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

It is given that,

P(getting a red ball) = $\frac{1}{4}$ and P(getting a blue ball) = $\frac{1}{3}$

Let P(getting an orange ball) be x.

Since, there are only 3 types of balls in the jar, the sum of probabilities of all the three balls must be 1.

 $$\begin{gathered} \therefore \frac{1}{4}+\frac{1}{3}+x=1 \\ \Rightarrow x=1-\frac{1}{4}-\frac{1}{3} \\ \Rightarrow x=\frac{12-3-4}{12} \\ \Rightarrow x=\frac{5}{12} \end{gathered}$$\
∴ P(getting an orange ball) = $\frac{5}{12}$

Let the total number of balls in the jar be n.

∴ P(getting an orange ball) = $\frac{10}{n}$
$$\begin{gathered} \Rightarrow \frac{10}{n}=\frac{5}{12} \\ \Rightarrow n=24 \end{gathered}$$

Thus, the total number of balls in the jar is 24.