Question

22.In what time, will 175 amount to 189 at 16% per annum, if the interest is compounded half-yearly?

Answer 1

To find the time required for an amount of 175 to grow to 189 at an interest rate of 16% per annum, compounded half-yearly, we can use the formula for compound interest:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (the initial amount of money).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time in years.

Given:
- \( A = 189 \)
- \( P = 175 \)
- \( r = 16\% = 0.16 \)
- Interest is compounded half-yearly, so \( n = 2 \).

We can substitute these values into the formula:

\[
189 = 175 \left(1 + \frac{0.16}{2}\right)^{2t}
\]

Calculating the term inside the parentheses:

\[
1 + \frac{0.16}{2} = 1 + 0.08 = 1.08
\]

So our equation becomes:

\[
189 = 175 (1.08)^{2t}
\]

Next, we need to isolate \((1.08)^{2t}\):

\[
\frac{189}{175} = (1.08)^{2t}
\]

Calculating the left-hand side:

\[
\frac{189}{175} = 1.08
\]

So we have:

\[
1.08 = (1.08)^{2t}
\]

Since the bases are the same, we can set the exponents equal to each other:

\[
1 = 2t
\]

Now solving for \(t\):

\[
t = \frac{1}{2} = 0.5 \text{ years}
\]

Thus, the time required for 175 to amount to 189 at a rate of 16% per annum, compounded half-yearly, is **0.5 years** or **6 months**.