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# Fractions

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Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is \(2\) 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 \(1/10\). What is the value of the first fraction he wrote?

May 14, 2022

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Supppose the first fraction is \(\dfrac{n + 2}n\). Then the other two fractions are \(\dfrac{n + 3}{n + 1}\) and \(\dfrac{n + 4}{n + 2}\) respectively.

Then

\(\dfrac{(n + 2)(n + 3)(n + 4)}{n(n + 1)(n + 2)} = \dfrac1{10}\\ n(n + 1)= 10(n + 3)(n + 4)\)

Now solve the quadratic equation for the positive integer value of n. Substituting back into (n + 2)/n gives the value of the first fraction he wrote.

Edit: There are no positive integer solutions for n. I think there are some problems in the question design.

Edit: I guess n = -5 counts because the resulting fractions are indeed positive, but the numerator and the denominator of the other fractions aren't increased by 1, but instead decreased by 1.

May 14, 2022
edited by MaxWong  May 14, 2022
edited by MaxWong  May 14, 2022