# A lady has only $25$ paisa and $50$ paise coins in her purse. She has a total of $120$ coins and the total amount is Rs$50$. The number of coins of each type in her purse, respectively, is:

A. $90,30$

B. $60,60$

C. $40,80$

D. $70,50$

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**Hint:**The problem can be solved by converting the given statements into mathematical equations. Further, we can solve those equations to get the desired solutions. The unknown quantity like the number of coins should be replaced with variables and this will allow us to find their exact values.

**Complete step-by-step answer:**

It is given that a lady has only $25$ paisa and $50$ paise coins in her purse but their quantity is unknown.

We can assume their quantities to be some variables so that it will allow us to approach the problem in a mathematical way.

So, let the number of $25$ paise coins $ = x$

Number of $50$paise coins $ = y$

Now the equation states that she has the total of $120$ coins. This will be equal to the sum of $25$ paisa and $50$ paise coins.

Hence, $x + y = 120$ $ - - - - - - \left( 1 \right)$

Further, it is said that the total amount is Rs $50$. This means that the total worth of $x$ number of $25$ paise coins and $y$ number of $50$ paise coins is Rs$50$.

So it can be concluded that the total amount will be equal to the sum of the products of the number of coins and the worth of the single coin.

We know that $1$paise $ = 0.01$rupee

So, $25$ paise $ = 0.25$rupee

And $50$ paise $ = 0.50$ rupee

Hence the above statements can be expressed in the mathematical form as,

$\left( {x \times 0.25} \right) + \left( {y \times 0.50} \right) = 50$

$0.25x + 0.50y = 50$

$\dfrac{{25}}{{100}}x + \dfrac{{50}}{{100}}y = 50$

$\dfrac{{x + 2y}}{4} = 50$

$x + 2y = 200$ $ - - - - - - \left( 2 \right)$

We can solve these simultaneous linear equations $\left( 1 \right)$ and $\left( 2 \right)$ to calculate the values of $x$ and $y$.

$x + y = 120$ $x + 2y = 200$

We will subtract both the equations (1) and (2) we get,

$\left( {x + 2y} \right) - \left( {x + y} \right) = 200 - 120$

$x + 2y - x - y = 80$

$y = 80$

Similarly, $x + y = 120$

$x + 80 = 120$

$x = 120 - 80$

$x = 40$

Hence, The number of $25$ paise coins $ = 40$

The number of $50$ paise coins $ = 80$

**So, the correct answer is “Option C”.**

**Note:**The student should always remember that we should aim to convert the given statements and relations into mathematical form. This will help the student to approach towards the required solution.