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# The average age of the mother and her six children is $12years$ which is reduced by $5\text{ }years$ if the age of mother is excluded. How old is the mother?A) $40years$B) $42years$C) $48years$D) $50years$

Hint: Assume any variable for the unknowns in the statement given. So, We will First assume the mother's age. Then use here the formula to find average$\left[ Average=\dfrac{Sum\text{ }of\text{ }quantity}{number\text{ }of\text{ }quantity} \right]$ to calculate total age of seven persons and again average of $6$ persons.

Complete step-by-step solution:
Given: Average age of a mother and her six children is $12years$, if mother’s age is excluded then the average of ages will reduce by $5\text{ }years$.
Let the age of the mother be $x\text{ }years$.
The average age of a mother and her six children is $12$.
Hence, sum of age of the mother and her six children is $12\times 7=84$
$\left[ Average=\dfrac{Sum\text{ }of\text{ }quantity}{number\text{ }of\text{ }quantity} \right]$ here number of quantity 7 and average is 12.
If we reduce the age of mother then, Sum of age of six children = $84-x$
Now average of age of six children = $\dfrac{84-x}{6}$
Put the values, and simplify -
According to question: \begin{align} &\Rightarrow \,12-5=\dfrac{84-x}{6} \\ & \Rightarrow 7=\dfrac{84-x}{6}\text{ }\!\![\!\!\text{ 6 from denominator goes to the numerator when change from RHS to LHS }\!\!]\!\!\text{ } \\ & \Rightarrow 7\times 6=84-x \\ \end{align}
\begin{align} & \Rightarrow 42=84-x \\ & \therefore x=84-42 \\ & \therefore x=42 \\ \end{align}
Hence, The age of the mother is $42\ \text{years}$
Therefore, from the given multiple choices,

Hence option B is correct.

Note: In these types of problems first assume the unknown quantity, here it is mother’s age. Then use the assumed variable to form an equation and compare it with the assigned value in question and solve. It will give us the required value. Find the mathematical expression from the word statement and find correlation between the unknowns.