奧數 [要有步驟]

甲、乙、丙、丁付出 x, y, z, w
x + y + z + w = 5400
x = (y + z + w)/4
y = (x + z + w)/2
z = (x + y + w )/3
------------------------------------
x + y + z + w = 5400 ------------- (1)
4x = y + z + w ------------ (2)
2y = x + z + w ------------ (3)
3z = x + y + w ------------ (4)
-----------------------------
代 (2) 入 (1), x + 4x = 5400
x = 1080
代 (3) 入 (1), y + 2y = 5400
y = 1800
代 (4) 入 (1), z + 3z = 5400
z = 1350
1080 + 1800 + 1350 + w = 5400
w = 1170
丁付 1170


2007-10-26 16:32:07 補充：
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甲、乙、丙、丁共同購買一部價值5400元的電腦,甲付出的款項是其餘三人付出的總和的四分之一,乙付出的款項是其餘三人付出的總和的50%,丙付出的款項是其餘三人付出的總和的三分之一,問丁付出多少元？

Let A be the amount of money contributed by 甲
Let B be the amount of money contributed by 乙
Let C be the amount of money contributed by 丙
Let D be the amount of money contributed by 丁

Then, A + B + C + D = 5400 ---------------------(1)
A = (1/4)(B+C+D) --------------------(2)
B = (1/2)(A+C+D) ---------------------(3)
C = (1/3)(A+B+D) ---------------------(4)

From (2), B+C+D = 4A. By substituting it into (1), we have A + 4A = 5400 or A = 1080
From (3), A+C+D = 2B. By substituting it into (1), we have B + 2B = 5400 or B = 1800
From (4), A+B+D = 3C. By substituting it into (1), we have C + 3C = 5400 or B = 1350
Then, D = 5400 - A - B - C = 5400 - 1080 - 1800 - 1350 = 1170

設 $a 為甲付出款項
$b 為乙付出款項
$c 為丙付出款項
$d 為丁付出款項

題目說：a + b + c + d = 5400

甲付出的款項是其餘三人付出的總和的四分之一
所以...
a = 1/4(b+c+d)
4a = b + c + d
4a = 5400 - a (<- 由於 a+b+c+d = 5400, b+c+d = 5400-a)
5a = 5400
a = 1080

b = 1/2(a+c+d)
2b = a + c + d
2b = 5400 - b (<- 由於 a+b+c+d = 5400, a+c+d = 5400-b)
3b = 5400
b = 1800

c = 1/3(a+b+d)
3c = a + b + d
3c = 5400 - c (<- 由於 a+b+c+d = 5400, a+b+d = 5400-c)
4c = 5400
c = 1350

d = 5400 - 1080 - 1800 - 1350 (<- 由於 a+b+c+d = 5400, d = 5400-a-b-c)
d = 1170