Depth · one concept from first principles · step 2 of 4
Row 6 starts with 1/6, and below it are 1/7 and 1/A. So 1/6 = 1/7 + 1/A; subtract to undo the sum, using the common denominator 42.
So A = 42.
6.NS.B.4Work BackwardsSince is the sum of the two numbers below it, and , the missing number is , a single unit fraction.
If , taking back off leaves alone, because subtraction undoes addition.
The difference is exactly one forty-second.
Renaming both in forty-seconds gives and without changing their sizes.
Seven forty-seconds take away six forty-seconds leaves one forty-second, since same-size pieces are just counted.
Below the 1/7 that starts row 7 are 1/8 and 1/B.
So B = 56. The pattern: next to 1/n sits 1/(n(n-1)).
7.NS.A.1Look For A PatternBelow 1/A = 1/42 are 1/B = 1/56 and 1/C. The least common multiple of 42 and 56 is 168.
So C = 168.
6.NS.B.4Work BackwardsAdd the three denominators.
A+B+C = 266.
7.NS.A.1Work BackwardsIf a rule adds two numbers to make a third, subtracting runs it backwards.
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The same shape shows up later on the AMC, as Arithmetic Expression Decomposition, Geometric Series Share, Lattice / Grid Pattern from Small Cases. Why the shapes repeat →