12 Days of Christmas GMAT Competition - Day 5: Amy and Beth, both work

(1) Amy alone can complete the same task in 8 days
This statement alone is insufficient because we don't know anything about Beth. Further, we also don't know anything about the total work.

(2) Beth alone can complete the same task in 20 days
This statement is insufficient because we don't know anything about Amy. Further, we also don't know anything about the total work.

Now combining 1 and 2
Let's assume total work to be 40 units
Amy per day efficiency= 5 units
Beth per day efficiency= 2 units
Now we know that they started working alternatively
If Amy starts then
Day 1 --> 5 units
Day 2 --> 2 units
Day 3 --> 5 units
Day 4 --> 2 units
Day 5 --> 5 units
Day 6 --> 2 units
Day 7 --> 5 units
Day 8 --> 2 units
Day 9 --> 5 units
Day 10 --> 2 units
Day 11--> 5 units
Work gets completed on 11th day itself.
If Beth starts
Day 1 --> 2 units
Day 2 --> 5 units
Day 3 --> 2 units
Day 4 --> 5 units
Day 5 --> 2 units
Day 6 --> 5 units
Day 7 --> 2 units
Day 8 --> 5 units
Day 9 --> 2 units
Day 10 --> 5 units
Day 11--> 2 units
Day 12--> 3 units
work gets completed on day 12
Therefore combining 1 and 2 is also insufficient
Option E

Given: Amy and Beth, both working at their respective constant rates, were assigned a task that they had to complete together. However, they decided to individually work on the task on alternate days till the task gets completed.
Asked: If they started the task from 1st of January, then on which day in January will the task get completed?


Since days in which Beth alone can complete the same task is unknown.
NOT SUFFICIENT


Since days in which Amy alone can complete the same task is unknown.
NOT SUFFICIENT

(1) + (2)



Case 1: Amy started on 1st of January
Task completion = 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8
Task completion requires 11 days and will be completed on 11th January.

Case 2: Beth started on 1st of January
Task completion = 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 1/8 + 1/20 + 3/40
Task completion requires 12 days and will be completed on 12th January.

NOT SUFFICIENT

IMO E

Condition 1:
We know Amy alone can complete within 8 days, but we don't know how long Beth can take. Then we can't find when the work will complete. Eliminate A and D.

Condition 2:
We know Beth alone can complete within 20 days, but we don't know how long Amy can take. Then we can't find when the work will complete. Eliminate B.

Condition 1 & 2:
We know amount of work both Amy and can do in a day. So,

Every 2 days, the amount of work that can complete is:
1/8 +1/20 = (5+2)/40 = 7/40


But still, we don't know who will start the work.
After 5x2= 10 days, the amount of work completed will be 5x7/40 = 35/40.
If Amy works on 11th day, then the total amount is 35/40+ 1/8 = 1. The work will be done on 11th January.
But, if Beth works on 11th day, then the total amount of work 35/40+1/20 = 37/40. The work will not complete 11th January, but will complete on 12th January.

So 2 answers are possible and we can't say when the work will complete. Eliminate option C.

Hence E is the best answer choice.

case 1:- Amy alone can complete the same task in 8 days --> alone insufficient (Beth rate is unknown)
case 2 :- Beth alone can complete the same task in 20 days --> alone insufficient (Amy rate is unknown)

combined case 1 & 2 insufficient. Ans E
we don't know who start the task. this information is important
if Amy starts the task. the task can be completed a day earlier.

Let Total work to be completed be '40T'

From Statement 1: A complete 5T in 1 day Insufficient

From Statement 2: B complete 2T in 1 day Insufficient

Now on combining statement 1 and statement 2 we get,

7T of work will be done by them alternatively in 2 days, After 10th Day 35T of work will be completed.

If if A starts the work, remaining 5T of work can be completed by A in 1 day. After 11 days total work can be completed.

If B starts the work , B will complete 2T in 11th day, than A will complete remaining 3T in 12th day. After 12 days total work can be completed.

Hence IMO E .

Let Amy complete the task in 'a' days and Beth complete it in 'b' days.
When working individually, their one-day work will keep getting added.
Amy's one day work = 1/a
Beth's one day work = 1/b

(1/a) + (1/b) + (1/a) + (1/b) + (1/a) + (1/b) + (1/a) + (1/b) + ... >= 1
We need to know their individual rates of working and equate that sum to 1 to count how many days it'll take them.
Who starts first may matter. We will check if we need to.

(1) Amy alone can complete the same task in 8 days
a = 8. We still need b to get total number of days.

(2) Beth alone can complete the same task in 20 days
b = 20. We still need a to get total number of days.

Together: we can calculate using (1/a) + (1/b) + (1/a) + (1/b) + (1/a) + (1/b) + (1/a) + (1/b) + ... >= 1
Now, notice one thing:
1/8 = 0.125
1/20 = 0.05

1/8 > 1/20
If Amy starts on January 1, 0.125+0.05+0.125+0.05+0.125+0.05+0.125+0.05+0.125+0.05+0.125=1.00 (11 days)
If Beth starts on January 1, 0.05+0.125+0.05+0.125+0.05+0.125+0.05+0.125+0.05+0.125+0.05+0.125=1.05 (again 12)

Not sufficient.

When we speak about two people doing some work together, we need to know the respective speeds of both of them - otherwise, there's no way of calculating the time they need to finish the task. That being said, conditions 1 and 2 will obviously not suffice separately (options A, B and D are out).

However, will we have enough information from both conditions combined?
Amy's speed will be \(a=\frac{1}{8}=\frac{5}{40}\), and Beth's will look as follows: \(b=\frac{1}{20}=\frac{2}{40}\)

Together, in any two consecutive days, they will manage to do \(a+b=\frac{5}{40} +\frac{ 2}{40} =\frac{ 7}{40}\) of their task. Therefore, after 10 days of work, they will finish \(5*\frac{7}{40} = \frac{35}{40}\).

And here's what the trouble is: now it becomes really important, who starts to work on January 1st. If it's Amy, then on January 11th she'll finish the job.
However, if it's Beth's turn, she will only do a small part, and the task will be finished only the following day by Amy, with \(\frac{35}{40}+\frac{2}{40}+\frac{5}{40} = \frac{42}{40}\).

Therefore, even two options together aren't sufficient, so the answer is E.

Let R_a & R_b be the rates at which A and B work.
Total work be W

Statement 1:
Ra = 1/8
Insufficient

Statement 2:
Rb = 1/20
Insufficient

Consider both statements together.
If say A start working on Jan 1st then in 2 days both would have finished (1/8 + 1/20) 7/40 of the work
so in 10 days, they will finish 35/40 of the total work
On the 11th day, A will work (5/40 in a day) and will finish the task

On the contrary, if B starts on Jan 1st then it'll take 12 days.

ANSWER E

(1) Amy alone can complete the same task in 8 days

By st1 not possible .since we don't know how many days Beth takes to complete the task

(2) Beth alone can complete the same task in 20 days

for similar reason not possible by st2

St1 and st2

let the total work be 40
work done by Amy in one day =5
work done by Beth in one day 2

we don't know who will start the task ..therefore cant say

OA:E

Answer is E
Amy and Beth, both working at their respective constant rates, were assigned a task that they had to complete together. However, they decided to individually work on the task on alternate days till the task gets completed. If they started the task from 1st of January, then on which day in January will the task get completed?

(1) Amy alone can complete the same task in 8 days Insufficient. No info about Beth
(2) Beth alone can complete the same task in 20 days Insufficient. No info about Amy
(1) + (2) Insufficient. Don't know who works first... E

Amy and Beth, both working at their respective constant rates, were assigned a task that they had to complete together. However, they decided to individually work on the task on alternate days till the task gets completed. If they started the task from 1st of January, then on which day in January will the task get completed?

(1) Amy alone can complete the same task in 8 days
No information about B's efficiency..
So, work done on one half of alternate day pair can't be determined.

Hence, insufficent

(2) Beth alone can complete the same task in 20 days

No information about A's efficiency..
Similarly, work done on one half of alternate day pair can't be determined.

Hence, insufficent

(1) & (2)

A alone does work in 8 days
B alone does work in 20 days

Let work = 40 units
A's efficiency = 5 units/day
B's efficiency = 2 units/day

If alternate day's work ABABAB...so on, then work finished in = 2days x 5 + 1 day = 11 days ie, on 11 th January
If alternate day's work BABABA...so on, then work finished in = 2days x 5 + 1 (3/5) days = 11 (3/5 days) ie, on 12 th January

Since, ABABAB... OR BABABA... pattern not known from question prompt,
the answer can't be uniquely determined

So, (E) is the correct answer

1 and 2 are clearly insuff.

3)
Convert rates to LCM.
A=5/40, B=2/40
If Beth starts:

2/40, 7/40, 9/40, 14/40, 16/40, 21/40, 23/40, 28/40, 30/40, 35/40, 37/40, 42/40. 12 days to complete.

If Amy starts:
5/40, 7/40, 12/40, 14/40, 19/40, 21/40, 26/40, 28/40, 33/40, 35/40, 40/40. 11 days.

We cannot determine the date, as we have no idea who starts.

E.

E is the answer!
Amy and Beth, both working at their respective constant rates, were assigned a task that they had to complete together. However, they decided to individually work on the task on alternate days till the task gets completed. If they started the task from 1st of January, then on which day in January will the task get completed?

Interpret:
--Work problem--Here's the general info on how to solve them.
--Need to solve work problems using the rate in the form JOBS/DAY whether or not we ultimately want our answer in days/hours (time) or # of jobs done per time (not common and not what is asked for here..)
--Let A = # of job (portion there of)/hour Amy can do. Example if Amy takes 8 days to do a job then she can do 1/8 job in one day.
--Let B = same for Beth.
--A+B = jobs per day for both together. The inverse of this is the total number of days required to do the job.

Now, let's interpret the data:

(1) Amy alone can complete the same task in 8 days
Not sufficient.
1/8th of the job is done at the end of each day that she works.
We know nothing about the intervening days.

(2) Beth alone can complete the same task in 20 days
Not sufficient.
1/20th of the job is done at the end of each day that she works.
We know nothing about the intervening days.

Answer C vs. E

We don't know the answer with both pieces of information either. We need to know who works the first day...
If Amy starts:
1/8 + 1/20 + 1/8 + 1/20 + .... + 1/8 = one full job at the end of day 11
If Beth starts:
1/20 + 1/8 + 1/20 + 1/8 + ... + 1/20 (on the 11th day) = only 37/40 of the job is done. Amy must still do some work and completes the job in the middle of the 12th day.
E is the answer!

Thanks! Great work problem! Keep the variations coming GMATClub!

Both 1 and 2 individually are not sufficient as we don't have know the work rate of other. person.
Now lets combine them. In the hind sight it might look like we have the number of days as we have work rate of both employee. But "working ALTERNATE days" puts a twist.

You guessed it. We don't know who started first. Lets see if that actually impacts.
In 2 consecutive days, they together work 7/40 of task. With this, we know it will take around 5.7 set of consecutive days.

Now after 5 set of consecutive days, only 1/8 of work will be left.
So if Amy starts on 1st Jan, she will complete work on 11th Jan. But if Beth starts on 1st Jan, then she will be working on 11st Jan and work will complete on 12th Jan

We need to have the info of both participants to conclude anything. Hence A and B are eliminated.

Let's assume the work is of 80 units.

So Amy can complete 10 units in 1 day.
//y Beth can complete 4 units in 1 day.

Lets say that Amy starts first

ABAB....

For each 2 days the work completed is 14 units. Total days req = 2*(80/14) = ABABABABAB(70units done) If A comes next it will take 11 days.

Suppose B starts the work. BABABABABA(70 nits) + B(4units) + A(6 units) = So 12th day middle.

Hence we need info on who starts the work.

IMO E.

Amy and Beth, both working at their respective constant rates, were assigned a task that they had to complete together. However, they decided to individually work on the task on alternate days till the task gets completed. If they started the task from 1st of January, then on which day in January will the task get completed?

(1) Amy alone can complete the same task in 8 days
(2) Beth alone can complete the same task in 20 days

Let work be 40 units

(1) and (2) are not sufficient on a standalone basis.

(1) + (2)

Amy does 5 units of work per day and Beth does 2 units/day

If Amy starts and they work alternatively then work will be done in 15 days ((5+2)*7+5)
If Beth starts and they work alternatively then work will be done in 16 days ((2+5)*7+2+3)

E is the answer

Amy and Beth, both working at their respective constant rates, were assigned a task that they had to complete together. However, they decided to individually work on the task on alternate days till the task gets completed. If they started the task from 1st of January, then on which day in January will the task get completed?

(1) Amy alone can complete the same task in 8 days
(2) Beth alone can complete the same task in 20 days

Statement 1 : Rate of Beth is not given,
Insufficient

Statement 2 : Rate of Amy is not given,
Insufficient

Combining Statement 1 and Statement 2, we do not know who will work the first day
If Amy starts the work the first day and Beth the next day, working on alternate days then work will be completed in 6 days
If Beth starts the work the first day and Amy the next day, working on alternate days then work will be completed in more than 6 days

Insufficient

Ans : E