Question : The sum of a fraction and 4 times its reciprocal is $\frac{13}{3}$. What is the fraction?
Option 1: $\frac{4}{3}$
Option 2: $\frac{3}{4}$
Option 3: $\frac{5}{4}$
Option 4: $\frac{4}{5}$
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Correct Answer: $\frac{4}{3}$
Solution :
Let the fraction be $a$.
Given:
$a + \frac{4}{a} = \frac{13}{3}$
After solving this, we get
$({a}^{2} + {4}) × 3 = 13a$
$3{a}^{2} -13a + 12 = 0$
$3{a}^{2} -9a -4a+ 12 = 0$
$3{a}(a-3) -4(a - 3) = 0$
$(a-3)(3a - 4) = 0$
$(a-3)=0$ or $(3a - 4) = 0$
⇒ $a = 3, \frac{4}{3}$
So, the fraction = $\frac{4}{3}$
Hence, the correct answer is $\frac{4}{3}$.
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