At what rate percent will a sum of Rs 64000 be compounded to Rs 68921 in 3 years?

The compound interest on a sum of Rs. 64000 at 5% per annum for a certain period is Rs. 4921, while the interest is compounded half-yearly. What will be the compound interest on the same sum at the same rate for the same period, if the interest is compounded annually?

1. Rs. 4890
2. Rs. 4880
3. Rs. 4820
4. Rs. 4860

Answer (Detailed Solution Below)

Option 2 : Rs. 4880

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Given:

Principal = Rs. 64000

Compound Interest = Rs. 4921

Rate = 5%

Concept used:

When the sum is compounded half-yearly, then the rate of interest becomes half and time becomes double.

Formula used:

A = P(1 + R/100)T

C.I = A - P

A = P + C.I

Where, A = Amount, P = Principal, T = Time, C.I = Compound interest and R = rate of interest

Calculation:

A = P + C.I

⇒ A = 64000 + 4921 = Rs. 68921

According to the question,

Compounded half-yearly,

A = P(1 + R/100)T

⇒ 68921 = 64000[1 + 5/(2 ×100)]2T

⇒ 68921/64000 = (41/40)2n

⇒ (41/40)3 = (41/40)2n

⇒ 2n = 3

⇒ n = 3/2

Annual

Now, compounded annually

P = Rs. 64000 n = 3/2 = \({1\frac{1}{2}}\) years, and r = 5%

A = P(1 + R/100)T

⇒ A = 64000(1 + 5/100)\({1\frac{1}{2}}\)

⇒ A = 64000(1 + 5/100)1 × (1 + 5/100)1/2

⇒ A = 64000 × (1 + 5/100)(1 + 5/2 × 100)

⇒ A = 64000 × 21/20 × 41/40 = Rs. 688,80

C.I = A - P

⇒ C.I = 68880 - 64000 = Rs. 4880

∴ The compound interest is Rs. 4880.

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Solution

The correct option is A 112 years
Here, Principal P = Rs. 64000, Amount A = Rs 68921, rate R=5% per annum.

Since the interest is compounded half - yearly.

∴A=P(1+R200)2n

⇒68921=64000(1+5200)2n

⇒6892164000=(4140)2n

⇒(4140)3=(4140)2n
Since the base are same, equating the powers we get
⇒3=2n
⇒n=32 years=112years

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