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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.

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Hint- Total number of balls in the jar will be the sum of all colour balls. Consider some variable for the total number of balls, and then proceed further by the use of basic definition of probability.

Let the total number of balls in the given jar be $ = x$
As the probability of getting a red ball is $\dfrac{1}{4}$.
So total number of red ball in terms of variable $x$ is
$
   = \dfrac{1}{4} \times x \\
   = \dfrac{x}{4} \\
 $
Also the probability of getting a blue ball is $\dfrac{1}{3}$ .
So total number of red ball in terms of variable $x$ is
$
   = \dfrac{1}{3} \times x \\
   = \dfrac{x}{3} \\
 $
And the number of orange balls is $10$.
As we know that
Total number of balls in the jar = Number of red balls + Number of blue balls + Number of orange balls
$ \Rightarrow x = \dfrac{x}{4} + \dfrac{x}{3} + 10$
Taking LCM on the R.H.S. and then solving the algebraic equation.
$
   \Rightarrow x = \dfrac{{4x + 3x + 120}}{{12}} \\
   \Rightarrow 12x = 7x + 120 \\
   \Rightarrow 5x = 120 \\
   \Rightarrow x = 24 \\
$
Hence, the total number of balls in the jar is $24$ .

Note- Such a question as in above is easier to solve with the help of algebraic equations. Just we have to consider the unknown value as some variables and proceed. This problem can also be done in another way by first finding the probability of selection of orange balls by subtracting the sum of probabilities of red and blue balls from one and then comparing it with the number of orange balls.
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