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

work done is alternate days
start day is 1st Jan
need to find day when work will end

#1
Amy alone can complete the same task in 8 days
rate of A ; 1/8
Beth rate unknown
insufficient
#2 Beth alone can complete the same task in 20 days

similarly rate of Amy not known
from 1 &2
we can say LCM of A &B ; 2^3 ; 2^2 * 5 ; 40

case 1 :
let work be 40 units
so A rate 5 days and B 2 days
number of days it will take to complete
40- ( ( 5+2) *5 + 5 )
i.e. 11 days
case 2 ;
let work be 80 units
A per day s in 10 days and B in 4 days
80- ( ( 14)*5 + 10)
again 11 days
case 3 :
let work be 120 units
A rate ( 15 ) and B is 6
120 - ( ( 21) * 5 + 15 ))
again 11 day
sufficient to say that on 11th day work will be completed if both A & B work together
no info who starts work first
we get 11th and 12th day ...
OPTION E is correct

Note: they had to complete together. However, they decided to individually work on the task on alternate days till the task gets completed.

a, b alone are not sufficient

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

Answer can be found if we take a+b work for 2 days individually as total of 1 day. No of days can be calculates. Ans: C

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

Work pattern is ABABABABA.... (alternate days)
S1) Amy completes W/8 portion in a 1st day, you cannot say what portion is completed the next day. (Insuff)

S2) You cannot say what portion is completed by Amy, Beth completes W/20 portion in next day (insuff)

S1+S2) W/8+W/20=6W/40 is completed in 2 days. So from here you can conclude when the work is getting completed. (suff)

Hence C)

Only by combining we can find the answer ...

Hence C is right

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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.

1. We know nothing about the other person s rate NS
2. We know nothing about the other person's rate NS
Combined- we know both their rates so we can determine how many days it will take- Sufficient
Ans C

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.

Neither statement alone is enough. Since we require both the efficiencies. Take statement 1&2. Calculate efficiency for Amy and Beth. Let total work = 40. Amy efficiency= 5. Beth efficiency= 2
Working alternate days=5+2+5+2+5+2+5+2+5+2+5. Hence total days=11. Work will be completed on 11th Jan. But if Beth begins on the first day, we will end up taking longer than 11 days. Hence option E.

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(1) Amy alone can complete the same task in 8 days
In this statement we don't know how long it will take Beth to complete task
So Not sufficient
(2) Beth alone can complete the same task in 20 days
as above. It makes no sense as long as we don't know Amy's capacity
So Not sufficient
LET'S combine the above 2 statements

Amy - 8 days=task 1 day=1/8task
Beth - 20 days=task 1 day=1/20task

1/8task*Xdays+1/20task*Xdays=1Task With that in mind, we can figure out how many days it takes Amy, Beth to complete the task together.

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.

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

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IMO C

Each statement alone cannot be sufficient as we need both the rates to determine the day.

We need both the rate of work of Amy and rate of work of Beth, to determine if they work on the task on alternate days, how many days it will take to complete the task.

Hence, IMO C.

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