Question

Deveshi has a total of $Rs.590$ as currency notes in the denominations of $Rs.50,Rs.20{\text{ and }}Rs.10.$ The ratio of the number of $Rs.50$ notes and $Rs.20$ notes is $3:5$. If she has a total of $25$ notes, how many notes of each denomination she has?

Answer 1

Hint: Here in this question, first we will try to write the possibilities of the denomination of the $Rs.50$ and $Rs.20$ notes. Then we will shortlist them according to our question and after shortlisting we take each shortlisted possibility and calculate the number of notes in that particular case.

Complete step-by-step answer:
As it is given that ratio of $Rs.50$ notes and $Rs.20$ notes is $3:5$. This means that either-
$Rs.50$ notes $ = 3$ and $Rs.20$ notes $ = 5$
$Rs.50$ notes $ = 6$ and $Rs.20$ notes $ = 10$
$Rs.50$ notes $ = 9$ and $Rs.20$ notes $ = 15$
And so on.
But the total amount of all notes is $Rs.590$, so the amount by alone $Rs.50$ and $Rs.20$ notes cannot be greater than $Rs.580$ as there must be at least one $Rs.10$ note as well as mentioned in the question. So, let’s calculate the sums in the above mentioned cases.
$Rs.50$ notes $ = 3$ and $Rs.20$ notes $ = 5$ ; Sum $ = 50 \times 3 + 20 \times 5 = Rs.250$
$Rs.50$ notes $ = 6$ and $Rs.20$ notes $ = 10$ ; Sum $ = 50 \times 6 + 20 \times 10 = Rs.500$
$Rs.50$ notes $ = 9$ and $Rs.20$ notes $ = 15$ ; Sum $ = 50 \times 9 + 20 \times 15 = Rs.750$ which is not possible.
For higher the number of notes cases, the sum will be higher. So, we are left with two cases only –
1) $Rs.50$ notes $ = 3$ and $Rs.20$ notes $ = 5$ ; Sum $ = 50 \times 3 + 20 \times 5 = Rs.250$:
For the amount to be $Rs.590$ we’ll be needing $34$ notes of $Rs.10$. So, we get total number of notes as $3 + 5 + 34 = 42$
2) $Rs.50$ notes $ = 6$ and $Rs.20$ notes $ = 10$ ; Sum $ = 50 \times 6 + 20 \times 10 = Rs.500$ :
For the amount to be $Rs.590$ we’ll be needing $9$ notes of $Rs.10$. So, we get a total number of notes as $6 + 10 + 9 = 25$ which is what we needed. So, this is the correct answer:
$Rs.50$notes - $6$
$Rs.20$ notes - $10$
$Rs.10$ notes - $9$

Note: Hence, there are many ways to solve this question, so you can choose any method whichever you feel easy and do have a stronghold on the concepts of that method. The method we use in order to solve this question is the concept of linear equations in two variables. This method involves substituting $y{\text{ }}\left( {{\text{or }}x} \right)$ as per your concern from one equation into the other equation. This simplifies the second equation and we can solve it easily.