A man wishes to save sh 200000 in 4years time . Find the sum of money he has to deposits now at 12\% p.a interest compound semi- annually to realise is goal ?

To find the sum of money the man has to deposit now, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:
A = the amount of money he wishes to save (sh 200000)
P = the principal amount (the sum of money he has to deposit now)
r = the annual interest rate (12% or 0.12)
n = the number of times the interest is compounded per year (2 for semi-annually)
t = the number of years (4)

Substitute the values into the formula:

200000 = P(1 + 0.12/2)^(2*4)

200000 = P(1 + 0.06)^8

200000 = P(1.06)^8

200000 = P(1.593848)

P = 200000 / 1.593848

P ≈ sh 125500.80

Therefore, the man has to deposit approximately sh 125500.80 now at 12% p.a interest compounded semi-annually to realize his goal of saving sh 200000 in 4 years.

Using the compound interest formula:

$$A = P(1 + \frac{r}{n})^{nt}$$

where:

- A is the future value
- P is the present value
- r is the annual interest rate
- n is the number of times per year that the interest is compounded
- t is the number of years

We have:

- A = sh 200000
- r = 12% = 0.12
- n = 2 (semi-annually)
- t = 4

Substituting these values into the formula, we get:

$$200000 = P(1 + \frac{0.12}{2})^{2 \times 4}$$

$$200000 = P(1 + 0.06)^{8}$$

$$200000 = P(1.56885)$$

$$P = \frac{200000}{1.56885}$$

$$P = \boxed{127438.14}$$

Therefore, the man needs to deposit sh 127438.14 now to achieve his goal of sh 200000 in 4 years at 12% p.a interest compounded semi-annually.