Hello, ramdrop!

There are only *three* circles in the Venn diagram!

Anything outside the Male circle is a female.

A mixed hospital ward contains 30 patients.

13 are 50 or older.

15 have back problems.

12 are male.

5 are males with back problems.

2 are males with back problems and are 50 or older.

5 are female, are 50 or older and do *not* have back problems.

Let $\displaystyle A$ = "Older" (50 or older).

. . .$\displaystyle B$ = "Back" (back problems)

. . .$\displaystyle M$ = "Male"

. . .$\displaystyle x \:=\:\text{(Male) } \cap \text{ (Older) } \cap \sim\!\text{(Back)}$

How many patients are female, under 50 and have no back problems?

Contruct the Venn diagram and insert the known quanatities and the $\displaystyle x.$

. . It looks like this . . .

Code:

*---------------------------*
| |
| *-----------* |
| | A | |
| | 5 *-------+---* |
| | | 6-x | B | |
| *---+---+---* | | |
| | | x | 2 | 3 | | |
| | *---+---+---* | |
| | | | 4+x | |
| | 7-x *---+-------* |
| | M | |
| *-----------* |
| 3 |
*---------------------------*

Add the quantities inside the three cicles:

. . $\displaystyle 5 + (-x) + (4+x) + x + 2 + 3 + (7-x) \;=\;27$

Since there are 30 patients on the ward,

. . there are 3 patients outside the three circles.

These are patients who are not male, are not 50 or older,

. . and do not have back problems.

These are *exactly* the patients the problem asked for!

. . $\displaystyle n(\text{female} \cap \text{under 50} \cap \text{no back problems}) \;=\;3$