小學份數數學問題

3/7 = 1/3 + 1/14 + 1/42
或
3/7 = 1/4 + 1/6 + 1/84


解釋:
3/7 = 6/14 = ... = 18/42 =... = 36/84 (擴分) = 1/?+1/?+1/?

>> 嘗試 18/42 = 1/?+1/?+1/? (三個不同分母)
18/42 = (??+ ?? + ??) /42 (三個不同是分子)

如果分子 ?? 是 42 的因數，那麼 ??/42 可約至 1/?
42的因數 ： 1, 2, 3, 6, 7, 14, 21, 42
(但分子 ?? 不會是6, 因為分母不是7而 6/42 = 1/7)
(由於分子的和是18, ?? < 18, 所以 ??不會是21 或 42)

18 = 14 + 3 + 1

所以 18/42
= (14 + 3 + 1) /42
= 14/42 + 3/42 + 1/42
= 1/3 + 1/14 + 1/42

>> 嘗試 36/84 = 1/?+1/?+1/? (三個不同分母)
36/84 = (??+ ?? + ??) /42 (三個不同是分子)

84的因數：1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42
(分子不可能是12, 21, 28, 42)
36 = 21 + 14 + 1
所以 36/84
= (21 + 14 + 1) /84
= 21/84 + 14/84 + 1/84
= 1/4 + 1/6 + 1/84

3/7=1/6+1/3+1/5

It is a question about Egyptian fraction

an unknown integer x that makes |3/7 - 1/x| minimum is x=2
consider 3/7 = 1/2 - 1/14

note that 1/n = 1/(n+1) + 1/n(n+1)
i.e. 3/7 = 1/2 - (1/(14+1) + 1/14(14+1))
3/7 = 1/2 - 1/15 - 1/210

i.e. the unknowns are 2, -15 and -210


2007-04-03 14:43:24 補充：
others' "answers":1/3+1/5+1/6 should be = 7/10 but not 3/7 wor...

如果係分母無限制ge話,,,,我會放小數落去


就會係:
1/10.5+ 1/21+ 1/3.5 = 3/7
(=2/21+ 1/21+ 6/3.5)
(=9/21)
=3/7