# Summands

We want to split the number 110 into three summands so that the first and the second summand are in ratio 4:5, and the third with the first are in ratio 7:3. Calculate the smallest summands.

a =  24
b =  30
c =  56

### Step-by-step explanation:

a+b+c=110
a=4/5·b
c=7/3·a

a+b+c = 110
5a-4b = 0
7a-3c = 0

Pivot: Row 1 ↔ Row 3
7a-3c = 0
5a-4b = 0
a+b+c = 110

Row 2 - 5/7 · Row 1 → Row 2
7a-3c = 0
-4b+2.143c = 0
a+b+c = 110

Row 3 - 1/7 · Row 1 → Row 3
7a-3c = 0
-4b+2.143c = 0
b+1.429c = 110

Row 3 - 1/-4 · Row 2 → Row 3
7a-3c = 0
-4b+2.143c = 0
1.964c = 110

c = 110/1.96428571 = 56
b = 0-2.1428571428571c/-4 = 0-2.14285714 · 56/-4 = 30
a = 0+3c/7 = 0+3 · 56/7 = 24

a = 24
b = 30
c = 56

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