MATHCOUNTS Question

Question

Mike wrote a list of six positive integers on his paper. He chose the first andsecond integers randomly, but the third integer was the sum of the first and

second, and each of the remaining integers was the sum of the two previous

integers in the list. He then found the sum of all six integers. What is the ratio

of the fifth integer in his list to this sum? Express your answer as a common

fraction.

Answer 1

Here is a start for you:

Let x = 1st

y = 2nd

x+y = 3rd

x+2y= 4th

2x+3y=5th

3x+5y=6th

Sum = 8x+12y

Now create a fraction with the expression for the 5th over the expression for the sum.

See if there is a greatest common factor for the sum that when factored out allows you to cancel the expression in the numerator, leaving the desired ratio (which does not involve x or y).