Dividing money

Imrich, Daniel, and Dezider shared an unknown amount in the ratio of 1:2:4, where Dezider received 750 euros more than Imrich, and Daniel got half as much as Dezider. Determine an unknown amount of money and determine the amount that got Imrich, Daniel, and Dezider.

Correct answer:

a = 1750
b = 250
c = 500
d = 1000

Step-by-step explanation:

a=b+c+d
b=1/(1+2+4)·a
c=2/(1+2+4)·a
d = 750+b

a-b-c-d = 0
0.142857a-b = 0
0.285714a-c = 0
b-d = -750

Row 2 - 0.14285714285714 · Row 1 → Row 2
a-b-c-d = 0
-0.8571b+0.1429c+0.1429d = 0
0.2857a-c = 0
b-d = -750

Row 3 - 0.28571428571429 · Row 1 → Row 3
a-b-c-d = 0
-0.8571b+0.1429c+0.1429d = 0
0.2857b-0.7143c+0.2857d = 0
b-d = -750

Pivot: Row 2 ↔ Row 4
a-b-c-d = 0
b-d = -750
0.2857b-0.7143c+0.2857d = 0
-0.8571b+0.1429c+0.1429d = 0

Row 3 - 0.28571428571429 · Row 2 → Row 3
a-b-c-d = 0
b-d = -750
-0.7143c+0.5714d = 214.2857
-0.8571b+0.1429c+0.1429d = 0

Row 4 - -0.85714285714286 · Row 2 → Row 4
a-b-c-d = 0
b-d = -750
-0.7143c+0.5714d = 214.2857
0.1429c-0.7143d = -642.8571

Row 4 - 0.14285714/-0.71428571 · Row 3 → Row 4
a-b-c-d = 0
b-d = -750
-0.7143c+0.5714d = 214.2857
-0.6d = -600

d = -600/-0.6 = 1000
c = 214.28571429-0.57142857142857d/-0.71428571 = 214.28571429-0.57142857 · 1000/-0.71428571 = 500
b = -750+d = -750+1000 = 250
a = 0+b+c+d = 0+250+500+1000 = 1750

a = 1750
b = 250
c = 500
d = 1000

Our linear equations calculator calculates it.