half years will be as follows.

For the first 2 half years, let's apply the net% effect.

Here, x = y = 10%

$\rm{Net\% \: Effect = x + y = \dfrac{xy}{100}} \\ \\ = 10 + 10 + \dfrac{10 + 10}{100} \: \: \: \: \: \: \: \: \: \\ \\ = 10 + 10 + \frac{ \cancel{10 \times 10}}{ \cancel{100}}^{ \large \: \: \: \: \: 1} \\ \\ = 10 + 10 + 1 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ = \bf\blue {21\%} \: \: \: \: \: \: \: \: \:$

Now let's take this 21% as x and 10% as y for the calculation of 3rd half year.

$= 21 + 10 + \dfrac{21 \times 10}{100} \\ \\ \bf = 33.1\% \: \: \: \: \: \: \: \: \: \:$

Similarly, let's take this 33.1% as x and 10% as y for the calculation of 4th half year.

$= 33.1 + 10 + \dfrac{33.1 \times 10}{100} \\ \\ = 43.1 + 3.31 \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ = \bf46.41\% \: \: \: \: \: \: \: \: \: \: \:$

Hence, Compound Rate is 46.41%.

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Traditional Method:

If interest is compounded half-yearly then time

(t) = 2 × 2 = 4;

R% = 20/2 = 10%

$A = [p( 1 + \dfrac{r}{100})^{ \bf t} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ = 15000[(1 + \frac{10}{100}) ^{ \bf 4} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ = 15000 \times \frac{11}{100} \times \frac{11}{100} \times \frac{11}{100} \times \frac{11}{100} \\ \\ = \bf\blue{ 21961.5} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

$\therefore CI = 21961.5 - 15000 \\ \\ = \bf\blue{₹6961.5} \: \: \: \: \: \: \: \: \: \: \: \: \:$

Hence, ₹6961.5 is be the compound interest accrued if the interest is compounded half-yearly.

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Shortcut Used:

P = Principal

R = Rate

T = Time

CI = Compound interest