MAT Exam 2025 Paper

Concept: Use the total probability rule: the sum of the probabilities of all colours equals the whole jar, and the orange balls are known.Explanation: Let the total number of balls be $x$.The number of red balls is $P(\text{red}) \times x = \frac{1}{7}x$.The number of blue balls is $P(\text{blue}) \times x = \frac{1}{6}x$.The number of orange balls is given as 29.Since the jar contains only red, blue, and orange balls, the total is their sum:$$x = \frac{1}{7}x + \frac{1}{6}x + 29.$$Take the LCM of 7 and 6, which is 42:$$x = \frac{6x + 7x}{42} + 29 = \frac{13x}{42} + 29.$$Bring the term with $x$ to the left-hand side:$$x - \frac{13x}{42} = 29.$$Simplify the left-hand side:$$\frac{42x - 13x}{42} = 29 \Rightarrow \frac{29x}{42} = 29.$$Cancel 29 from both the numerator and the denominator:$$\frac{x}{42} = 1 \Rightarrow x = 42.$$Thus the jar contains 42 balls in total, which matches option C.Answer: C. 42 balls.