33 Pearls ProblemDate: 11/14/96 at 20:19:53 From: James Fletcher Subject: 33 Pearls Problem Dear Dr. Math, I'm having a hard time with this problem, harder than any word problem I've ever done before, and I think it is because I do not understand the concept behind it. The problem goes like this: On a string of 33 pearls, the middle pearl is the largest and most expensive of all. Starting from one end, each pearl is worth $100 more than the one before, up to the middle. From the other end, each pearl is worth $150 more than the one before, up to the middle. The string of pearls is worth $65,000. What is the value of the middle pearl? Thank you very much! James Fletcher Date: 12/11/96 at 21:59:16 From: Doctor Daniel Subject: Re: 33 Pearls Problem Hi James, You're right that this isn't an easy problem, so we'll take it slow. We'll start with some basics. For starters, there are exactly 16 pearls to either side of the pearl in the center. We'll call the end where each pearl is worth $100 more than the previous one the left side, and the end where each pearl is worth $150 more than the previous one the right side. (That's just so it's easier for me to write about them.) So I've cut the necklace into 3 parts: the left side, the middle pearl, and the right side. Each of the two sides has 16 pearls. Say that the left side has value L, the middle pearl has value M, and the right side has value R. Then we know that L + M + R = 65000. It'd be great if we could solve for L and for R in terms of M. Then we could solve for M, and we'd be done, right? Well, what's L? We know that the pearl in the left side closest to the middle has value M-100. And the next one has value M-200, on to the last pearl which has value M-1600 (the easy way to convince yourself of this is to draw a picture of the whole necklace). So we have: L = (M-100) + (M-200) + (M-300) + ... + (M-1600). I'm going to collect terms here to get the M's separate: L = 16M - (100+200+300+...+1600) L = 16M - 100(1+2+3+...+16) Now, what's 1+2+...+16? You could show, just by brute force, that this is 136. But that's unsatisfying, so let's take a quick detour. 1+2+3+...+16 is the 16th triangle number. The general form for the nth triangle number is 1/2 * (n) * (n+1). Why? Group the sum instead as 1+16 + 2+15 + 3+14 + 4+13 + . . . Each of these small blocks will have sum equal to 17 (In the general case, it will have sum n+1) and there will be exactly 8 of them (in the general case, n/2). So the sum will be (n+1) * n/2, which is another way of stating that formula I just gave you. So back to our word problem: L = 16M - 100(1+2+...+16) L = 16M - 100(17)(16/2) L = 16M - 100*136 L = 16M - 13600. Now let's look at R: The pearls of R are worth M-150, M-300, and so on, all the way down to M - 16*150, which is M-2400. So the total value of the right side is: R = (M-150) + (M-300) + ... + (M-2400) R = 16M - (150+300+...+2400) R = 16M - 150(1+2+...+16) R = 16M - 150*136 R = 16M - 20400 Okay! Now we just need to plug these values for L and R into our overall equation for the value of the necklace: L+M+R = 65000 16M - 13600 + M + 16M - 20400 = 65000 33M - 34000 = 65000 33M = 99000 M = 3000 So the middle pearl is worth $3000. The pearls on the left side are worth $2900, $2800, down to $1400, while the ones on the right go in value from $2850, $2700, $2550, all the way down to $(3000-2400) = $600. If you add these values together, you do indeed get the right answer (which is always nice!). I hope this helps you with understanding this problem. Good luck! -Doctor Daniel, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 01/14/03 at 15:51:05 From: Doctor Ian Subject: Re: 33 Pearls Problem Hi James, Let's forget about the total number of pearls for a moment, and assume that we have one pearl, whose value is x. Then let's add pairs of pearls, following the formula given by the problem: x / \ (x-100) (x-150) <-- 2x - 250 | | (x-200) (x-300) <-- 2x - 500 | | (x-300) (x-450) <-- 2x - 750 | | (x-400) (x-600) <-- 2x - 1000 . . . . (x-100n) (x-150n) <-- 2x - 250n Add them all up, and for (2n+1) pearls you get total value = x + n(2x) - 250[1 + 2 + ... + n] = (2n+1)x - 250[n(n+1)/2] In this case, we have 33 pearls, so n=16; and the total value is $65,000; so 65,000 = (2*16 + 1)x - 250[16(17)/2] = 33x - 34,000 which gives a value of $3000 for the middle pearl. -Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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