AndrewN wrote:
Given 4☐3 < 5△2, if ☐ and △ denote different mathematical operations (+, -, *, /), how many ways are there to satisfy the inequality?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
Hello, everyone. It is interesting that as of this writing, fewer than half the people who have attempted this question have answered correctly. I suspect that the most common error stems from not reading the question closely enough, specifically the part about the symbols representing
different mathematical operations. As you can see in the image below, almost 60 percent of those who answered incorrectly chose (D), and the number 9 would indeed correspond to the
total number of valid inequalities
if the symbols could be the same.
Attachment:
Screen Shot 2021-07-20 at 11.41.42.png
To avoid recapitulating the solution offered by
chetan2u above, I will simply point out that you could just as easily work from the right-hand side, 5
△2, as well:
1)
5 + 2 = 7, and now the + sign is off limits for the left-hand side. Check the other signs:
4 - 3 < 7
√4 * 3 < 7
X4/3 < 7
√No real work is necessary, other than keeping information organized. Now let the
△ be a - sign.
2)
5 - 2 = 3 As before, check the other signs for the left-hand side:
4 + 3 < 3
X4 * 3 < 3
X4/3 < 3
√Again, we can use mental math, without having to worry about, say, the specific value of 4/3 (although you probably know it instantly anyway). Work through the other operations for the
△ and
☐ in the same way:
3)
5 * 2 = 104 + 3 < 10
√4 - 3 < 10
√4/3 < 10
√4)
5/2 = 2.54 + 3 < 2.5
X4 - 3 < 2.5
√4 * 3 < 2.5
XNow tally up the
valid results:
2 + 1 + 3 + 1 = 7
The answer must be (B).
Note that there are a few ways to go wayward here. First, as mentioned earlier, you could repeat signs:
4 + 3 < 5 + 2
X4 - 3 < 5 - 2
√4 * 3 < 5 * 2
X4/3 < 5/2
√One trap that no one has yet fallen into, but that is also plausible for someone in a hurry, is forgetting to
disqualify 7 < 7, a mistake that could lead to 10 apparently valid ways.
I hope you had fun with this one. I was sitting in a chair after work one day and just thought this one up, starting with a simple concept. Anyway, happy studies, everyone.
- Andrew
As of this writing, things are only going south. And the question is now in the category of 95% hard. Strange!