Question of the Week - 43 (While working individually, each of A, B)

given a,b,c,d are consective integers
so a=x ,b=x+1, c= x+2, d= x+3
also given
1/a + 1/c>1/2
use hit and trial to see at which value of x we get 1/a + 1/c>1/2
only at a=4 and c=6 we get >1/2
so a=4,b=5,c=6,d=7
working together
1/4+1/5+1/6+1/7 = 5*6*7+4*6*7+4*5*7+4*5*6/ (4*5*6*7 )
solve we get
638/840 ; time 840*60/638 = ~79 mins
IMO E

a + c > 2 and a<b<c<d,
means, a will take at least 4 hours to complete the work
in an hour, units of work completed,

for A, 60*1/4 = 15
B, 60*1/5 = 12
C, 60*1/6=10
D, 60*1/7 ~ 8

Total work = 15+12+10+8 = 45
remaining work= 60-45= 15 which 4th of total, so would need additional 4th of an hour, i.e. 15mins,

in all the minimum time would be greater than, 60 + 15 = 75, the only answer which is greater than 75, is E

For work to complete in 2 hours or less, A and C must complete half the work in an hour. The combined rate
of work of A and C cannot be greater than 30 units in an hour. B and D will complete more than 30 units/hr.
With increasing time, the rate at which they individually complete the work decreases.

Let us assume that A completes the work at 15 units/hr. A will take 4 hours (\(\frac{60}{15} = 4\))

Since the time at which they individually complete work is in consecutive order,
B will take 5 hours and the work done by A in an hour is \(\frac{60}{5} = 12\) units.
C will take 6 hours and the work done by A in an hour is \(\frac{60}{6} = 10\) units.
D will take 7 hours and the work done by A in an hour is \(\frac{60}{7} = 8.5\) units. (approximately)
In an hour, they will complete a total of \(15 + 12 + 10 + 8.5 = 45.5\) units.

Therefore, the time take to complete the work is \(\frac{60}{45.5} = 1.32\) hours (or) 79 minutes(Option E)

Because of the structure of the question, we can devise a shortcut to solving this problem – one that takes only a few seconds - by employing a bit of logic:

In solving questions like this we have to fix a minimum or a maximum time for the value to be determined (in this case, the time that all four will take to complete the job). In this particular case, however, we can only fix a minimum time because there is no maximum limit set: A & C could take a 100 or even a million hours to complete the job. It follows that when A, B, C and D all work together there is no maximum time limit and we can only fix a minimum value since there is a fixed relationship between the times that each take to finish the job while working alone.
It stands to reason that the minimum time in which A, B, C and D can complete the work must lie between the highest value (E) and the second highest value (D) thus making E the only possible answer because, if the minimum time is less (i.e. between any two of the lesser answer choices), all the answer choices which are more than the minimum time become possible. For example, if the minimum time is 50 minutes (i.e. between B and C), that would make C, D and E all possible but the question allows for only one correct answer. If the minimum time was more than 79 minutes (E) then the correct answer would have been "None" which is not among the answer choices. So the correct answer can only be E. On the other hand, if the minimum time is less than (A), all the options (A) through (E) could be possible and the correct answer would be "All" which is also not among the given options.
This shortcut could be prevented by changing the structure of the question to allow for more than one correct answer (e.g. D and E, A only, E only, all five, none, etc.).

Hi everyone,

I'm a bit confused and I was hoping to get some clarification on this.

I see an answer that states:
(1/a + a/c) > 1/2
Why does a has to be at least 4? If a were to be 1, (1/1 + 1/3) is indeed greater than 1/2.

Another answer states the opposite:
(1/a + 1/c) < 1/2
In this case, I do understand that a must be at least 4, but why are we posing (1/a + 1/c) less than 1/2 when we know that it takes a+c at least 2 hours to build 60 units?

Thanks so much for helping!
Mike

Hi,

let w,x,y,z be the respective work rates with x=a+1, y=a+2,..., we get from the text:

60/(w+y)>2
30>w+y, because w=60/a and y=60/(a+2) we get
30>60/a+60/(a+2), expanding RHS
30>(120a+120)/(a^2+2a)
30a^2+60a>120a+120, dividing by 30
a^2+2a>4a+4
a^2-2a>4
a(a-2)>4

a is an integer -> a>=4 -> the least possible work rates are w=60/4, x=60/5, ..., so the least time in hours they require to get the job done is given by:

60/(w+x+y+z)=420/319

-> 60*420/319=78.99min -> (E)

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