Question : Divide 50 into two parts so that the sum of their reciprocal is $\frac{1}{12}$.
Option 1: 35 and 15
Option 2: 20 and 30
Option 3: 24 and 36
Option 4: 28 and 22
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Correct Answer: 20 and 30
Solution :
Given:
The number is 50.
Let the first part of 50 be $\text{x}$.
The second part will be $50 - \text{x}$.
The sum of their reciprocal is $\frac{1}{12}$
$\Rightarrow \frac 1 {\text {x} }+ \frac 1 {\text {50-x} }=\frac 1 {\text {12} }$
$\Rightarrow \frac {50+\text {x}-\text {x}} {\text {50x} -{\text{x}^{2}} }=\frac {1}{12}$
$\Rightarrow \text {50x} -{\text{x}^{2}} = 600$
$\Rightarrow \text {50x} -{\text{x}^{2}} - 600=0$
$\Rightarrow {\text{x}^{2}}-\text {50x} + 600=0$
After factorising, we get
⇒ (x-30)(x-20) = 0
⇒ x = 20 and 30
Hence, the two parts are 20 and 30.
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