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?

Question

Nova Allen · Accepted Answer

#"male":"female" = 6:5# 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#

Nova Allen · Answer

#"males : females " = 6 : 5#

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#

Aurora Arias · Answer

undefined To find the ratio of male to female guests at the party, we first need to determine the proportion of male and female guests based on their average ages.

Let's denote the number of male guests as $ x $ and the number of female guests as $ y $.

Given:
Average age of all guests = 29 years
Average age of male guests = 34 years
Average age of female guests = 23 years

We can set up equations based on the average ages:

1. Total age of all guests = Average age × Total number of guests
$$ 29 \times (x + y) = 34x + 23y $$

Now, we can simplify this equation:

$$ 29x + 29y = 34x + 23y $$
$$ 6y = 5x $$
$$ y = \frac{5}{6}x $$

This equation gives us the relationship between the number of male guests (x) and female guests (y). The ratio of male to female guests is given by the fraction $ \frac{x}{y} $. Substituting the value of y in terms of x into this ratio:

$$ \frac{x}{y} = \frac{x}{\frac{5}{6}x} = \frac{6}{5} $$

So, the ratio of male to female guests at the party is 6:5.