Question 288037: [Problem]
On a string of 15 pearls, the centre pearl is the largest and most expensive. Starting from one end and including the centre pearl, each pearl is worth R50 more than the previous one. Starting from the other end and including the centre pearl, each pearl is R25 more than the previous one. The total value of the pearls is R 4 650. What is the value of the centre pearl?
[Thoughts]
Okay, I can see that, this is an arithmetic series problem.
I can see that the middle term is the centre pearl, the 8th pearl in the middle of the string of 15 pearls (n=8) and it is the most expensive, meaning that counting from one end (say, from left to right) with a common difference of d=50 (R50) , and including the centre pearl will only account for the first 8terms. Because, the middle term (8th term) is greater in price, hence the 9th term cannot be R50 more that the previous one (which is the 8th term)
The same occurs with the R25 difference, counting from one end (say, from right to left) with a common difference of d=25 (R25) ,and including the centre pearl will only account for the first 8terms. Because, the middle term (8th term) is greater in price, hence the 9th term (pearl) cannot be R25 more that the previous one (which is the 8th term).
Now , this leaves me with 2 sets of arithmetic calculations. First one : d=50, n=8 An=? (An is the centre pearl, counting from left to right) .
Using the equation: An = a1 + (n-1) d (a1 =denotes first term on the right end)
= a1 + 7 (50)
= a1 + 350
The second one is: d=25, n=8 An=? (Again , An is the centre pearl, from right to left) .
Using the equation: An = a + (n-1) d (a =denotes first term on the right end)
= a + 7 (25)
= a + 175
From the above 2 equations, I get that: An=a1 + 350= a + 175
a1 + 350= a + 175
a = a1 + 175. . . . . . . . . .(1)
The total value of the 15 pearls = 4650 , hence meaning that Sn = 4650 and therefore
Sn = n/2 [a1+An] + n/2 [a + An]
4650 = 8/2 [a1 + (a1 + 350)] + 8/2 [ a + (a + 175) ]
4650 = 8a1 + 1400 + 8a + 700
4650 = 8a1 + 8a + 2100
2550 = 8a1 + 8a
then, substituting eqtn (1), I get . . .
2550 = 8a1 + 8 (a1 + 175)
1150 = 16a1
a1 = 72
then, substituting in the eqtn : An = a1 + 350
= (72) + 350
= 422
Therefore, I get centre pearl to be R 422 .
[Difficulty]
My problem is , summing up ALL the terms does not give me R 4650 . Where did I go wrong?
Answer by toidayma(44)   (Show Source):
You can put this solution on YOUR website! You should count from the center for this problem.
Let the value of the center pearl be x
To the center pearl, we have
the 1st next left one and the 1st next right one has a total value of: (x-50) + (x-25) = 2x-75
the 2nd next left one and the 2nd next right one has a total value of: (x-2*50) + (x-2*25) = 2x-2*75
the 3rd next left one and the 3rd next right one has a total value of: (x-3*50) + (x-3*25) = 2x-3*75
.......
the 7th next left one and the 7th next right one has a total value of: (x-7*50) + (x-7*25) = 2x-7*75
So, total value of the string (not included the center one) is: 14x - 75*(1+2+...+7) = 14x - 28*75
Therefore, the total value of the string is: 15x - 28*75 and this equal to R4650
Thus, 15x = 4650 + 28*75 = 6750 or x = (R)450.
Your problem is that you count the value of the center twice.
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