Two-fifths 2763

Pupils of four grades saved n=45000 crowns together. Of this, the first year saved one-third, the second one-third of the remainder, the third two-fifths of the remainder, and the fourth the remainder.
How many crowns did each year save?

Correct answer:

a = 15000
b = 10000
c = 8000
d = 12000

Step-by-step explanation:

n = 45000
a+b+c+d=n
a = 1/3 · n
b = (n - a)/3
c = 2/5 · (n - a - b)

n = 45000
a+b+c+d-n = 0
3a-n = 0
a+3b-n = 0
2a+2b+5c-2n = 0

Pivot: Row 1 ↔ Row 3
3a-n = 0
a+b+c+d-n = 0
n = 45000
a+3b-n = 0
2a+2b+5c-2n = 0

Row 2 - 1/3 · Row 1 → Row 2
3a-n = 0
b+c+d-0.66667n = 0
n = 45000
a+3b-n = 0
2a+2b+5c-2n = 0

Row 4 - 1/3 · Row 1 → Row 4
3a-n = 0
b+c+d-0.66667n = 0
n = 45000
3b-0.66667n = 0
2a+2b+5c-2n = 0

Row 5 - 2/3 · Row 1 → Row 5
3a-n = 0
b+c+d-0.66667n = 0
n = 45000
3b-0.66667n = 0
2b+5c-1.33333n = 0

Pivot: Row 2 ↔ Row 4
3a-n = 0
3b-0.66667n = 0
n = 45000
b+c+d-0.66667n = 0
2b+5c-1.33333n = 0

Row 4 - 1/3 · Row 2 → Row 4
3a-n = 0
3b-0.66667n = 0
n = 45000
c+d-0.44444n = 0
2b+5c-1.33333n = 0

Row 5 - 2/3 · Row 2 → Row 5
3a-n = 0
3b-0.66667n = 0
n = 45000
c+d-0.44444n = 0
5c-0.88889n = 0

Pivot: Row 3 ↔ Row 5
3a-n = 0
3b-0.66667n = 0
5c-0.88889n = 0
c+d-0.44444n = 0
n = 45000

Row 4 - 1/5 · Row 3 → Row 4
3a-n = 0
3b-0.66667n = 0
5c-0.88889n = 0
d-0.26667n = 0
n = 45000

n = 45000/1 = 45000
d = 0+0.26666666666667n = 0+0.26666667 · 45000 = 12000
c = 0+0.88888888888889n/5 = 0+0.88888889 · 45000/5 = 8000
b = 0+0.66666666666667n/3 = 0+0.66666667 · 45000/3 = 10000
a = 0+n/3 = 0+45000/3 = 15000

a = 15000
b = 10000
c = 8000
d = 12000
n = 45000

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