Raxit85 wrote:
Three friends were working on a college project. The amount of time that the three friends worked on the project were in the ratio 4: 5: 7. If the total number of hours taken to complete the task is an integer less than 50 and the time spend by each friend is also an integer, then which of the following could be the time taken by the friend who spent least time on the project?
A) 3
B) 8
C) 10
D) 11
E) 16
Solution 1:
• Ratio of time taken by friends =\( 4: 5: 7.\)
o Let \(4x, 5x\), and \(7x\) be the time taken by friends.
• Total time taken by friends is less than 50 hours
o \(4x + 5x + 7x =16x < 50\)
• Times spend by each friend is an integer.
o \(4x, 5x\), and \(7x\), are integers.
• We need to find the least time taken by any of the friend.
o We need to find the value of \(4x\).
o Since \(4x\) is an integer, time taken by him will be a multiple of \(4\).
o In the answer choices, only options B and E are multiple of \(4\).
• Let's take the answer Option B, and E
• Option B: \(4x = 8, x = 2\)
o \(16*x =16*2 =32\), which is less than \(50\), \(4x = 8\) is a possible answer.
• Option E: \(4x = 16, x = 4\)
o \(16*4=64\), which is not less than \(50\), \(4x = 16\) is not possible.
Hence, the correct answer is
Option B.Solution 2:
• Ratio of time taken by friends =\( 4: 5: 7\).
o Let \(4x, 5x\), and \(7x\) be the time taken by friends.
• Total time taken by friends is less than 50 hours and time spend by each friend is an integer.
o \(4x + 5x + 7x =16x < 50\)
o \(x < \frac{50}{16}\)
o \(x < 3.125\), \(x\) can \(1, 2\), and \(3\).
we need to find the value of \(4x\),
o Possible values of \(4x\) are:
o \(4*1=4\)
o \(4*2=8\)
o \(4*3=12\),
Only \(8\) is given in the answer options,
Hence, the correct answer is
Option B. _________________
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