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The Circus Fee

A circus performance is witnessed by 120 people
who have have paid a total of $120.
The men pay $5, women pay $2,
and children pay 10 cents each.
How many of each went to the circus?

There are 17 men, 13 women, 90 children.

Let the number of men be m,
number of women be w,
number of children be c,
and m + w + c = 120 or m = 120 - w - c

Then value of men's admissions is 5m,
value of women's admissions is 2w,
value of children's admissions is 0.1c,
and 5m + 2w + 0.1c = 120

Then, 5(120 - w - c) + 2w + 0.1c = 120
600 - 5w - 5c +2w + 0.1c = 120
-3w -4.9c = -480
30w + 49c = 4800

By Trial & Error, let w=10
Then 300 + 49c = 4800
49c = 4500 or c = 4500/49 = 91.8 which is not an integer.

By Trial & Error, let w=13
Then 390 + 49c = 4800
49c = 4410 or c = 4410/49 = 90 is the number of children.

30w + 49(90) = 4800
30w = 4800 - 4410
30w = 390 or w = 390/30 = 13 is the number of women.

m + 13 + 90 = 120
m + 103 = 120
m = 17 is the number of men.