There are 4 card-processing machines in an office. The fastest of thes

Question

There are 4 card-processing machines in an office. The fastest of these machines processes x cards in 7 hours and the slowest processes x cards in 8 hours. Which of the following could NOT be the average time per machine for each of the 4 machines to process x cards?

(A) 7.2
(B) 7.3
(C) 7.5
(D) 7.6
(E) 7.7

(A) 7.2
(B) 7.3
(C) 7.5
(D) 7.6
(E) 7.7

u1983 · Accepted Answer

By, condition the machines those are not fastest or the slowest , can take any time between 7 & 8 hrs to process x cards.
The minimum possible timimg : 7,7,7,8 ; total time = 29 ; Avg time = 7.25 >7.2 ( min of the options) --Thus Option 1 is not possible , all others possible.
The maximum possible timimg : 7,8,8,8 ; total time = 31 ; Avg time = 7.75 >7.7( the max of the options)- Thus all cases possible.
 Ans A .

KarishmaB · Answer

Hey adkikani,

There is no calculation involved in this question.

Out of 4 values, we know two are 7 (least) and 8 (greatest).
The average must be between 7 and 8. All options are within the range 7 to 8. 
Then what would be the constraint? Th problem has to be the least or the greatest value since one of those would lie outside the possible values range.

Think of weighted averages. If 3 machines take 7 hrs each and 1 machine takes 8 hrs, the average would be 1/4 away from 7 i.e. at 7.25.
Average time cannot be less than 7.25 hrs so 7.2 hrs is not possible.

Answer (A)

mayokanmi · Answer

The extremes of the other 2 machines are either all 7 per hour in which case the average will be 
(7+7+7+8)/4= 7.25

Or

all 8per hour making the overall average to be 
(7+8+8+8)/4= 31/4=7.75

Hence the answer cannot be lower than 7.25 per hour making A the answer

Posted from my mobile device

adkikani · Answer

VeritasKarishma   chetan2u   niks18   gmatbusters   Gladiator59   GMATPrepNow   EMPOWERgmatRichC

I understood the above solutions; my query is:
Is there a way to skip long division? The answers seems so close by to approximate.

EMPOWERgmatRichC · Answer

Hi adkikani,

Since the answer choices are so 'close' to one another, we have to do enough 'precision' math to make sure that we have the correct answer (if you approximate the answer incorrectly, then you might convince yourself that one of the wrong answers is correct). Thankfully, the work here isn't too bad - and you really just have to think about no more than two possibilities:

We know the exact rates of 2 of the 4 machines. The FASTEST processes 7 cards in an hour and the SLOWEST processes 8 cards in an hour. We don't know the exact speeds of the other two machines, but we know that they are some rate BETWEEN 7 cards and 8 cards per hour. Thus, there will be a limit to how fast or how slow the overall average could be. By extension, the correct answer will either be TOO FAST or TOO SLOW for what that group is capable of averaging (meaning that the correct answer will be Answer A or Answer E).

Even though we can't have the other two machines working at 7 cards/hour, IF they did work at that rate, then the average would be (7+7+7+8)/4 = 29/4 = 7.25 cards/hour. There is NO POSSIBLE OPTION for the average to be a lower number than that (those other two numbers can only get bigger than 7). At this point, we can stop working; 7.2 is NOT a possibility, so it has to be the answer to this question.

GMAT assassins aren't born, they're made,
Rich

ThatDudeKnows · Answer

The average of the two machines that we know is 7.5.
We know we need an extreme value, so B, C, and D are out.
7.2 deviates from 7.5 by 0.3.
7.7 deviates from 7.5 by 0.2.
How would it be possible to get to 7.7 but not be possible to get to 7.2? E is out.

Answer choice A.

Usernamevisible · Answer

4 machines working together will take much much lesser than 7 hours if they need to process only x machines (and not 4x machines), what am I missing ?

I took LCM of 56 
thus if 7,7,7,8 gives 31 units thus time taken - 56/31 
if 7,8,8,8 then time taken 56/29

what am I missing

Bunuel · Answer

You are finding the combined time for 4 machines working together.

The question asks for the average of the 4 separate individual times.

Have you checked the discussion above?

Usernamevisible · Answer

Two learnings

ONE-LINE MEMORY RULE
Average of the measurements themselves → arithmetic mean.
Average of rates → compute using total output ÷ total input, not by averaging the rate numbers.

DOUBT 
"I learned that I cannot average times/speeds directly. But in this question we averaged the machine times. Why is that allowed?"

---

WHAT CAUSED THE CONFUSION?
You mixed up two different concepts:
Concept A = Average of observations
Concept B = Average rate/speed/work rate

They sound similar but are different.
---

CONCEPT A: AVERAGE OF OBSERVATIONS
When the question literally asks:
* average age
* average salary
* average score
* average time taken by employees
* average processing time of machines

you simply use arithmetic mean.

Example:
Times = 7, 7.5, 7.8, 8
Average time
= (7+7.5+7.8+8)/4
This is valid.
Why? Because we are averaging the observations themselves.

---

CONCEPT B: AVERAGE RATE / SPEED / EFFICIENCY

When the question asks:
* average speed
* average work rate
* average production rate

you usually cannot average the rate numbers directly.

Example:
60 km at 30 km/h
60 km at 60 km/h
Wrong:
(30+60)/2 = 45

Correct:
Average speed
= Total Distance / Total Time
= 120 / 3
= 40

Why?
Because rates combine through work or distance, not through simple averaging.

---

SECOND DOUBT 
"I can make the two unknown machine times any decimals between 7 and 8. Why doesn't that method apply?"

---

ANSWER
It DOES apply.
In fact it is the fastest method.
Machines:
7, ?, ?, 8
The two unknowns must satisfy:
7 ≤ ? ≤ 8
Therefore
14 ≤ (?+?) ≤ 16
Adding the fixed machines:
29 ≤ Total ≤ 31
Dividing by 4:
7.25 ≤ Average ≤ 7.75
Therefore 7.2 is impossible.
Done.

---

HOW TO RECOGNIZE WHICH CONCEPT TO USE
Ask:
What is being averaged?"
If answer is:
"people's ages, salaries, times, scores"
→ Arithmetic Mean

If answer is:
"speed, efficiency, production rate, work rate"
→ Usually NOT arithmetic mean
→ Use Total Work / Total Time or Total Distance / Total Time

There are 4 card-processing machines in an office. The fastest of thes

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

There are 4 card-processing machines in an office. The fastest of thes

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards