Bunuel wrote:
Four staff members at a certain company worked on a project. The amounts of time that the four staff members worked on the project were in the ratio 2 to 3 to 5 to 6. If one of the four staff members worked on the project for 30 hours, which of the following CANNOT be the total number of hours that the four staff members worked on the project?
(A) 80
(B) 96
(C) 160
(D) 192
(E) 240
We are given that the amounts of time that the four members worked on the project were in the ratio 2 to 3 to 5 to 6. A straightforward approach is to create a ratio with x multipliers. The ratio becomes: 2x : 3x : 5x : 6x, in which 2x was person 1’s time, 3x was person 2’s time, 5x was person 3’s time, and 6x was person 4’s time.
From this information, we can determine that the total time worked by all the members is the sum of our ratios: 2x + 3x + 5x + 6x = 16x.
We are also given that one of the members worked for 30 hours. Thus, we can create 4 different equations to get 4 different possible x values.
Option 1) If Person 1 was the individual who worked 30 hours, then 2x = 30 and x = 15
Option 2) If Person 2 was the individual who worked 30 hours, then 3x = 30 and x = 10
Option 3) If Person 3 was the individual who worked 30 hours, then 5x = 30 and x = 6
Option 4) If Person 4 was the individual who worked 30 hours, then 6x = 30 and x = 5
The above results show the 4 different options for the total number of hours an individual staff member worked.
Now, remember that the entire group worked for 16x hours. We substitute each of the 4 possible values for x into this expression:
Option 1: 16x = (16)(15) = 240 hours
Option 2: 16x = (16)(10) = 160 hours
Option 3: 16x = (16)(6) = 96 hours
Option 4: 16x = (16)(5) = 80 hours
The only value that we did not get was 192 hours, so D is the correct answer.
Answer: D
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