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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?

 Nov 18, 2022
 #1
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The first fraction is 14/5.

 Nov 18, 2022
 #2
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Did you read this part?   "the numerator is 1 greater than the denominator" 

Guest Nov 19, 2022
 #3
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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.  

.

 Nov 19, 2022

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