You have three categories in one group. Draw a rectangle to represent your "universe", being the forty boys, and draw three overlapping circles inside to represent the three categories. Use the sort of arrangement displayed in the second example on

this page. Label the circles with the three types of team game.

You are given that twenty-four play football, so, on the outside of the circle, label the "football" circle with "24". Label the "soccer" circle with "22". You can't label the "basketball" circle with any numbers yet.

Since fifteen play both football and soccer, draw an arrow indicating the overlap of the "football" and "soccer" circles, and label this as "15".

Since ten are on all three teams, write "10" inside the overlap of all three circles. Then the overlap between the "football" and "soccer" circles, that you'd previously indicated held "15" in total, must have only 15 - 10 = 5 in

just the "football and soccer" overlap. That is, there must be five who play football and soccer but not basketball.

Since fourteen are on the football and basketball teams, you can label the remaining portion of the overlap between the "football" and "basketball" circles with "4", since the triple-overlap portion in the middle is already labelled with "10".

You now have all overlap portions of the "football" circle filled in. Subtract these portions from the "football" total to find the number who only played football.

You should have three sections unfilled: soccer only (label this "S"), basketball only (label this "B"), and two teams (being soccer and basketball; label this "T").

They give you the total for soccer, which means you have 5 + 10 + T + S = 22.

They also give you the overall total, which means you have 5 + 5 + 10 + 4 + T + S + B = 40.

Solve 5 + 10 + T + S = 22 for the value of T + S. (You will

not be finding the value of T or S!)

Plug this number in for "T + S" in the second equation. Solve for B.

If you get stuck, please reply showing how far you have gotten. Thank you!