In what time will 4400 become 4576 at 8% per annum interest compounded half-yearly?

Solution : Principal `(P)=Rs.4400`<br>Amount `(A)=Rs. 4576`<br>Rate `(R)=8%` or `4%` half-yearly<br>Let time per half-years`=n`<br>`A/P=(1+R/100)^n`<br>`4576/4400=(1+4/100)^n`<br>`104/100=(26/25)^n`<br>`(26/25)^1=(26/25)^n`<br>On comparing both the sides, we get:<br>`n = 1`<br>`therefore` The time per half-years`=1` years

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question thank you

In what time will Rs. 4400 become Rs. 4576 at 8% per annum interest compounded half yearly?

1. 6 months
2. 2 years
3. 7 months
4. 1 year

Answer (Detailed Solution Below)

Option 1 : 6 months

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Let P = principal, R = rate of interest and N = time

Amount = P[1 + (R/2) /100]2n

⇒ 4576 = 4400(1 + 4/100)2n

⇒ 1.04 = 1.042n

⇒ 2n = 1

⇒ N = 1/2 years

∴ Time = 6 months

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Given details are,

Rate = 8% per annum = 8/2 = 4% (half yearly)

A = Rs 4576

Principal (P) = Rs 4400

Let n be ‘2T’

By using the formula,

A = P (1 + R/100)n

4576 = 4400 (1 + 4/100)2T

4576 = 4400 (104/100)2T

(104/100)2T = 4576/4400

(104/100)2T = 26/25

(26/25)2T = (26/25)1

So on comparing both the sides, n = 2T = 1

∴ Time required is ½ year.

In what time will Rs 4400 become Rs 4576 at 8% per annum interest compounded half-yearly?

Let the time period be n years.
R = 8 % = 4 % (Half - yearly)
Thus, we have: 
\[A = P \left( 1 + \frac{R}{100} \right)^n \]
\[4, 576 = 4, 400 \left( 1 + \frac{4}{100} \right)^n \]
\[4, 576 = 4, 400 \left( 1 . 04 \right)^n \]
\[ \left( 1 . 04 \right)^n = \frac{4, 576}{4, 000}\]
\[ \left( 1 . 04 \right)^n = 1 . 04\]
\[ \left( 1 . 04 \right)^n = 1 . {04}^1 \]
On comparing both the sides, we get: 
n = 1
Thus, the required time is half a year.