Classifying and Sorting

Two groups make four regions, not two: only the first, only the second, both, and neither. Here eleven students play piano alone, seven play both, seven play guitar alone, and five play neither, which totals thirty. The trap is the word piano on its own: the number who play piano is eleven plus seven, which is eighteen — the whole circle, including the overlap, not just the part outside it. Almost every wrong answer on this kind of question is exactly one region out, so name the region you want before you start adding.

The four regions always add to the total, so once you know three of them the fourth follows. Adding the three regions inside the circles gives twenty-five students who play at least one instrument, and thirty take twenty-five leaves five who play neither. That neither group sits outside both circles but still inside the box, so it is part of the total and is easy to forget. Whenever a question gives you a total, check whether it is asking about the whole box or only the circles.

Adding the two group sizes gives thirty-two, but only twenty-five students play anything at all. The extra seven are the students in the overlap, who were counted once as piano players and once again as guitar players. Subtract them back exactly once: eighteen plus fourteen minus seven is twenty-five. That is the rule worth memorising — A plus B minus both gives the number in at least one group. And it works as a check in the other direction too: if a total comes out bigger than it should be, you have double-counted the overlap.

Three groups make seven regions, and the order you fill them in matters. Always start with the centre — the people in all three — then the pairs, then the singles. Working middle-out means each number you write is final. Working outside-in forces you to keep subtracting from numbers you have already written, and that is where mistakes come from. Watch the wording too: runs and swims normally means the pair region plus the centre, because anyone who does all three does also run and swim. Only the word only cuts the centre out.

When a question hands you the actual items rather than counts, a tick grid is faster than a diagram. One row per item, one column per property. A row with two ticks is in the overlap, a row with one tick belongs to only that group, and a row with no ticks is the neither group. Here P and R have both ticks, C is curved only, and E and T are straight only. That is the whole lesson: four regions rather than two, remember neither, subtract the overlap once, and fill a three-group diagram from the middle outwards. Quiz and exercises now.