(PrepSwift) Quantitative Strategy > Making Deductions 2 > Quiz Question1

GRE Quant Problem Solving
1

The question in the quiz is: In a company, there are only managers and workers in the ratio 2:5. It can be shown that if there are between p and q workers (both inclusive) in the company, the number of employees in the company can be uniquely determined. What is the maximum value of q−p?

The solution I have come up with was as follows.

  1. p <= 5x <= q , where x can be any integer value
  2. thus, p + q must be a multiple of 5, as both p and q have to be multiples of 5 in order for x to be an integer
  3. since the only way we have one multiple of 5 in a list of numbers is when we have five integers (through brute force checking of 1,2,3,4,5, and then 22,23,24,25,25, and so on), I deduced that the highest possible value for q-p must be 4. If they are different by at least 5 or more, we will have two possibilities (for instance 20,21,22,23,24,25), here both 20 and 25 can be answers and thus not unique.

I have a few questions. The solution on the page says that to satisfy the request, the range must satisfy: 5(k−1) +1 and 5(k+1)−1. I am not sure how you go to this equation. Is this something I can practice?
If we have q-p max =8 for instance, would the list 22,23,24,25,26,27,28,29,30 be invalid since we won’t have one unique solution?

2

That would be invalid (because you have two solutions 25 and 30), but 21-29 would work as you have only one solution, namely 25.

3

I understand but so would 22,23,24,25,26 right? A difference of 4 will ALWAYS work. but if we say 8 works but doesn’t 9 also work?

4

Yes, and we want to maximise this. “22,23,24,25,26” works. 8 works. Can you find an example where 9 works?

5

would this not work for 9 - “22,23,24,25,26,27,28,29,30,31”. Looking at this, maybe q-p can go up to 12?

6

That would not work because the number of workers could be 25 or 30.

7

ah right okay. that slipped my mind for a second. so for this question, we can say it is valid for some 8 range numbers so 8 is the max possible?

8

Indeed.