Question

The number of the Rs 5 notes is twice the number of 10 rupee notes in the purse. If the number of 10 rupee notes and the number of five rupee notes are the roots of the quadratic equation ${{x}^{2}}-6x+c=0$, then
[a] Find the value of c.
[b] The number of 5 rupee notes and the number of 10 rupee notes.
[c] If a note from the purse is drawn at random, determine the probability that the note is a 5 Rupee note and the probability that the note is a 10 rupee note.

Answer 1

Hint: Assume that the number of 10 rupee notes is m and the number of 5 rupee notes is 2m. Use the fact that the sum of roots of the quadratic expression $a{{x}^{2}}+bx+c=0$ is given by $\dfrac{-b}{a}$ and the product of the roots is given by $\dfrac{c}{a}$. Hence find the value of m and the value of c. Hence find the number of 5 rupee notes in the purse and the number of 10 rupee notes in the purse. Use the fact that if E is an event corresponding to a random experiment and S is the sample space of the random experiment, then $P\left( E \right)=\dfrac{n\left( E \right)}{n\left( S \right)}$. Hence find the probability that the note is drawn is a 5 rupee note and the probability that the note is drawn is a 10 rupee note.

Complete step-by-step solution:
Let the number of 10 rupee notes in the purse be m.
Since the number of 5 rupee notes is twice the number of 10 rupee notes, we have the number of 5 rupee notes is 2m.
Hence, we have m and 2m are the roots of the quadratic equation ${{x}^{2}}-6x+c=0$
We know that the sum of the roots of the quadratic expression $a{{x}^{2}}+bx+c=0$ is given by $\dfrac{-b}{a}$. Hence, we have
$\begin{align}
  & 2m+m=\dfrac{-\left( -6 \right)}{1}=6 \\
 & \Rightarrow 3m=6 \\
\end{align}$
Dividing both sides by 3, we get
$m=2$
Hence the number of 10 rupee notes is 2 and the number of 5 rupee notes is $2\times 2=4$
We know that the product of roots of the quadratic expression $a{{x}^{2}}+bx+c=0$ is given by $\dfrac{c}{a}$
Hence, we have
$\begin{align}
  & \dfrac{c}{1}=2\times 4 \\
 & \Rightarrow c=8 \\
\end{align}$
Hence the value of c is 8.
Let E be the event that the note drawn is a 5 rupee note and F be the event that the note drawn is a 10 rupee note.
Since there are four 5 rupees notes, we have $n\left( E \right)=4$
Since there are two 10 rupees notes, we have $n\left( F \right)=2$
Also, since there are $4+2 = 6$ notes in total in the purse, we have $n\left( S \right)=6$
We know that if E is an event corresponding to a random experiment and S is the sample space of the random experiment, then $P\left( E \right)=\dfrac{n\left( E \right)}{n\left( S \right)}$.
Hence, we have
$P\left( E \right)=\dfrac{4}{6}=\dfrac{2}{3}$
Similarly, we have $n\left( F \right)=2$
Hence, we have
$P\left( F \right)=\dfrac{2}{6}=\dfrac{1}{3}$
Hence the probability that the note is drawn is a 5 rupee note is $\dfrac{2}{3}$ and the probability that the note is drawn is a 10 rupee note is $\dfrac{1}{3}$

Note: [1] The drawing out of notes uniformly at random is important for the application of the above formula for probability. If the draw is not uniform then there is a bias factor in drawing out. Such questions are solved using the concept of conditional probability.
[2] Here E and F are mutually exclusive as well as exhaustive events. So the sum of their probabilities should be 1, which can be verified directly.