Puzzle for June 25, 2019 ( )
Scratchpad
Find the 4-digit number ABCD by solving the following equations:
A, B, C, and D each represent a one-digit non-negative integer.
Scratchpad
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Hint #1
Add C to both sides of eq.2: B - C + C = A + C + C which becomes B = A + 2×C In eq.3, replace B with A + 2×C: A - C + D = A + 2×C + C In the above equation, add C to both sides, and subtract A from each side: A - C + D + C - A = A + 2×C + C + C - A which simplifies to D = 4×C
Hint #2
In eq.4, replace D with 4×C: 4×C - A = A - C Add A and C to both sides of the above equation: 4×C - A + A + C = A - C + A + C which simplifies to 5×C = 2×A Divide both sides by 2: 5×C ÷ 2 = 2×A ÷ 2 which makes 2½×C = A
Hint #3
In eq.2, substitute 2½×C for A: B - C = 2½×C + C Add C to both sides of the equation above: B - C + C = 2½×C + C + C which makes B = 4½×C
Solution
Substitute 2½×C for A, 4½×C for B, and 4×C for D in eq.1: 2½×C + 4½×C + C + 4×C = 24 which becomes 12×C = 24 Divide both sides by 12: 12×C ÷ 12 = 24 ÷ 12 which means C = 2 making A = 2½×C = 2½ × 2 = 5 B = 4½×C = 4½ × 2 = 9 D = 4×C = 4 × 2 = 8 and ABCD = 5928