# Guppies for sale

Paul had a bowl of guppies for sale. Four customers were milling around the store.
1. Rod told paul - I'll take half the guppies in the bowl, plus had a guppy.
2. Heather said - I'll take half of what you have, plus half a guppy.
Nancy, the third customer, said - I'll take half of what you have left plus half a guppy.
Michele, the fourth customer, said - I'll take half of what you have left plus half a guppy.

All four customers left with live guppies, and Pual was happy to have sold all fish. How many guppies were there in the bowl when he started? How many did each customer take?

H =  4
M =  1
N =  2
R =  7
n =  14

### Step-by-step explanation:

R = n/2
H = (n-R)/2+0.5
N = (n-R-H)/2+0.5
M = (n-R-H-N)/2+0.5
R+H+N+M=n

2R-n = 0
2H+R-n = 1
H+2N+R-n = 1
H+2M+N+R-n = 1
H+M+N+R-n = 0

Pivot: Row 1 ↔ Row 2
2H+R-n = 1
2R-n = 0
H+2N+R-n = 1
H+2M+N+R-n = 1
H+M+N+R-n = 0

Row 3 - 1/2 · Row 1 → Row 3
2H+R-n = 1
2R-n = 0
2N+0.5R-0.5n = 0.5
H+2M+N+R-n = 1
H+M+N+R-n = 0

Row 4 - 1/2 · Row 1 → Row 4
2H+R-n = 1
2R-n = 0
2N+0.5R-0.5n = 0.5
2M+N+0.5R-0.5n = 0.5
H+M+N+R-n = 0

Row 5 - 1/2 · Row 1 → Row 5
2H+R-n = 1
2R-n = 0
2N+0.5R-0.5n = 0.5
2M+N+0.5R-0.5n = 0.5
M+N+0.5R-0.5n = -0.5

Pivot: Row 2 ↔ Row 4
2H+R-n = 1
2M+N+0.5R-0.5n = 0.5
2N+0.5R-0.5n = 0.5
2R-n = 0
M+N+0.5R-0.5n = -0.5

Row 5 - 1/2 · Row 2 → Row 5
2H+R-n = 1
2M+N+0.5R-0.5n = 0.5
2N+0.5R-0.5n = 0.5
2R-n = 0
0.5N+0.25R-0.25n = -0.75

Row 5 - 0.5/2 · Row 3 → Row 5
2H+R-n = 1
2M+N+0.5R-0.5n = 0.5
2N+0.5R-0.5n = 0.5
2R-n = 0
0.125R-0.125n = -0.875

Row 5 - 0.125/2 · Row 4 → Row 5
2H+R-n = 1
2M+N+0.5R-0.5n = 0.5
2N+0.5R-0.5n = 0.5
2R-n = 0
-0.0625n = -0.875

n = -0.875/-0.0625 = 14
R = 0+n/2 = 0+14/2 = 7
N = 0.5-0.5R+0.5n/2 = 0.5-0.5 · 7+0.5 · 14/2 = 2
M = 0.5-N-0.5R+0.5n/2 = 0.5-2-0.5 · 7+0.5 · 14/2 = 1
H = 1-R+n/2 = 1-7+14/2 = 4

H = 4
M = 1
N = 2
R = 7
n = 14

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