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Range question

by tamiri » Sat Jun 01, 2013 9:36 am
A school staff consists of senior teachers and junior teachers. If the range of the salaries of the senior teachers is $4500 and the range of the salaries of the junior teachers is $2000, what is the least possible range of the salaries for the entire school's staff?
a. $0
b. $2000
c. $2500
d. $4500
e. $6500

I thought it would be answer A and I am told it is D. Why?

Thanks,
Tamir
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by fcabanski » Sat Jun 01, 2013 11:30 am
Identify - range problem.

Set Up:

Write what you know. Range is the highest value minus the lowest value.
Write what the problem tells you: Range of seniors is 4500. Range of juniors is 2000.

Plug in some numbers.

Senior teachers: Highest salary is $6500 and lowest is $2000. ($6500 - $2000 = $4500)
Junior teachers: Highest salary is $4000 and lowest is $2000. ($4000 - $2000 = $2000)

Make sure: What does the problem ask? It asks for the lowest possible range for all teachers' salaries.

Execute - solve the problem. The range for all salaries in the example (with plugged in numbers) is $6500 - $4000 = $2500.

If the junior teachers' highest salary is lower than the senior teachers' lowest, then the overall range becomes larger. For example: junior $1999 is lowest and $2999 is highest. Senior salaries are our original example. Overall range is then $6500 - $1999 = $4501. If the juniors' salaries are even lower, the range grows even more.

If the seniors' lowest is higher than the juniors' highest, the overall range also increases. Example:
senior $2001 is lowest and $6501 is highest. Junior salaries are our original example. Overall range is then $6501 = $2000 = 4501. If the senior salaries are even higher, the range grows even more.

The lowest possible range is the original example, when the low salaries for juniors and seniors are the same. The smallest range is the seniors' range, which is $4500.
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by tamiri » Sun Jun 02, 2013 7:24 pm
Great , Thank you!
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by faraz_jeddah » Mon Jun 03, 2013 2:47 am
fcabanski wrote:Identify - range problem.

Set Up:

Write what you know. Range is the highest value minus the lowest value.
Write what the problem tells you: Range of seniors is 4500. Range of juniors is 2000.

Plug in some numbers.

Senior teachers: Highest salary is $6500 and lowest is $2000. ($6500 - $2000 = $4500)
Junior teachers: Highest salary is $4000 and lowest is $2000. ($4000 - $2000 = $2000)

Make sure: What does the problem ask? It asks for the lowest possible range for all teachers' salaries.

Execute - solve the problem. The range for all salaries in the example (with plugged in numbers) is $6500 - $4000 = $2500.

If the junior teachers' highest salary is lower than the senior teachers' lowest, then the overall range becomes larger. For example: junior $1999 is lowest and $2999 is highest. Senior salaries are our original example. Overall range is then $6500 - $1999 = $4501. If the juniors' salaries are even lower, the range grows even more.

If the seniors' lowest is higher than the juniors' highest, the overall range also increases. Example:
senior $2001 is lowest and $6501 is highest. Junior salaries are our original example. Overall range is then $6501 = $2000 = 4501. If the senior salaries are even higher, the range grows even more.

The lowest possible range is the original example, when the low salaries for juniors and seniors are the same. The smallest range is the seniors' range, which is $4500.
but the question asks us what is the least possible range? Should it not be 2500 as you proved it.
4500 > 2500

I feel lost.
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by Matt@VeritasPrep » Mon Jun 03, 2013 10:16 am
faraz_jeddah wrote: but the question asks us what is the least possible range? Should it not be 2500 as you proved it.
4500 > 2500

I feel lost.
Here's one way of thinking about it.

Let's say the highest paid member of the senior staff is A, and that the lowest paid member of the senior staff is B. We know that A = B + $4,500.

Let's also the highest paid member of the junior staff is C, and that the lowest paid member of the junior staff is D. We know that C = D + $2,000.

Now we have a few possibilities to consider.

If B makes more money than C, our salaries go A > B > C > D. Since A = B + $4,500 and B > C = D + $2,000, A is at least $6,500 greater than D, so our range is huge.

Now let's say B makes less than C but more than D. Then our salaries go A > C > B > D. Since A is still equal to B + $4,500, and D is less than B, the range is $4,500 PLUS the difference between B and D. We don't know what the range of this set is, but whatever it is, it's greater than $4,500.

Now let's say that both C and D make more money than B, but less money than A. Then our salaries go A > C > D > B, and our range is exactly $4,500.

Now let's say that we live my private heaven, where the junior staff actually make more than the executives :D

If C makes more than A, then our salaries go either C > A > D > B or C > A > B > D. In either case, the range is greater than $4,500, since we have the difference between A and B ($4,500) plus the difference between C and A (in both cases) and between D and B (in the first case). So again our range exceeds $4,500.

The last case is the truly remarkable one where C > D > A > B ... and again the range exceeds $4,500.

An easy way of summing all this up is that the range is difference between the greatest and least terms in a set, so that range will never shrink if you add more terms to that set, since that difference still exists. Since the range between A and B is $4,500, adding more junior members will either increase the range (as we saw in four of the five cases above) or keep it the same (if A > C > D > B).
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by fcabanski » Mon Jun 03, 2013 11:15 am
":but the question asks us what is the least possible range? Should it not be 2500 as you proved it.
4500 > 2500 "

The question asks for the least possible range amongst all the salaries.

That range cannot be only 2500, since the range of senior salaries is 4500. The lowest senior salary is $4500 less than the highest. The junior salaries cannot shrink the senior salary range.

Try a very simple example.

Seniors: $10,000, $14,500 range = 4500
Juniors: $12,000, $14,000 range = 2000

Take all four together: $10,000, 12,000, 14,000, 14,500 range = 14500 - 10000 = 4500

That's the smallest possible range when junior and senior teachers' salaries are in one big list.

If any of the junior salaries fall below the lowest senior salary, the range increases.

Juniors $9,000, $11,000 range = 2000

Now the combined list becomes: 9000, 10000, 11000, 14,500 range = 5500

In any two sets of numbers A and B, where each has a different range rangeA > rangeB, the smallest possible value of the combined set's range (rangeAB) is the larger range (range A). That happens when the set with the smaller range, B, consists of values that are within A's range.
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