Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 1 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator of the previous fraction by 1. He then multiplies all his fractions together. He has 20 fractions, and their product equals 3. What is the value of the first fraction he wrote?
+23245 Call the first numerator: x
Then, the first denominator is: x - 1
Decreasing each numerator and denominator by 1 gives the following expression:
(x)/(x - 1) · (x - 1)/(x - 2) · (x - 2)/(x - 3) · (x - 3)/(x - 4) · ... · (x - 19)/(x - 20)
Cancelling all the factors that we can cancel leaves us with: x / (x - 20)
Therefore: x / (x - 20) = 3 ---> x = 3(x - 20) ---> x = 3x - 60
-2x = -60
x = 30
The first fraction was: 30 / 29
