# Puzzles/Arithmetical puzzles/Luxury Cars/Solution

Puzzles | Arithmetical puzzles | Luxury Cars | Solution

Los Angeles - 256
New York - 192
San Francisco - 144
Boston - 108
Miami - 320

## Reasoning

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
```