Question

The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is $\dfrac{1}{4}$ . The probability of selecting a blue ball at random, from the same jar is $\dfrac{1}{3}$ . If the jar contains 10 orange balls, find the total number of balls in the jar.

Answer 1

Hint: We will suppose that the total number of balls in the jar be x and then we will write the probability in terms of total number of blue and red balls in x. then, we will use the sum that orange balls + red balls + blue balls = total number balls in jar = x to calculate the total number if balls in the jar.

Complete step-by-step answer:
 We are given the probability of selecting a red ball from a jar is $\dfrac{1}{4}$.
The probability of selecting a blue ball from the jar is $\dfrac{1}{3}$.
The total number of orange balls in the jar = 10.
Let the total number of balls in the jar be x.
We are told that the jar only contains red, blue and orange balls.
Hence, we can write: orange balls + red balls + blue balls = total number balls in jar = x – (1)
Now, the total number of red balls can be calculated as: probability of getting a red ball from jar$ \times $ total number of balls in the jar
$ \Rightarrow $ Total number of red balls = $\dfrac{1}{3} \times x = \dfrac{x}{3}$
Similarly, the total number of blue balls in the jar = $\dfrac{1}{4}x = \dfrac{x}{4}$
Substituting these values in the equation (1), we get
$ \Rightarrow 10 + \dfrac{x}{3} + \dfrac{x}{4} = x$
Solving it for the value of x, we get
$
   \Rightarrow \dfrac{{120 + 4x + 3x}}{{12}} = x \\
   \Rightarrow 120 + 4x + 3x = 12x \\
   \Rightarrow 120 + 7x = 12x \\
   \Rightarrow 120 = 5x \\
   \Rightarrow x = 24 \\
 $
Therefore, the total number of balls in the jar is 24.

Note: In this question, you may go wrong while converting the probability of blue and red balls into the total number of balls coloured red and blue using the total number of balls in the jar i.e., x. Be careful while solving for x using the sum of all the three coloured balls using the definition of the probability.