# Honza + Alice + Tonda

Honza + Alice + Tonda have a total of 111 USD. The ratio between Honza and Alena is 5:6, and the ratio between Alice and Tonda is 4:5

How much money has each of them had?

H =  30
A =  36
T =  45

## Step-by-step explanation:

h+a+t = 111
6h = 5a
5a = 4t

h+a+t = 111
6·h = 5·a
5·a = 4·t

a+h+t = 111
5a-6h = 0
5a-4t = 0

Pivot: Row 1 ↔ Row 2
5a-6h = 0
a+h+t = 111
5a-4t = 0

Row 2 - 1/5 · Row 1 → Row 2
5a-6h = 0
2.2h+t = 111
5a-4t = 0

Row 3 - Row 1 → Row 3
5a-6h = 0
2.2h+t = 111
6h-4t = 0

Pivot: Row 2 ↔ Row 3
5a-6h = 0
6h-4t = 0
2.2h+t = 111

Row 3 - 2.2/6 · Row 2 → Row 3
5a-6h = 0
6h-4t = 0
2.467t = 111

t = 111/2.46666667 = 45
h = 0+4t/6 = 0+4 · 45/6 = 30
a = 0+6h/5 = 0+6 · 30/5 = 36

a = 36
h = 30
t = 45

Our linear equations calculator calculates it.

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