x y + y z + z x x y z \frac{xy+yz+zx}{xyz}

Algebra Level 2

Positive numbers a a , b b , c c , x x , y y and z z satisfy a b c = 125 , a x = b y = c x = 5 11 . abc=125, a^x=b^y=c^x=5 ^{11}. What is the value of x y + y z + z x x y z ? \frac{xy+yz+zx}{xyz}?

2 11 \frac{2}{11} 1 11 \frac{1}{11} 4 11 \frac{4}{11} 3 11 \frac{3}{11}

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2 solutions

Anubhav Sharma
Apr 4, 2014

Here, It is said that abc = 125 If we factorise 125 then we will find that the possible factors are 5, 5 and again 5. It's because
\­( 5^{3}\­) = 125

So, a = b = c = 5

Now, it is given that \­( a^{x}\­) = \­( b^{y}\­) = \­( c^{z}\­) = \­( 5^{11}\­)

We know that a = b = c = 5. So, the only possible value for x, y, z is 11.

Hence, x = y = z = 11

Finally , our work is to simplify the expression and it would be easy as all the values are known.

\­( \frac{xy + yz + zx}{ xyz }\­)

= \­( \frac{11 * 11 + 11* 11 + 11* 11}{ 11 * 11* 11 }\­)

= \­( \frac{121 + 121 + 121}{ 1331 }\­)

= \­( \frac{ 363 }{ 1331 }\­)

= \­( \frac{ 3 }{ 11 }\­)

Hence, Final answer is = \­( \frac{ 3 }{ 11 }\­)

Nilesh Shelar
Mar 17, 2014

given x=11,y=11,z=11

:: (11 11+11 11+11 11) / (11 11*11)

::3(11 11) / (11 11*11)

::3/11

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