# Find the answer when $\dfrac{6}{13}$ is multiplied by the reciprocal of $\dfrac{12}{5}$.

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Hint: Interchange the numerator and denominator of $\dfrac{12}{5}$ to get the reciprocated number, then multiply this number with $\dfrac{6}{13}$ to get the answer. Multiply 6 with 5 and 13 with 12.

In mathematics, a multiplicative inverse or reciprocal of a number $x$, denoted by $\dfrac{1}{x}\text{ or }{{x}^{-1}}$, is a number which when multiplied by $x$ yields the multiplicative identity, 1. The multiplicative inverse of a fraction $\dfrac{a}{b}$ is $\dfrac{b}{a}$. To find the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is $\dfrac{1}{5}$ or 0.2 and the reciprocal of 0.5 is $\dfrac{1}{0.5}$ or 2. The reciprocal function, the function $f(x)$ that maps $x$ to $\dfrac{1}{x}$, is one of the simplest examples of a function which is its own inverse. The reciprocal term is common to describe two numbers whose product is 1. In the real numbers, zero does not have a reciprocal because no real number multiplied by zero produces 1. With the exception of zero, reciprocal of every real number is real. Reciprocal of every rational number are rational and reciprocal of every complex number are complex.

Now, come to the question. We have to find the reciprocal of $\dfrac{12}{5}$ which will be \[\dfrac{5}{12}\]. Now, we multiply it with $\dfrac{6}{13}$ to get the answer. Multiply 6 with 5 in the numerator and 13 with 12 in the denominator.

$\therefore \dfrac{6}{13}\times \dfrac{5}{12}=\dfrac{30}{156}$. Now, 6 is the common factor in both numerator and denominator, so, cancel it out.

$\therefore \dfrac{30}{156}=\dfrac{6\times 5}{6\times 26}=\dfrac{5}{26}$. Hence, $\dfrac{5}{26}$ is the required answer.

Note: We have multiplied numerator with numerator and denominator with denominator because this is the rule of multiplication. Never multiply numerator with denominator. Also, in the last, if possible, we must cancel the common factor. Here, 6 was the common factor so it got cancelled.

__Complete-step-by-step answer:__In mathematics, a multiplicative inverse or reciprocal of a number $x$, denoted by $\dfrac{1}{x}\text{ or }{{x}^{-1}}$, is a number which when multiplied by $x$ yields the multiplicative identity, 1. The multiplicative inverse of a fraction $\dfrac{a}{b}$ is $\dfrac{b}{a}$. To find the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is $\dfrac{1}{5}$ or 0.2 and the reciprocal of 0.5 is $\dfrac{1}{0.5}$ or 2. The reciprocal function, the function $f(x)$ that maps $x$ to $\dfrac{1}{x}$, is one of the simplest examples of a function which is its own inverse. The reciprocal term is common to describe two numbers whose product is 1. In the real numbers, zero does not have a reciprocal because no real number multiplied by zero produces 1. With the exception of zero, reciprocal of every real number is real. Reciprocal of every rational number are rational and reciprocal of every complex number are complex.

Now, come to the question. We have to find the reciprocal of $\dfrac{12}{5}$ which will be \[\dfrac{5}{12}\]. Now, we multiply it with $\dfrac{6}{13}$ to get the answer. Multiply 6 with 5 in the numerator and 13 with 12 in the denominator.

$\therefore \dfrac{6}{13}\times \dfrac{5}{12}=\dfrac{30}{156}$. Now, 6 is the common factor in both numerator and denominator, so, cancel it out.

$\therefore \dfrac{30}{156}=\dfrac{6\times 5}{6\times 26}=\dfrac{5}{26}$. Hence, $\dfrac{5}{26}$ is the required answer.

Note: We have multiplied numerator with numerator and denominator with denominator because this is the rule of multiplication. Never multiply numerator with denominator. Also, in the last, if possible, we must cancel the common factor. Here, 6 was the common factor so it got cancelled.

Last updated date: 24th Sep 2023

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