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**Hint:**Let us assume the \[500\] and \[100\] denomination notes as \[x\] and \[y\] after assuming the total number of notes and their monetary value equivalence we form two equations that is product of total money amounting from \[500\] denomination added to that of the denomination amounting for \[100\] rupees. After that we will equate the equations and then subtract to find the value of \[x\] and then the

value of \[y\].

**Complete step by step solution:**

As given in the question, the value of \[500\] notes and \[100\] notes total to be \[150\] number of notes. The amount withdrawn from the ATM is valued at \[Rs.25000\].

Hence, let us assume that the total number of \[500\] notes are \[x\]. And assume that the total number of \[100\] notes are \[y\]. Therefore, the sum of \[500\] denomination and \[100\] denomination notes are to written as:

\[x\text{ }+\text{ }y\text{ }=\text{ }150\]

The total amount of \[500\] denomination amounts to Rs \[500x\] and the total amount of \[100\] denomination amounts to Rs. \[100y\].

Hence adding the total amount will form an Equation of \[500x+100y=25000\]

Subtracting the Equation based on the two situations, we get the value of \[y\] as:

\[\Rightarrow \begin{matrix}

x\text{ }+\text{ }y\text{ }=\text{ }150\text{ } \\

500x+100y=25000 \\

\end{matrix}\]

Multiplying the base equation with the above equation we get the two equation as:

\[\Rightarrow \begin{matrix}

500x+\text{500}y=75000\text{ } \\

500x+100y=25000\text{ } \\

\end{matrix}\]

\[\Rightarrow \text{400}y=50000\]

\[\Rightarrow y=\dfrac{50000}{400}\]

\[\Rightarrow y=125\]

And after getting the value of \[y\] we put the value in the equation to get the value of \[x\] by placing the value of \[y\] in the equation \[x\text{ }+\text{ }y\text{ }=\text{ }150\], we get the value of \[x\] as:

\[\Rightarrow x\text{ }+\text{ }125\text{ }=\text{ }150\]

\[\Rightarrow x=25\]

**Therefore, the total number of \[500\] denominations is given as and the number of \100\ denomination is given as \[25\] and \[125\] respectively.**

**Note:**Another method to find the number of notes is that we assume that the denomination of \[500\] notes as \[x\] and the number of \[100\] denominations as \[150-x\]. Hence, the equation for the total sum and the number of denomination is:

\[\Rightarrow 500x+100\left( 150-x \right)=Rs.25000\]

\[\Rightarrow 500x+15000-100x=Rs.25000\]

\[\Rightarrow 400x=Rs.\left( 25000-15000 \right)\]

\[\Rightarrow x=\dfrac{Rs.\left( 25000-15000 \right)}{400}\]

\[\Rightarrow x=\dfrac{Rs.10000}{400}\]

\[\Rightarrow x=25\]

Therefore, the number of \[500\] notes as \[25\] and \[100\] notes as \[125\].

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