如何不用通分
算出(1/3+1/5+1/7+1/9)/(1/9+1/11+1/13+1/15)?

因為不小心砍到了回答只能在這裡敘述.

方法1: 國中以上可用
用公式 1/(X-Y) + 1/(X+Y) = 2xX / (XxX-YxY)

(1/3+1/5+1/7+1/9)
= (1/(6-3)+1/(6-1)+1/(6+1)+1/(6+3))
= 2x6/(6x6-3x3)+2x6/(6x6-1x1)
= 2x6x(1/27+1/35)
= 2x6x(1/(31-4)+1/(31+4))
= 2x6x2x31/(31x31-4x4)
= 2x6x2x31/945

(1/9+1/11+1/13+1/15)
= (1/(12-3)+1/(12-1)+1/(12+1)+1/(12+3))
= 2x12/(12x12-3x3)+2x12/(12x12-1x1)
= 2x12x(1/135+1/143)
= 2x12x(1/(139-4)+1/(139+4))
= 2x12x2x139x(139x139-4x4)
= 2x12x2x139/19305

(1/3+1/5+1/7+1/9)/(1/9+1/11+1/13+1/15)
= (2x6x2x31/945)/(2x12x2x139/19305)
= (31x19305)/(2x139x945) ---> 求分子分母質數
= (31x3x3x3x5x11x13) / (2x139x3x3x3x5x7) ---> 分子分母皆為質數
= (31x11x13)/(2x139x7)
= 4433/1946

方法2: 小學生可用
用乘法與加法,

(1/3+1/5+1/7+1/9)/(1/9+1/11+1/13+1/15)
= ((1/3+1/5+1/7+1/9))x3x5x7x9x11x13x15) / ((1/9+1/11+1/13+1/15)x3x5x7x9x11x13x15)
= ((5x7x9+3x7x9+3x5x9+3x5x7)x11x13x15) / ((11x13x15+9x13x15+9x11x15+9x11x13)x3x5x7)
= ((5x7x9+3x7x9+3x5x9+3x5x7)x11x13) / ((11x13x15+9x13x15+9x11x15+9x11x13)x7)
= ((5x7x9+3x7x9+3x5x9+3x5x7)x11x13) / ((11x13x5+9x13x5+9x11x5+3x11x13)x3x7)
= ((315+189+135+105)x11x13) / (715+585+495+429)x3x7)
= (744x11x13) / (2224x3x7) ---> 求分子分母質數
= (2x2x2x3x31x11x13) / (2x2x2x2x139x3x7) ---> 分子分母皆為質數
= (11x13x31)/(2x7x139)
= 4433/1946