Question

The five numbers 4, 9, 15, 20, and 27 are placed in the boxes below to make a true statement. What is the sum of the numbers placed in the boxes in the numerators on the left-hand side: \[\frac{\boxed{\phantom{99}}}{\boxed{\phantom{99}}} + \frac{\boxed{\phantom{99}}}{\boxed{\phantom{99}}} = \frac{9}{\boxed{\phantom{99}}}.\]

Answer 1

Answer:

Is that from AOPS

Step-by-step explanation:

Answer 2

Answer:

36

Step-By-Step Explanation:

This is more of a trial, rather than a computation. I am starting with the total number 9/4.

First, we note that 9/4 = 2 1/4. Next, at least one of the fractions on the left must have a denominator that is a multiple of 4, so 20 must be in one of the denominators. If we place 27 in the other denominator, then both fractions are less than 1, which means the sum can't reach 2. So, 27 must be in a numerator. First, we try placing it over the 20. Since 27/20 = 1 7/20 and 7/20 > 1/4, the other fraction must be less than 1. 27 must be in the numerator of the other fraction (not over 20). If 9 is the denominator of that fraction, then the fraction equals 3, which is too high. That leaves only one possibility:

9/20 + 27/15 = 9/20 + 9/5 = 9/20 + 36/20 = 45/20 = 9/4

Success! And we've exhausted all the other possibilities, so we know that the desired numerator sum is 9+27= 36.