Right now I'm reading the GMAT club math book and I'm having trouble on the advanced overlapping sets problems because no matter how much I read the definitions, I don't understand when to use each formula, which are Total=A+B+C-(sum of 2 group overlaps)+all three+neither and Total=A+B+C-(sum of EXACTLY 2 group overlaps)-2*(all three)+neither. Like what hints do you need to look for in the problem to know when to add all three or to subtract by all three times 2? Thanks for your help!
If you have 2-group overlap use the first formula and if you ahve EXACTLY 2-group overlap use the second formula.Example 1:Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?Translating:
"are placed on at least one team": members of none =0;
"20 are on the marketing team": M=20;
"30 are on the Sales team": S=30;
"40 are on the Vision team": V=40;
"5 workers are on both the Marketing and Sales teams": MnS=5, note here that some from these 5 can be the members of Vision team as well, MnS is sections d
on the diagram (assuming Marketing = A, Sales = B and Vision = C);
"6 workers are on both the Sales and Vision teams": SnV=6 (the same as above sections f
"9 workers are on both the Marketing and Vision teams": MnV=9.
"4 workers are on all three teams": MnSnV=4, section 4.Question:
Applying first formula as we have intersections of two groups and not the number of only (exactly) 2 group members:
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.Example 2:Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?Translating:
"Each of the 59 members
in a high school class is required to sign up for a minimum of one
and a maximum of three
academic clubs": Total=59, Neither=0 (as members are required to sign up for a minimum of one
"22 students sign up for the poetry club": P=22;
"27 students for the history club": H=27;
"28 students for the writing club": W=28;
"6 students sign up for exactly two clubs
": (sum of EXACTLY 2-group overlaps)=6, so the sum of sections d
, and f
is given to be 6, (among these 6 students there are no one who is a member of ALL 3 clubs) Question:
: "How many students sign up for all three clubs?" --> \(PnHnW=g=?\)
Apply second formula: \(Total=P+H+W -(sum \ of \ EXACTLY \ 2-group \ overlaps)-2*PnHnW + Neither\)
--> \(59=22+27+28-6-2*x+0\) --> \(x=6\).Answer: 6.
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There are 9 more examples discussed here: advanced-overlapping-sets-problems-144260.html
Hope this helps.
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