At the Northgate distribution center, a sorting line can run in one of

Question

At the Northgate distribution center, a sorting line can run in one of two modes: automated mode, in which it processes 180 parcels per hour, and manual-check mode, in which it processes 45 parcels per hour. For a certain batch, was the line’s average (arithmetic mean) processing rate greater than 120 parcels per hour?

(1) The batch contained 240 parcels.
 (2) More than 3/4 of the parcels in the batch were processed in automated mode.

Bunuel · Accepted Answer

GMAT Club Official Solution:

At the Northgate distribution center, a sorting line can run in one of two modes: automated mode, in which it processes 180 parcels per hour, and manual-check mode, in which it processes 45 parcels per hour. For a certain batch, was the line’s average (arithmetic mean) processing rate greater than 120 parcels per hour?

Let the total number of parcels be n, and suppose a fraction x of the parcels was processed in automated mode. Then 1 - x was processed in manual-check mode.

The total time would then be (work)/(rate):

nx/180 + n(1 - x)/45

So the average processing rate would be (work)/(total time):

n/(nx/180 + n(1 - x)/45)

= 1/(x/180 + (1 - x)/45)

= 1/((x + 4 - 4x)/180)

= 1/((4 - 3x)/180)

= 180/(4 - 3x)

We are asked whether this is greater than 120:

Is 180 / (4 - 3x) > 120?

Since 0 <= x <= 1, we have 4 - 3x > 0, so we can multiply both sides without changing the direction of the inequality:

Is 180 > 120(4 - 3x)?

Is 180 > 480 - 360x?

Is 360x > 300?

Is x > 5/6?

Thus, the question essentially asks whether the fraction of parcels processed in automated mode was greater than 5/6 of the total.

(1) The batch contained 240 parcels.

This gives only the total number of parcels, which does not determine whether x > 5/6. Not sufficient.

(2) More than 3/4 of the parcels in the batch were processed in automated mode.

Since 3/4 is less than 5/6, the fraction processed in automated mode could still have been less than 5/6 or greater than 5/6.

(1)+(2) Statement (1) still adds nothing useful, and statement (2) still does not tell us whether x > 5/6. Not sufficient.

Answer: E.

Adit_ · Answer

Statement 1:
Clearly INSUFFICIENT.

Statement 2:
Lets take 3x for automated and x for manual.
Avg = Total goods / Total time
Avg = 3x+x/(3x/180+x/45)

= 4x/(7x/180)
= 180*4/7
= 720/7 = 103

And 103<120.
This value will increase as we increase the amount of goods doing automated increases. Thus could be >120 per hour too.
INSUFFICIENT.

Combining the two:
Its clear that we get all the data using the ratios only.
Knowing just the total amount of good here is of now help the same 3:1 ratio can be split from the total 240 and yield same results.
INSUFFICIENT.

Answer:  Option E

MartyMurray · Answer

At the Northgate distribution center, a sorting line can run in one of two modes: automated mode, in which it processes 180 parcels per hour, and manual-check mode, in which it processes 45 parcels per hour. For a certain batch, was the line’s average (arithmetic mean) processing rate greater than 120 parcels per hour?

(1) The batch contained 240 parcels.

The fact that the batch contained 240 parcels indicates nothing about the rate at which the parcels were processed.

Insufficient.

(2) More than 3/4 of the parcels in the batch were processed in automated mode.

3/4 is the low end boundary value for the proportion of the parcels in the batch processed in automated mode.

So, to determine whether this statement is sufficient, we can calculate the average rate at this boundary value since we know that the average rate will be higher than the average so calculated.

Average Rate = Total Parcels/Total Time

To find the average rate when 3/4 are processed at the higher rate, try 3 parcels at the higher rate and 1 at the lower rate.

Time per package at automated rate: 1/180 hour

Time per package at manual-check rate: 1/45 hour

Total Time of Example = 3/180 + 1/45 = 7/180

Average Rate of Example = 4/(7/180) = 720/7 ≈ 103/hour

103/hour < 120/hour

More than 3/4 could have been processed at the faster rate. So, the rate could also have been faster than 120/hour.

Insufficient.

Statements (1) and (2) combined

Adding to the information provided by statement (2) information on the number of parcels from statement (1) makes no difference. After all, regardless of the number of parcels, the ratio of packages processed at the two rates remains the same, meaning that the average could have been as low as 103/hour and could have been higher than 120/hour.

Insufficient.

Correct answer:   E

Adit_ · Answer

They have asked the average rate which is total jobs per total time and taking the direct weighted averages in this case won't work IMO.
Just like average speed situation where we have total distance per total time.

MartyMurray · Answer

Oh, right. Of course, that makes sense.

At the Northgate distribution center, a sorting line can run in one of

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