Bunuel wrote:
Both the average (arithmetic mean) and the median of a set of 7 numbers equal 20. If the smallest number in the set is 5 less than half the largest number, what is the largest possible number in the set?
(A) 40
(B) 38
(C) 33
(D) 32
(E) 30
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:Out of the letters A through E, which one is your favorite?
You may be thinking, “Huh? What a weird question. I don’t have a favorite.”
I don’t have one in the real world either, but I do for the GMAT, and you should, too. When you get stuck, you’re going to need to be able to let go, guess, and move on. If you haven’t been able to narrow down the answers at all, then you’ll have to make a random guess—in which case, you want to have your favorite letter ready to go.
If you have to think about what your favorite letter is, then you don’t have one yet. Pick it right now.
I’m serious. I’m not going to continue until you pick your favorite letter. Got it?
From now on, when you realize that you’re lost and you need to let go, pick your favorite letter immediately and move on. Don’t even think about it.
(This assumes, of course, that your favorite letter is still in the mix. If you were able to narrow down the answers, and you crossed off your favorite letter, then obviously don’t pick that one!)
Okay, let’s solve this thing. What did you do first?
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I glanced at the beginning of the text and saw that it said average (arithmetic mean). I then glanced down to the answers to see how precise I was going to have to be with calculations. The answers are very close together, so estimation isn’t going to work. I’m going to have to do real math.
Okay, time to read the problem and jot down the given info.
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Because the problem gives the median, arrange the numbers in order from smallest to largest (this is a general requirement whenever calculating or displaying a median).
Hmm. That second sentence of the problem is going to take some work. It’s a relationship between the first and last number…but I don’t know the actual value of either one.
Time to set a variable: let’s call something x. My natural inclination would be to call the smallest number x, but I’ve learned to ask myself one important question before arbitrarily assigning that variable: what am I trying to find?
In this case, they asked for the largest possible number, so it’s better to call the last number x. That way, I’ll be solving for the thing that they want; I’m less likely to make a mistake and accidentally solve for x = the smallest number.
If the largest number is x, then the smallest is (1/2)x – 5:
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What next? If the average is 20 and there are 7 numbers, then the sum must be (Average)(number of terms) = (20)(7) = 140.
Pause for a moment and look at everything you’ve got. What’s the best plan from here?
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They want to maximize the largest number, that x. Only one number is set, the middle one (20). All 7 numbers have to add up to 140.
So what needs to happen in order to make x as large as possible?
Since the sum is fixed, the first 6 numbers need to be minimized in order to get the last number to be as large as possible. This step is the key to all max/min problems: figure out what you can influence and either maximize or minimize (as needed) to get to the desired outcome.
Look back at your notes to remind yourself of the restrictions. The fifth and sixth numbers cannot be less than the median of 20. They could equal the median, though, so set these values to 20:
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That leaves a sum of 140 – 20(3) = 80 for the remaining 4 numbers.
Now, what about the second and third values? Again, they can’t be smaller than the first one, but they could equal it. Set all three equal to (1/2)x – 5.
Check it out: you now have a way to express each of the remaining four numbers in terms of that single variable x. Time to set up an equation and solve!
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The largest possible number in the set is 38.
The correct answer is (B).
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