Question

:4. In the Department of Natural Sciences, 14 faculty members have a PhD, and 30 faculty members do not have a PhD. In the Department, the number of female faculty who do not have a PhD is 10 more than the number of females who have a PhD. If a third of the male faculty in the Department have a PhD, then what is the number of female faculty in the Department with a PhD?​

Answer 1

The number of female faculty in the Department with a PhD is 8.

Explanation:

There are 14 + 30 = 44 faculty members.

Of those, x are male and y are female.

Then

x + y = 44.

The number of female faculty who do not have a PhD is 10 more than the number of females who have a PhD.

y = z + w

z is the number of females with PhD.

w is the number of females without PhD.

w = z + 10

If a third of the male faculty in the Department have a PhD

`(x)/(3) + z = 14`

Now, we can write all variables as functions of z, which is the number of female faculty in the Department with PhD.

The objective is:

To find z from the first equation, that is:

`x + y = 44`

To do this, we have to write x and y as functions of z.

Writing x and y as functions z.

`(x)/(3) + z = 14`

`(x)/(3) = 14 - x`

`x = 3(14 - z)`

`x = 42 - 3z`

And

`y = z + w`

In which

`w = 10 + z`

So

`y = z + 10 + z`

`y = 2z + 10`

Replacing:

`x + y = 44`

`42 - 3z + 2z + 10 = 44`

`-z + 52 = 44`

`z = 52 - 44`

`z = 8`

The number of female faculty in the Department with a PhD is 8.