Question

Durga’s mother gave some $10$ rupee notes and some \[5\] rupee notes to her, which amounts to Rs. $190$. Durga said “If the number of $10$ rupee notes and \[5\] rupee notes would have been interchanged, I would have Rs. $185$ in my hand.” So, how many notes of rupee $10$ and rupee \[5\] were given to Durga?
(A) Number of rupee $10$ note $ = 10$, Number of rupee $5$ note $ = 10$
(B) Number of rupee $10$ note $ = 12$, Number of rupee $5$ note $ = 13$
(C) Number of rupee $10$ note $ = 13$, Number of rupee $5$ note $ = 12$
(D) Number of rupee $10$ note $ = 16$, Number of rupee $5$ note $ = 6$

Answer 1

Hint: This question can be solved by assuming the number of notes as variables such as X and Y then by converting these statements in form of equations having two variables X and Y. There are two equations possible, then we solve these equations and find the number of the notes as X and y.

Complete step-by-step answer:
Let us assume the number of rupee $10$ notes be X and the number of rupee $5$ note be Y.
Then the question can be written in the form of equations having two variables. So, we get,
The first equation as-
$10X + 5Y = 190$
And now according to the question if the number of $10$ rupee notes and \[5\] rupee notes are interchanged then; we get the second equation as-
$5X + 10Y = 185$
Now Multiplying the first equation by 2 we get,
The third equation as-
$20X + 10Y = 380$
Now subtracting the second equation equation with the third equation we get,
$
\Rightarrow \left( {20 - 5} \right)X + \left( {10 - 10} \right)Y = \left( {380 - 185} \right)\\
\Rightarrow 15X = 195\\
\Rightarrow X = \dfrac{{195}}{{15}}\\
\Rightarrow X = 13
$
Putting $X = 13$ in the first equation we get,
$
\Rightarrow \left( {10 \times 13} \right) + 5Y = 190\\
\Rightarrow 5Y = 190 - 130\\
\Rightarrow 5Y = 60\\
\Rightarrow Y = 12
$
Therefore, the Number of rupee $10$ note $ = 13$,
And the Number of rupee $5$ note $ = 12$

So, the correct answer is “Option C”.

Note: Alternate method to solve this question is to find the value of variable X in terms of Y from the first equation and then substitute this value in the second equation and solve.
So, from the first equation we get the value of X in terms of Y as-
\[
\Rightarrow 10X = \left( {190 - 5Y} \right)\\
\Rightarrow X = \dfrac{{\left( {190 - 5Y} \right)}}{{10}}\\
\Rightarrow X = \left( {19 - \dfrac{Y}{2}} \right)
\]
Now substituting this value of X in the second equation we get,
\[
\Rightarrow 5\left( {19 - \dfrac{Y}{2}} \right) + 10Y = 185\\
\Rightarrow 95 - \dfrac{{5Y}}{2} + 10Y = 185\\
\Rightarrow \dfrac{{\left( {20 - 5} \right)Y}}{2} = 185 - 95\\
15Y = 180
\]
Solving this we get,
$Y = 12$
And
\[
 \Rightarrow X = \left( {19 - \dfrac{{12}}{2}} \right)\\
 = \left( {19 - 6} \right)\\
 = 13
\]