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I began with the same as John E. Franklin.
1/3 + 1/6 = 1/2
That means only 1/2 remains.
So, I added the two largest fractions whose sum will not surpass 1/2:
1/4 + 1/5 = 9/20
9/20 is only 1/20 less the 1/2 that remained.
So, I added two numbers together that would get me really close to 1/20, but leave a little for the final fraction.
I tried 1/40 + 1/30 = 7/120. That didn't work because 1/20 = 6/120.
I changed 1/30 to 1/60, so I could still have 120 as my denominator.
1/40 + 1/60 = 5/120, which is only 1/120 away from the 1/20 that remained.
Therefore
1/3 + 1/6 + 1/4 + 1/5 + 1/40 + 1/60 + 1/120 = 1
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