# Puzzles/Arithmetical puzzles/Luxury Cars/Solution

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Puzzles | Arithmetical puzzles | Luxury Cars | Solution

Los Angeles - 256

New York - 192

San Francisco - 144

Boston - 108

Miami - 320

## Reasoning[edit]

Let L, N, S, B, M be the numbers of cars sold at Los Angeles, New York, San Francisco, Boston and Miami respectively.

The statement of the puzzle gives these equations:

L = (L + N + S + B + M)/4 + 1 N = (N + S + B + M)/4 + 1 S = (S + B + M)/4 + 1 B = (B + M)/4 + 1 L + S = B + N + 100 (a)

The first four equations can be simplified:

4L = L + N + S + B + M + 4 4N = N + S + B + M + 4 4S = S + B + M + 4 4B = B + M + 4

3L = N + S + B + M + 4 3N = S + B + M + 4 3S = B + M + 4 3B = M + 4

3L = N + 3N = 4N (b) 3N = S + 3S = 4S (c) 3S = B + 3B = 4B (d) 3B = M + 4 (e)

From equation (a) we have

4L + 4S = 4B + 4N + 400 => 4L + 4S = 3S + 3L + 400 => L + S = 400 => L = 400 - S

Combining this equation with equations (b) and (c) gives

3(400 - S) = 4N => 9(400 - S) = 12N = 16S => 3600 - 9S = 16S => 25S = 3600 => S = 144

From this result and equations (b) to (e), the remaining values easily fall into place:

L = 256 N = 192 S = 144 B = 108 M = 320