Question

A sum Rs. 500 is in the form of denominations of Rs. 5 and Rs. 10. If the total number of notes is 90 find the number of notes of each denomination. (Hint: let the number of 5 rupee notes be \['x'\] , then number of 10 rupee notes = \[90 - x\] )

Answer 1

Hint: Here, we have to find the number of notes of each denomination. We will first assume the denominations of Rs 5 and 10 to be any variable. Then we will find the total amount in terms of denomination and add them. We will then equate it with the given sum and form an equation. We will further solve the equation to get the required values.

Complete step-by-step answer:
Let the number of 5 rupee notes be \[x\] and the number of 10 rupee notes be \[y\] .
Since the total number of notes is 90, we get
\[x + y = 90\]
\[ \Rightarrow y = 90 - x\]
So, the number of 90 rupee notes be \[90 - x\] .
Now we will find total amount which is in the form of denominations as Rs. 5 and Rs. 10. Therefore, we get
The total amount in Rs. 500 is in the form of denominations as Rs. 5 \[ = 5 \times \]number of rs. 5 notes
\[ \Rightarrow \]The total amount in Rs. 500 is in the form of denominations as Rs. 5 \[ = 5x\]
Now, the total amount in Rs. 500 is in the form of denominations as Rs. 10 \[ = 10 \times \]number of rs. 10 notes
\[ \Rightarrow \] The total amount in Rs. 500 is in the form of denominations as Rs. 10 \[ = 10(90 - x)\]
Now, we have to find the number of notes of each denomination.
We know that the total sum is Rs. 500. So, we will add the total amount in the form of denominations as Rs. 5 and Rs. 10 and equate it to 500. Therefore, we get
\[5x + 10(90 - x) = 500\]
Multiplying the terms, we get
\[ \Rightarrow 5x + 900 - 10x = 500\]
Subtracting the like terms, we get
\[ \Rightarrow 400 = 5x\]
Dividing both the sides by \[5\] , we get
\[ \Rightarrow \dfrac{{5x}}{5} = \dfrac{{400}}{5}\]
\[ \Rightarrow x = 80\]
The total number of notes with the denomination of Rs. 5 is 80.
Substituting the value of \[x\] in \[y = 90 - x\], we get
\[ \Rightarrow y = 90 - 80\]
Subtracting the terms, we get
\[ \Rightarrow y = 10\]
The total number of notes with the denomination of Rs. 10 is 10.
Therefore, the number of notes of each denomination of Rs. 5 and Rs.10 is 80 and 90 respectively.

Note: We have to know that every currency note or coin has a monetary value. So, the number of notes we have multiplied with the monetary value of the note gives us the total money we got. We can verify the answer by multiplying the number of notes with the denomination, we will get the total sum.
Substituting the value of \[x\] and \[y\] in the equation \[5x + 10(90 - x) = 500\] , we get
\[ \Rightarrow 5\left( {80} \right) + 10(90 - 80) = 500\]
Simplifying the expression, we get
\[ \Rightarrow 400 + 10\left( {10} \right) = 500\]
Multiplying 10 by 10, we get
\[ \Rightarrow 400 + 100 = 500\]
Adding the terms on LHS, we get
\[ \Rightarrow 500 = 500\]
The left hand side of the equation is equal to the right hand side, so our answer is correct.
We can follow the same method of calculation irrespective of the denomination.