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02 Mar 2021, 05:30
Kudos
Bookmarks
B
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Difficulty:
85%
(hard)
Question Stats:
31% (01:28) correct
69%
(01:33)
wrong
based on 177
sessions
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A line of length L is cut at random point. What is the probability that the ratio of the shorter to the longer segment is less than 1/3 ?
A. 2/3
B. 1/2
C. 1/3
D. 1/4
E. 1/6
Are You Up For the Challenge: 700 Level Questions: 700 Level Questions
A. 2/3
B. 1/2
C. 1/3
D. 1/4
E. 1/6
Are You Up For the Challenge: 700 Level Questions: 700 Level Questions
02 Mar 2021, 09:05
Kudos
Bookmarks
Bunuel
Let the total length be x, so shorter = total-longer.
Therefore, \(\frac{Total-longer}{longer}=\frac{x}{longer}-1<\frac{1}{3}.........\frac{x}{longer}<\frac{4}{3}........longer>\frac{3x}{4}\)
Or shorter length<\(\frac{x}{4}\)
So shorter has to be 1/4 of total, but the side will be called shorter if it is less than 1/2 of total.
Thus, we are looking of probability of a length to be <1/4, if it is given as less than 1/2 of total = \(\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1*2}{4*1}=\frac{1}{2}\)
B
20 Mar 2021, 13:21
Kudos
Bookmarks
Bunuel
Solution:
We can let L be 12 inches. If the shorter segment has a length of 3 inches and the longer segment has a length of 9 inches, then the shorter segment is exactly 1/3 of the length of the longer segment. Now mark the 12-inch line at the 3-inch and 9-inch locations from one end of the line. That is, there are three non-overlapping intervals of the line: the first 3 inches, the middle 6 inches and the last 3 inches. If the line is cut in the first or the last interval, then the ratio of the shorter segment to the longer segment is less than 1/3 (for example, if we cut the line at the midpoint of the first interval, then we have the shorter segment of 1.5 inches and the longer segment of 10.5 inches, and the ratio would then be 1.5/10.5 = 1/7). Since we can cut at any point of the first interval or the last interval and since the total length of these two intervals is 3 + 3 = 6 inches out of the total length of the 12-inch line, the probability is 6/12 = 1/2.
Answer: B
