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16 Jul 2021, 16:31
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
(hard)
Question Stats:
35% (01:55) correct
65%
(02:00)
wrong
based on 268
sessions
History
Date
Time
Result
Not Attempted Yet
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
Source: Self-made question
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
Source: Self-made question
16 Jul 2021, 23:18
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AndrewN
Clearly the largest value will be when the mathematical operations are *
4☐3 =4*3=12
5△2 = 5*2=10
Minimum will be when mathematical operations are - or /.
4☐3 =4-3=1
5△2 = 5/2=2.5
Let us take different operations for ☐
1) ☐ = -
4☐3 =4-3=1, BUT minimum value of 5△2 is greater than 1 in every case.
Thus when ☐ = -, △ can be of the FOUR ops........3 ways as both are DISTINCT.
2) ☐ = /
4☐3 =4/3=1.33, BUT minimum value of 5△2 is greater than 1 in every case.
Thus when ☐ = /, △ can be of the FOUR ops........3 ways as both are DISTINCT.
3) ☐ = +
4☐3 =4+3=7, value of 5△2 is greater than 7 only when it is *.
Thus when ☐ = +, △ can be only *........1 way
4) ☐ = *
4☐3 =4*3=12, BUT maximum value of 5△2 is less than 12 in every case.
Thus when ☐ = *, 0 ways.
Total 3+3+1+=7
B
20 Jul 2021, 11:27
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AndrewN
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 [ 35.91 KiB | Viewed 3734 times ]
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 X
4/3 < 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 X
4 * 3 < 3 X
4/3 < 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 = 10
4 + 3 < 10 √
4 - 3 < 10 √
4/3 < 10 √
4) 5/2 = 2.5
4 + 3 < 2.5 X
4 - 3 < 2.5 √
4 * 3 < 2.5 X
Now 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 X
4 - 3 < 5 - 2 √
4 * 3 < 5 * 2 X
4/3 < 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
