Prize money of K10,000 is divided amongst 10 competitors so that second prize is half of first price, third prize is half the second prize and so on. Find the value of the first and the tenth prizes?

Let the first prize be x.

The second prize is half of the first prize, so it is x/2.
The third prize is half of the second prize, so it is (x/2)/2 = x/4.
Continuing this pattern, the tenth prize is x/(2^9) = x/512.

The total prize money is the sum of all the prizes:
x + x/2 + x/4 + ... + x/512 = 10000

This is a geometric series with a common ratio of 1/2 and 10 terms. The sum of a geometric series is given by the formula:
S = a(1 - r^n) / (1 - r)

Plugging in the values:
x(1 - (1/2)^10) / (1 - 1/2) = 10000
x(1 - 1/1024) / (1/2) = 10000
x(1023/1024) / (1/2) = 10000
x(1023/512) = 10000
x = 10000 * (512/1023)
x = 4975.86

Therefore, the value of the first prize is approximately K4975.86.

The value of the tenth prize is x/512 = 4975.86 / 512 = 9.72

Therefore, the value of the tenth prize is approximately K9.72.