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Divide $32$ into two parts such that if the larger is divided by the smaller, the quotient is $2$ and the remainder is $5$.
(A) The smaller number is $6$ and the larger number is $30$.
(B) The smaller number is $9$ and larger number is $23$.
(C) The smaller number is $8$ and the larger number is $14$.
(D) The smaller number is $3$ and the larger number is $19$.

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Last updated date: 13th Jun 2024
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Answer
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Hint: We know that if $x$ and $y$ are two parts of the number $z$ then we can write $x + y = z$. We will use this information and given data. Also we will use the division algorithm to solve this problem. We will get two linear equations. Then, we will use a simple elimination method to solve those equations.

Complete step-by-step answer:
We know that if $x$ and $y$ are two parts of the number $z$ then we can write $x + y = z$. We will use this information and given data. Also we will use the division algorithm to solve this problem. We will get two linear equations. Then, we will use a simple elimination method to solve those equations.

So, the correct answer is “Option C”.
In the given problem, we need to find two parts of the number $32$ such that the larger part is divided by the smaller part then the quotient will be $2$ and remainder will be $5$. For this, let us assume that the larger part of the number $32$ is $x$ and the smaller part of the number $32$ is $y$. As we assume that $x$ and $y$ are two parts of the number $32$, we can write $x + y = 32 \cdots \cdots \left( 1 \right)$.
It is given that when the larger part $x$ is divided by the smaller part $y$ then the quotient is $2$ and remainder is $5$. Note that when $x$ is divided by $y$ then $x$ is called dividend and $y$ is called divisor. Division algorithm states that Dividend $ = $ (Divisor $ \times $ Quotient) $ + $ Remainder. So, by using division algorithm and given information we can write
$x = \left( {y \times 2} \right) + 5$
$ \Rightarrow x = 2y + 5$
$ \Rightarrow x - 2y = 5 \cdots \cdots \left( 2 \right)$
Let us solve the equations $\left( 1 \right)$ and $\left( 2 \right)$ to find values of $x$ and $y$. Let us subtract equation $\left( 2 \right)$ from equation $\left( 1 \right)$. So, we can write
$\left( {x + y} \right) - \left( {x - 2y} \right) = 32 - 5$
$ \Rightarrow x + y - x + 2y = 27$
$ \Rightarrow 3y = 27$
$ \Rightarrow y = \dfrac{{27}}{3}$
$ \Rightarrow y = 9$
Now we will substitute the value $y = 9$ in equation $\left( 1 \right)$ and simplify the obtained equation. So, we can write
$x + 9 = 32$
$ \Rightarrow x = 32 - 9$
$ \Rightarrow x = 23$
Hence the required parts are $23$ and $9$. Hence, we can say that the larger part of the number $32$ is $23$ and the smaller part of the number $32$ is $9$ such that when the larger part is divided by the smaller part then the quotient is $2$ and remainder is $5$. Hence, option B is correct.

So, the correct answer is “Option C”.

Note: In the given problem, we need to find values of two unknown variables. So, we need only two equations. We can solve two linear equations by using a simple elimination method. To find values of $n$ unknowns, we need $n$ equations.