已知a,b,c为实数,且ab/a+b=1/3,bc/b+c=1/4,ca/c+a=1/5.求abc/ab+bc+ca的值
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![已知a,b,c为实数,且ab/a+b=1/3,bc/b+c=1/4,ca/c+a=1/5.求abc/ab+bc+ca的值](/uploads/image/z/970517-29-7.jpg?t=%E5%B7%B2%E7%9F%A5a%2Cb%2Cc%E4%B8%BA%E5%AE%9E%E6%95%B0%2C%E4%B8%94ab%2Fa%2Bb%3D1%2F3%2Cbc%2Fb%2Bc%3D1%2F4%2Cca%2Fc%2Ba%3D1%2F5.%E6%B1%82abc%2Fab%2Bbc%2Bca%E7%9A%84%E5%80%BC)
已知a,b,c为实数,且ab/a+b=1/3,bc/b+c=1/4,ca/c+a=1/5.求abc/ab+bc+ca的值
已知a,b,c为实数,且ab/a+b=1/3,bc/b+c=1/4,ca/c+a=1/5.求abc/ab+bc+ca的值
已知a,b,c为实数,且ab/a+b=1/3,bc/b+c=1/4,ca/c+a=1/5.求abc/ab+bc+ca的值
将已知条件倒置,即有
1/a+1/b=3
1/b+1/c=4
1/c+1/a=5
从而,得到
1/a+1/b+1/c=6
故abc/ab+bc+ca=1/(1/a+1/b+1/c)=1/6
1/a+1/b=(a+b)/ab=3;1/b+1/c=(b+c)/bc=4;1/c+1/a=(c+a)/ca=5
2(1/a+1/b+1/c)=3+4+5=12
1/a+1/b+1/c=6
abc/(ab+bc+ca)=1/(1/a+1/b+1/c)=1/6
ab/a+b=1/3,bc/b+c=1/4,ca/c+a=1/5
则a+b/ab=3 b+c/bc=4,c+a/ca=5 a+b/ab+b+c/bc+c+a/ca=12
则2(ab+bc+ca)/abc=12 ab+bc+ca)/abc=6
abc/ab+bc+ca=1/6
1/6