Question

Arpit has 100, 50 and 20 rupee notes in his wallet in the ratio \[2:3:5\] then, find the total amount of money in his wallet if he has a total of 50 notes of all the three denominations.

Answer 1

Hint: We solve this problem by using the ratio definition.
If a number \[n\] is divided in the ratio \[a:b:c\] then there exists a number \[x\] such that
\[\Rightarrow ax+bx+cx=n\]
Here, the three divided numbers will be \[ax,bx,cx\] respectively.
By using this theorem we find the number of notes of each denomination to find the total amount in the wallet.

Complete step-by-step solution
We are given that Arpit has 100, 50, and 20 rupee notes in his wallet
We are given that the ratio of the number of notes as \[2:3:5\]
We are given that there are a total of 50 notes.
Let us assume that the total number of notes as
\[\Rightarrow n=50\]
We know that If a number \[n\] is divided in the ratio \[a:b:c\] then there exists a number \[x\] such that
\[\Rightarrow ax+bx+cx=n\]
Here, the three divided numbers will be \[ax,bx,cx\] respectively.
By using this theorem to the given question that is 50 notes are divided in the ratio \[2:3:5\] we get
\[\begin{align}
  & \Rightarrow 2x+3x+5x=50 \\
 & \Rightarrow 10x=50 \\
 & \Rightarrow x=5 \\
\end{align}\]
Now, let us assume that the number of 100 rupee notes as \[{{n}_{1}}\]
From the definition of ratios we have the number of 100 rupee notes as
\[\begin{align}
  & \Rightarrow {{n}_{1}}=2x \\
 & \Rightarrow {{n}_{1}}=2\times 5=10 \\
\end{align}\]
Now, let us assume that the number of 50 rupee notes as \[{{n}_{2}}\]
From the definition of ratios we have the number of 50 rupee notes as
\[\begin{align}
  & \Rightarrow {{n}_{2}}=3x \\
 & \Rightarrow {{n}_{2}}=3\times 5=15 \\
\end{align}\]
Now, let us assume that the number of 20 rupee notes as \[{{n}_{3}}\]
From the definition of ratios we have the number of 100 rupee notes as
\[\begin{align}
  & \Rightarrow {{n}_{3}}=5x \\
 & \Rightarrow {{n}_{3}}=5\times 5=25 \\
\end{align}\]
Let us assume that the total amount of money that is in the wallet as \[T\]
We know that if there are \[n\] notes of \[k\] rupee notes then the amount will be \[n\times k\]
By using this result we get the total amount in the wallet as
\[\Rightarrow T={{n}_{1}}\times 100+{{n}_{2}}\times 50+{{n}_{3}}\times 20\]
Now, by substituting the required values in the above equation we get
\[\begin{align}
  & \Rightarrow T=10\times 100+15\times 50+25\times 20 \\
 & \Rightarrow T=1000+750+500 \\
 & \Rightarrow T=2250 \\
\end{align}\]
Therefore we can conclude that the total amount in the wallet is 2250 rupees.

Note: We can get the number of each denomination using the direct formula.
If a number \[n\] is divided in the ratio \[a:b:c\] then we have
\[\text{First number}=\dfrac{a}{a+b+c}\times n\]
\[\text{Second number}=\dfrac{b}{a+b+c}\times n\]
\[\text{Third number}=\dfrac{c}{a+b+c}\times n\]
We are given that the ratio of 100, 50 and 20 rupee notes as \[2:3:5\]
Now, by using the above formula we get the number of 100 rupee notes \[{{n}_{1}}\] as
\[\begin{align}
  & \Rightarrow {{n}_{1}}=\dfrac{2}{2+3+5}\times 50 \\
 & \Rightarrow {{n}_{1}}=\dfrac{2}{10}\times 50=10 \\
\end{align}\]
Similarly, we get the number of 50 rupee notes \[{{n}_{2}}\] as
\[\begin{align}
  & \Rightarrow {{n}_{2}}=\dfrac{3}{2+3+5}\times 50 \\
 & \Rightarrow {{n}_{2}}=\dfrac{3}{10}\times 50=15 \\
\end{align}\]
Similarly, we get the number of 50 rupee notes \[{{n}_{3}}\] as
\[\begin{align}
  & \Rightarrow {{n}_{3}}=\dfrac{5}{2+3+5}\times 50 \\
 & \Rightarrow {{n}_{3}}=\dfrac{5}{10}\times 50=25 \\
\end{align}\]
Let us assume that the total amount of money that is in the wallet as \[T\]
We know that if there are \[n\] notes of \[k\] rupee notes then the amount will be \[n\times k\]
By using this result we get the total amount in the wallet as
\[\Rightarrow T={{n}_{1}}\times 100+{{n}_{2}}\times 50+{{n}_{3}}\times 20\]
Now, by substituting the required values in the above equation we get
\[\begin{align}
  & \Rightarrow T=10\times 100+15\times 50+25\times 20 \\
 & \Rightarrow T=1000+750+500 \\
 & \Rightarrow T=2250 \\
\end{align}\]
Therefore we can conclude that the total amount in the wallet is 2250 rupees.