Math Word Problems with Logic?

Question

These problems is based on logic thinking. Draw Venn diagrams (example: http://www.learnnc.org/lp/media/lessons/DebbieFox2112003991/VennDiagram1.JPG ) to help you solve the problems.

1. Mark asked 80 students whether they had tried skiing or snowboarding. He found that 35 had tried skiing, 29 had tried snowboarding, and 12 had tried both. How many students in Mark's survey had tried at least one of the two sports?

2. Sergio asked 40 students whether they buy music on tape or compact disc. 7 bought tape, 12 bought the disc, and 10 bought both. How many bought neither discs nor tapes?

3. Tom asked 100 adults whether they had ever collected stamps, coins, or baseball cards. 16 collected stamps, 7 collected coins, 22 collected cards, 4 collected cards and stamps, 1 collected coins and cards, 5 collected stamps and coins, and 5 collected all. How many had collected none of the items?

Please explain clearly and thoroughly!

Thanks for helping me!

Answer 1

For 1: We count the 35 who've tried skiing and add the 29 who tried snowboarding.  But then we've counted the ones who've tried both twice, so we subtract that from our sum, giving us: 35 + 29 - 12, or 52.

For 2: The ones who bought neither discs nor tapes are the ones who did NOT buy at least one of tapes or discs.  We calculate that as above, giving us 7 + 12 - 10 = 9.  If 9 bought at least one, then 40 - 9 = 31 bought neither.

For 3: Let's count up all of the ones who collected one of those things at one point.  We first add up all the ones who collected any:

16 + 7 + 22 = 45.

We subtract the ones who collected two of them (since we added them twice in already)

45 - (4 + 1 + 5) = 35.

We then add in the ones who collected all three (since when we subtracted out the ones who collected two of them, we took them out by accident more than once):

35 + 5 = 40.

So 40 collected at least one, and so 100 - 40 = 60 collected none.