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 3 fractions, and their product equals 10. What is the value of the first fraction he wrote?

Guest Nov 18, 2022

#3**0 **

*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 3 fractions, and their product equals 10. What is the value of the first fraction he wrote?*

I tried solving this algebraically, using x as the denominator in the first fraction.

When you set it equal to ten, the final answer is **x = 1/3** which isn't what we want.

(x+1) (x+1+1) (x+1+1+1)

It looked like this. –––– • ––––––– • –––––––––––

(x) (x+1) (x+1+1)

~~ (x+1) ~~ ~~ (x+1+1) ~~ (x+1+1+1) (x+3)

You may notice that some –––– • ––––––– • ––––––––––– = –––––

of those terms will cancel. (x) ~~ (x+1) ~~ ~~ (x+1+1) ~~ (x)

Call it y = (x+3) / (x) and take it to www.desmos.com/calculator and you'll find that the

only time that y ever equals 10, at that point x is between 0 and 1,

about in the right spot where x = 1/3, which I'd found algebraically.

Therefore, the problem as stated has **no solution** which is an integer.

_{.}

Guest Nov 19, 2022