In a group of men, women, boys, and girls, there were 10 more men than boys
M = B+10
and 5 more women than girls.
W = G+5
If there are twice as many boys as girls
B = 2G
and 105 people in all,
M+W+B+G = 105
(1) M = B+10
(2) W = G+5
(3) B = 2G
(4) M+W+B+G = 105
Using (3) substitute 2G for B in (1) and (4)
(5) M = 2G+10
M+W+2G+G = 105
(6) M+W+3G = 105
So now our system is
(5) M = 2G+10
(2) W = G+5
(6) M+W+3G = 105
Using (2) substitute G+5 for W in (6)
(6) M+W+3G = 105
M+G+5+3G = 105
(7) M+4G = 100
Using (5), substitute 2G+10 for M in (7):
(7) M+4G = 100
2G+10+4G = 100
6G = 90
(8) G = 15
Using (8), substitute 15 for G in (7)
(7) M+4G = 100
M+4(15) = 100
M+60 = 100
(9) M = 40
how many are men?
That's the answer. But to check we need W and B
Using (8), substitute 15 for G in (2)
(2) W = G+5
W = 15+5
W = 20
Using (8), substitute 15 for G in (3)
(3) B = 2G
B = 2(15)
B = 30
Answer: 40 men, 20 women, 30 boys, and 15 girls.
Checking:
In a group of men, women, boys, and girls, there were 10 more men than boys
That checks, because 40 is 10 more than 30
and 5 more women than girls.
That checks, because 20 is 5 more than 15
If there are twice as many boys as girls
That checks, because 30 is twice as many as 15
and 105 people in all,
That checks because 40+20+30+15 = 105
Edwin