# BW-BS balls

Adam has a full box of large or small balls, black or white. The ratio between large and small balls is 5:3. Within the large balls, the ratio of black to white is 1:2, and between small balls, the ratio of black to white is 1:8.

What is the ratio of all-black to all-white balls?

x =  0.33

### Step-by-step explanation:

a=1
a+b = 5/3·(c+d)
a = 1/2·b
c = 1/8 · d

a = 1
3a+3b-5c-5d = 0
2a-b = 0
8c-d = 0

Pivot: Row 1 ↔ Row 2
3a+3b-5c-5d = 0
a = 1
2a-b = 0
8c-d = 0

Row 2 - 1/3 · Row 1 → Row 2
3a+3b-5c-5d = 0
-b+1.6667c+1.6667d = 1
2a-b = 0
8c-d = 0

Row 3 - 2/3 · Row 1 → Row 3
3a+3b-5c-5d = 0
-b+1.6667c+1.6667d = 1
-3b+3.3333c+3.3333d = 0
8c-d = 0

Pivot: Row 2 ↔ Row 3
3a+3b-5c-5d = 0
-3b+3.3333c+3.3333d = 0
-b+1.6667c+1.6667d = 1
8c-d = 0

Row 3 - -1/-3 · Row 2 → Row 3
3a+3b-5c-5d = 0
-3b+3.3333c+3.3333d = 0
0.5556c+0.5556d = 1
8c-d = 0

Pivot: Row 3 ↔ Row 4
3a+3b-5c-5d = 0
-3b+3.3333c+3.3333d = 0
8c-d = 0
0.5556c+0.5556d = 1

Row 4 - 0.55555556/8 · Row 3 → Row 4
3a+3b-5c-5d = 0
-3b+3.3333c+3.3333d = 0
8c-d = 0
0.625d = 1

d = 1/0.625 = 1.6
c = 0+d/8 = 0+1.6/8 = 0.2
b = 0-3.3333333333333c-3.3333333333333d/-3 = 0-3.33333333 · 0.2-3.33333333 · 1.6/-3 = 2
a = 0-3b+5c+5d/3 = 0-3 · 2+5 · 0.2+5 · 1.6/3 = 1

a = 1
b = 2
c = 1/5 = 0.2
d = 8/5 = 1.6

Our linear equations calculator calculates it.

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