Question

The average age of the adults that participate at Yosief Birthday is 29 years. Knowing that the average age of the male and female guests was 34, 23 years respectively, find the ratio of male to female guests at the party?

Answer 1

`"male":"female" = 6:5`

Explanation:

Let `M` be the number of males
and `F` be the number of females

The sum of the ages of the males is `34M`
The sum of the ages of the females is `23F`

The sum of the ages of the males and females together is `34M+23F`

The number of males and females together is `M+F`

The average age of males and females together is
`color(white)("XXX")(34M+23F)/(M+F) = 29`

`color(white)("XXX")34M+23F=29M+29F`

`color(white)("XXX")34M=29M+6F`

`color(white)("XXX")5M=6F`

`color(white)("XXX")M/F = 6/5`

Answer 2

`"males : females " = 6 : 5`

Explanation:

When working with averages (means), remember that we can add sums and numbers, but we cannot add averages.

(An exception would be if there were the same number of males and females - in this case we may add the averages and divide by 2)

Let the number of females be `x`.
Let the number of males be `y`

Let's work with the `color(red)("whole group first:")`
The total number of people at the party is `color(red)(x+y)`
The sum of all their ages is `color(red)((x+y) xx 29)`

Now let's work with `color(blue)("males and females separately.")`
The sum of the ages of all the females = `23 xx x = color(blue)(23x)`
The sum of the ages of all the males = `34xx y = color(blue)(34y)`

The sum of the ages of all the people = `color(blue)(23x + 34y)`

The sum of the ages of all the people = `color(red)(29(x+y))`

We now have 2 different expressions for the same information, so we can make an equation.

`color(red)(29(x+y)) = color(blue)(23x + 34y)`

`29x + 29y = 23x +34y`

`34y -29y =29x-23x`

`5y = 6x " we need to compare " y : x`

`y = (6x)/5`

`y/x = 6/5`

`y:x = 6:5`

Notice that although we do not know the actual number of people at the party, we are able to determine the ratio.

`"males : females " = 6 : 5`