# One-fifth 21003

John, Paul, Peter, and Jan earned 42 thousand. John and Paul have the same. Peter is 20% more than Paul. George was one-fifth less than Paul. How much did everyone earn?

A =  10500
I =  8400
J =  10500
P =  12600

### Step-by-step explanation:

J+A+P+I=42000
J=A
P=1.2·A
I=A-A/5

A+I+J+P = 42000
A-J = 0
1.2A-P = 0
4A-5I = 0

Pivot: Row 1 ↔ Row 4
4A-5I = 0
A-J = 0
1.2A-P = 0
A+I+J+P = 42000

Row 2 - 1/4 · Row 1 → Row 2
4A-5I = 0
1.25I-J = 0
1.2A-P = 0
A+I+J+P = 42000

Row 3 - 1.2/4 · Row 1 → Row 3
4A-5I = 0
1.25I-J = 0
1.5I-P = 0
A+I+J+P = 42000

Row 4 - 1/4 · Row 1 → Row 4
4A-5I = 0
1.25I-J = 0
1.5I-P = 0
2.25I+J+P = 42000

Pivot: Row 2 ↔ Row 4
4A-5I = 0
2.25I+J+P = 42000
1.5I-P = 0
1.25I-J = 0

Row 3 - 1.5/2.25 · Row 2 → Row 3
4A-5I = 0
2.25I+J+P = 42000
-0.6667J-1.6667P = -28000
1.25I-J = 0

Row 4 - 1.25/2.25 · Row 2 → Row 4
4A-5I = 0
2.25I+J+P = 42000
-0.6667J-1.6667P = -28000
-1.5556J-0.5556P = -23333.3333

Pivot: Row 3 ↔ Row 4
4A-5I = 0
2.25I+J+P = 42000
-1.5556J-0.5556P = -23333.3333
-0.6667J-1.6667P = -28000

Row 4 - -0.66666667/-1.55555556 · Row 3 → Row 4
4A-5I = 0
2.25I+J+P = 42000
-1.5556J-0.5556P = -23333.3333
-1.4286P = -18000

P = -18000/-1.42857143 = 12600
J = -23333.33333333+0.55555555555556P/-1.55555556 = -23333.33333333+0.55555556 · 12600/-1.55555556 = 10500
I = 42000-J-P/2.25 = 42000-10500-12600/2.25 = 8400
A = 0+5I/4 = 0+5 · 8400/4 = 10500

A = 10500
I = 8400
J = 10500
P = 12600

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