Patterns & Reasoning

Depth · one concept from first principles · step 2 of 4

Peel off whole parts and flip the remainder

Write 83/13 as a stack of fractions: a whole number a plus 1 over (b plus 1 over (c plus 1 over (d + 1/2))). Find the whole numbers and then a+b+c-d.
Number system
Your answer
How to solve
Strategy Identify Subproblems — Break the problem into levels. At each level: take the whole-number part, keep the leftover proper fraction, and flip it (a proper fraction p/q equals 1 ÷ q/p). Repeat the same move until the last level shows d + 1/2.
1STEP 1

Level 1: find a

Divide 83 by 13: 13 × 6 = 78 with remainder 5.

83/13 = 6 + 5/13
2STEP 2

Level 2: flip and find b

A fraction equals 1 divided by its reciprocal.

5/13 = 1/(13/5), 13/5 = 2 + 3/5
3STEP 3

Level 3: flip again and find c

Repeat the same move with 3/5.

3/5 = 1/(5/3), 5/3 = 1 + 2/3
4STEP 4

Level 4: find d

Flip 2/3 once more.

2/3 = 1/(3/2), 3/2 = 1 + 1/2
5STEP 5

Combine

Substitute a = 6, b = 2, c = 1, d = 1.

6 + 2 + 1 - 1 = 8
Answer
8
Rebuild from the bottom: 1 + 1/2 = 3/2; 1 + 2/3 = 5/3; 2 + 3/5 = 13/5; 6 + 5/13 = 83/13. It returns the original fraction.
Takeaway

Take the whole part, flip what is left, repeat: that is all a stacked fraction asks.

  • Level 1: find a
  • Level 2: flip and find b
  • Level 3: flip again and find c
  • Level 4: find d
  • Combine
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