Question

A sum of ₹ 8192 becomes ₹ 9826 at 12.5% per annum compounded half-yearly. Find the time period.

Answer 1

Given:

• Principal (P) = ₹ 8192
• Amount (A) = ₹ 9826
• Rate of interest (per annum) = 12.5%
• Interest is compounded half-yearly.
• Need to find the time period (t).

Step-wise calculation:

1. Since the interest is compounded half-yearly, the rate per half-year = `12.5/2` = 6.25%

2. Let the number of half-years be n.

The amount formula for compound interest is:

`A = P xx (1 + r/100)^n`

Substitute the given values: 

`9826 = 8192 xx (1 + 6.25/100)^n`

9826 = 8192 × (1.0625)ⁿ

3. Divide both sides by 8192:

`9826/8192 = (1.0625)^n`

1.199 = (1.0625)ⁿ

4. Take the logarithm on both sides:

log(1.199) = n × log(1.0625)

`n = (log(1.199))/(log(1.0625))`

5. Calculate values using logarithm base 10: 

log(1.199) ≈ 0.0782,

log(1.0625) ≈ 0.0266

`n = 0.0782/0.0266 ≈ 2.94 ≈ 3` half-years

6. Since n = number of half-years, the time period in years:

`t = n/2`

`t = 3/2`

t = 1.5 years

The time period for the sum of ₹ 8192 to become ₹ 9826 at 12.5% per annum compounded half-yearly is 1.5 years.