# Unions now

The plant has 1,250 employees. The Trade Unions (former ROH, OH) is 18% male and 23% female. Plant management reported that 20% of all employees are in the trade union. How many men and women are in the plant, and how many are men and women in trade unions?

a =  135
b =  115
m =  750
w =  500

### Step-by-step explanation:

m+w = 1250
a + b = 0.20 · 1250
a=0.18 m
b =0.23w

m+w = 1250
a + b = 0.20 · 1250
a=0.18·m
b =0.23·w

m+w = 1250
a+b = 250
a-0.18m = 0
b-0.23w = 0

Pivot: Row 1 ↔ Row 2
a+b = 250
m+w = 1250
a-0.18m = 0
b-0.23w = 0

Row 3 - Row 1 → Row 3
a+b = 250
m+w = 1250
-b-0.18m = -250
b-0.23w = 0

Pivot: Row 2 ↔ Row 3
a+b = 250
-b-0.18m = -250
m+w = 1250
b-0.23w = 0

Row 4 + Row 2 → Row 4
a+b = 250
-b-0.18m = -250
m+w = 1250
-0.18m-0.23w = -250

Row 4 - -0.18 · Row 3 → Row 4
a+b = 250
-b-0.18m = -250
m+w = 1250
-0.05w = -25

w = -25/-0.05 = 500
m = 1250-w = 1250-500 = 750
b = -250+0.18m/-1 = -250+0.18 · 750/-1 = 115
a = 250-b = 250-115 = 135

a = 135
b = 115
m = 750
w = 500

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