Question

A amount of Rs. 3270 is made up with the notes of denominations Rs. 100, Rs. 50, Rs. 20, and Rs. 10. The number of Rs. 10 notes is 5 times the number of Rs. 20 notes. The number of Rs. 20 notes are 4 times the number of Rs. 50 notes. The number of Rs. 50 notes is 3 times the number of Rs. 100 notes. How many total notes of denomination Rs. 20 are there?

Answer 1

Hint: Here, we need to find the number of notes of denomination Rs. 20. We will assume the number of notes of denomination Rs. 100 to be \[x\]. Using the given information we will find the number of notes of other denominations in terms of \[x\]. Then, we will form an equation and find the value of \[x\]. We will use the value of \[x\] to the number of notes of denomination Rs. 20.

Complete step-by-step answer:
Let the number of notes of denomination Rs. 100 be \[x\].
First, we will find the number of notes of other denominations in terms of \[x\].
The number of Rs. 50 notes is 3 times the number of Rs. 100 notes.
Therefore, we get
Number of notes of denomination Rs. 50 \[ = 3 \times x = 3x\]notes
The number of Rs. 20 notes is 4 times the number of Rs. 50 notes.
Therefore, we get
Number of notes of denomination Rs. 20 \[ = 4 \times 3x = 12x\]notes
The number of Rs. 10 notes is 5 times the number of Rs. 20 notes.
Therefore, we get
Number of notes of denomination Rs. 10 \[ = 5 \times 12x = 60x\]notes
Now, we will find the amount made up by the notes of different denominations.
The amount made up by the notes of denomination Rs. 100 is equal to the product of the number of notes of Rs. 100, and the value of 1 note.
Therefore, we get
Amount made up by \[x\] notes of Rs. 100 \[ = {\rm{Rs}}{\rm{. }}100 \times x = {\rm{Rs}}{\rm{. }}100x\] rupees
The amount made up by the notes of denomination Rs. 50 is equal to the product of the number of notes of Rs. 50, and the value of 1 note.
Therefore, we get
Amount made up by \[3x\] notes of Rs. 50 \[ = {\rm{Rs}}{\rm{. }}50 \times 3x = {\rm{Rs}}{\rm{. }}150x\] rupees
The amount made up by the notes of denomination Rs. 20 is equal to the product of the number of notes of Rs. 20, and the value of 1 note.
Therefore, we get
Amount made up by \[12x\] notes of Rs. 20 \[ = {\rm{Rs}}{\rm{. }}20 \times 12x = {\rm{Rs}}{\rm{. }}240x\] rupees
The amount made up by the notes of denomination Rs. 10 is equal to the product of the number of notes of Rs. 10, and the value of 1 note.
Therefore, we get
Amount made up by \[60x\] notes of Rs. 10 \[ = {\rm{Rs}}{\rm{. 1}}0 \times 60x = {\rm{Rs}}{\rm{. }}600x\] rupees
Now, the total amount will be the sum of the amounts made by the notes of different denominations.
Therefore, we get
Total amount made up by all notes \[ = 100x + 150x + 240x + 600x\]
We know that the total amount made up by all the notes is Rs. 3270.
Thus, we get the equation
\[ \Rightarrow 100x + 150x + 240x + 600x = 3270\]
We will now solve the above equation to find the value of \[x\].
Adding the like terms in the equation, we get
\[ \Rightarrow 1090x = 3270\]
Dividing both sides of the equation by 1090, we get
\[\begin{array}{l} \Rightarrow \dfrac{{1090x}}{{1090}} = \dfrac{{3270}}{{1090}}\\ \Rightarrow x = 3\end{array}\]
Therefore, the value of \[x\] is 3.
Substituting \[x = 3\] in the number of Rs. 20 notes, we get
Number of notes of denomination Rs. 20 \[ = 12x = 12\left( 3 \right) = 36\]notes
Therefore, there are 36 notes of denomination Rs. 20.

Note: Here, we observe that Rs. 3270 is the total amount made up by all the notes of different denominations. A common mistake is to use 3270 as the total number of notes and equate it to \[x + 3x + 12x + 60x\]. This will give us incorrect answers. In this question, we have formed a linear equation with one variable to find the number of notes. Linear equations are those equations where the highest degree of the variable is 1.