Calculation of Compound Interest

To calculate the compound interest on half-yearly rests, we need to consider the interest rate for each year and the compounding frequency. Let's break down the given information and calculate the compound interest step by step.

Given:
Principal amount (P) = $10,000
Rate of interest for the first and second years (R1) = 6% per annum
Rate of interest for the third year (R2) = 9% per annum

Formula:
The compound interest formula with half-yearly rests is:
A = P(1 + r/n)^(nt)

Where:
A = Final amount (including interest)
P = Principal amount
r = Rate of interest per period
n = Number of compounding periods per year
t = Number of years

Calculation:
First and Second Years:
Rate of interest per period (r1) = R1/2 = 6%/2 = 3% per half-year
Number of compounding periods per year (n1) = 2 (half-yearly rests)
Number of years (t1) = 2 (first and second years)

Third Year:
Rate of interest per period (r2) = R2/2 = 9%/2 = 4.5% per half-year
Number of compounding periods per year (n2) = 2 (half-yearly rests)
Number of years (t2) = 1 (third year)

First and Second Years:
A1 = P(1 + r1/n1)^(n1*t1)
= 10000(1 + 0.03/2)^(2*2)
= 10000(1 + 0.015)^4
= 10000(1.015)^4
= 10000 * 1.06167625
= $10,616.76

Third Year:
A2 = A1(1 + r2/n2)^(n2*t2)
= 10616.76(1 + 0.045/2)^(2*1)
= 10616.76(1 + 0.0225)^2
= 10616.76(1.0225)^2
= 10616.76 * 1.04500625
= $11,081.67

Compound Interest:
Compound interest (CI) = A2 - P
= 11081.67 - 10000
= $1,081.67

Therefore, the compound interest on $10,000 with half-yearly rests and the given interest rates is $1,081.67.

We know that in half year time is double and rate is half
so 6% will 6/200 for 1 year
and 9% will 9/200 for 1 year

10000(1+6/200)(1+9/200)=10609
so C.I.=Amt.-Principal=10609-10000=609