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Here's a boring shmoop-ily unvoiced question you'll find on the exam...

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At a local high school, the mean age of its ten-member glee club is 16.

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With the start of the new school year, the club is expecting the enrollment of

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five additional new members, with the mean age of the first four new recruits as 14.

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If the larger 15-member club turns out to have a mean age of 15.4,

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how old was the final person to join?

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Here are the possible answers:

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For this problem, we really have to know what mean... means.

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Because it has nothing to do with that seriously unfriendly janitor,

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Spyder, who works the night shift at your high school.

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Mean just means... average.

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Yep, we're dealing with an AVERAGING problem.

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Remember Mrs. Slobovitz' words of wisdom:

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Add up all the numbers, then divide by the number of items you added.

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And get your feet off the desk.

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So this question is asking for an average,

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but there are really 2 parts to the... summing we'll have to do.

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Yeah... we're gonna get sum.

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The first part is the ten 16-year-old glee guys and gals.

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16 times 10 is 160, which is how long they've all on this Earth, combined.

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But that's only the first part. Now come the interlopers.

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The second part tells us that 4 recruits have a mean age of 14.

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Do you care if they were 10, 18, 6, and 22? Nah, not really.

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We just want to solve the problem, we're not looking to start a Little League team.

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We only care that the average age of those 4 recruits is 14.

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So now we have 14 of the 15 already in the bag... age-wise...

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...we know the numbers...

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...so we'll be adding 10 times 16, or 160... to 4 times 14, or 56.

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Just like the grocery story check out lady says, Your subtotal is 216, ma'am.

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Only problem is that we're missing one body.

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Let's just step back a sec for a reality check here.

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What's our average up to this point? Well, we have 14 bodies and a total of 216...

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so if we forgot about the last man standing, we could get the average age thus far if we wanted.

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We have 216 divided by 14 which is 15.42... ish

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That's an interesting number because they tell us that the overall average is 15.4.

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Let's say we totally didn't get how to solve this problem. Like... no clue at all.

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If we've got a calculator, the answer is at our fingertips.

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Isn't it about time we gave up and let the machines win?

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Let's look at the answer choices.

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If the last one was 14, then we'd have:

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216 + 14 = 230

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Then we'd divide by 15 and get 15.33.

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If the last one was 15 we'd have:

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216+15 = 231

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Then we'd divide by 15 and get 15.4. Hm...we'll come back to you, buddy...

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If the last one was 16 we'd have:

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216+16= 232

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Then we'd divide by 15 and get 15.466.

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If the last one was 17 we'd have:

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216+17 = 233

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Then we'd divide by 15 and get 15.533.

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The formal name for this method of solving this kind of problem?

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BRUTE FORCE.

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And yeah - it's a tough way to make a living. This result leaves the age of the last recruit,

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which we assume to be x.

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The total sum of all 15 members' ages is 15.4 times 15 or 77 fifths times 15, which equals

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231... so the equation in terms of the data sums is x + 160 + 56 = 231.

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160 plus 56 is 216. If we subtract 216 from both sides, we're left with x on the left

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side and 15 on the right side... so the age of the last member is 15.

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