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(D) Let the two numbers be $a$ and $b$. Since one number is the reciprocal of the other,we have $b = \frac{1}{a}$. The arithmetic mean of the two numb

Vedclass · Accepted Answer

$\frac{3}{2}, \frac{2}{3}$

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(D) Let the two numbers be $a$ and $b$. Since one number is the reciprocal of the other,we have $b = \frac{1}{a}$. The arithmetic mean of the two numbers is given by $\frac{a + b}{2} = \frac{13}{12}$. Substituting $b = \frac{1}{a}$,we get $\frac{a + \frac{1}{a}}{2} = \frac{13}{12}$. Multiplying by $2$,we get $a + \frac{1}{a} = \frac{13}{6}$. Multiplying the entire equation by $6a$,we get $6a^2 + 6 = 13a$,which simplifies to the quadratic equation $6a^2 - 13a + 6 = 0$. Factoring the quadratic equation: $6a^2 - 9a - 4a + 6 = 0 \Rightarrow 3a(2a - 3) - 2(2a - 3) = 0$. This gives $(3a - 2)(2a - 3) = 0$. Thus,$a = \frac{2}{3}$ or $a = \frac{3}{2}$. If $a = \frac{3}{2}$,then $b = \frac{2}{3}$. If $a = \frac{2}{3}$,then $b = \frac{3}{2}$. Therefore,the numbers are $\frac{3}{2}$ and $\frac{2}{3}$.

$A$ number is the reciprocal of the other. If the arithmetic mean of the two numbers is $\frac{13}{12}$,then the numbers are:

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