First, let's list the facts we know about the problem:
2/5 of the people were men.
3 times as many women as children.
45 more men than children.
How many people total?
Let's call x the total number of people. The total number of people equals the men + women + children. Let's express the relationship between each of these groups and the total, mathematically:
2/5 * (x) = men
3 * (children) = women
(men) - 45 = children
Then, let's substitute the equation for men into the equation for children. Then, we'll substitute the equation for children into the equation for women.
.4x = men
(.4x-45) = children
3 (.4x-45) = women
Now, let's place this in an equation where the total number of people (x) equals the sum of each category:
x = .4x + (.4x-45) + 3(.4x-45)
Now, let's solve this problem with algebra. First, we'll distribute:
x = .4x + .4x - 45 + 1.2x - 135
Now let's simplify by combining like factors:
x = 2x - 180
Now, we simply add 180 to both sides, while subtracting x from both sides:
x -x + 180 = 2x -x -180 + 180
180 = x
The total number of people at the concert was 180.
Last, let's do a sanity check. Does this make sense?
2/5 of the total number of people are men = 72 men
subtract 45 for the number of children = 27 children
multiply number of children by 3 to obtain number of women = 81 women
72 + 27 + 81 = 180
Yup, the math makes sense and we have whole numbers of people!
