All of the students of Music High School

[ShaneL361]

All of the students of Music High School are in the band, the orchestra, or both. 80 percent of the students are in only one group. There are 119 students in the band. If 50 percent of the students are in the band only, how many students are in the orchestra only?

30
51
60
85
119


What's the best way to determine that its better to use a Venn diagram than a double set matrix here? Simply because there are no students that are in neither group? When I see a set with two groups I'm typically going straight to the double set matrix but time-wise that is clearly not the best path for this question.

Thanks,

Shane

[tim]

Here's how to decide: always use a double set matrix if you can. No exceptions. Zero. Ever.

What makes you think "that is clearly not the best path for this question"?

[RonPurewal]

For 2 overlapping sets, "Plan A" should be the matrix. If you get well and truly stuck, then perhaps try a Venn diagram as "Plan B".

As Tim mentioned, though, starting out with a Venn diagram is generally unwise. It's much less efficient in general.

--

If you happen to be given a problem with three overlapping criteria, THEN you'll want to use a Venn diagram, with 3 circles.

These (rare) problems are specifically engineered to work well with Venn diagrams.
The equivalent of the matrix for 3 sets would be a three-dimensional 3x3x3 cube. They definitely don't expect you to draw three-dimensional organizational devices, so they'll engineer any 3-set problems accordingly.

[tim]

Exactly. Use a Venn Diagram iff you cannot use a double set matrix - either you get stuck, or the problem gives you more than two overlapping sets.

[RonPurewal]

^^ "iff" = "if and only if"

[FrankY886]

Here's how to decide: always use a double set matrix if you can. No exceptions. Zero. Ever.

What makes you think "that is clearly not the best path for this question"?

Because the answer explanation explicitly says so:

"Note: It is also possible to set up algebraic equations or use a double-set matrix but the math is much more tedious (so much so that you might want to decide to guess and move on instead)."

So, I too would like some clarity as to why we would be instructed to use a venn diagram, or why you guys think it is better to use a Matrix Box. Also, i think it would be helpful to see how one would solve this question using a matrix box.

Thanks

[RonPurewal]

yeah, i'd use the matrix too.

here's the deal:

• every region in a venn diagram IS ALSO a cell in the matrix. (the middle 'football shape' is the top left box in the matrix; the two crescent moon shapes are the top middle and middle left boxes; and the region outside the two circles is the box in the middle of the grid.)
therefore, there is no possible mathematical advantage in using a venn diagram, since EVERYTHING that's represented in a venn diagram is also represented in the matrix.

BUT
• the 'total' cells (from the matrix) are NOT explicitly represented in a venn diagram.

thus the matrix has 2 distinct advantages:
1/ it can represent more quantities explicitly (= anything involving a 'total'), as discussed above.
2/ the math is much more 'automatic', since the rows and columns are always additive (first thing + second thing = third thing). with the venn diagram, by contrast, you have to keep thinking actively about how the math is supposed to work.

...so there's basically no good reason to use a venn diagram for 2 overlapping sets, UNLESS there's a significant VISUAL advantage in using it.
if you're a very visual learner, THIS may be an advantage of the venn diagram: it allows you to visualize the overlap, inclusion, exclusion, etc. (as opposed to the matrix, which is basically just a 'spreadsheet').

[RonPurewal]

for this problem, let's say the matrix looks like this:

..............B.....not B.....total
O
not O
total

since this forum doesn't use a uniform-width font, i can't really get the matrix to line up correctly. so, from this point forward, i'll just describe the content in words.

All of the students of Music High School are in the band, the orchestra, or both
--> this means the middle box ('not O, not B') is zero.

80 percent of the students are in only one group
--> 'only one group' is the combination of 'B, not O' (middle left) and 'O, not B' (top middle). this is not terribly helpful—but it also means that the REMAINING 20 percent are in BOTH 'B' and 'O'.

so, if 'T' is the total, then '0.2T' is in the TOP LEFT box.

There are 119 students in the band
--> 119 in the bottom left cell ('B, total').

50 percent of the students are in the band only
--> 0.5T goes in the cell marked 'B, not O' (middle left).

we now have all three cells of the left-hand column filled in, so we can get:
0.2T + 0.5T = 119
0.7T = 119
7T = 1190
T = 170

--

FINALLY...
we know that %80 of the students are in exactly one group.
since %50 of them are in only the band, we know that the other %80 – %50 = %30 of them are in only the orchestra.
so that's 30 percent of 170, or 51.

[RonPurewal]

by the way, i'll just be frank here and admit that this problem has too many steps. the official problems will not have this many steps.