The probability of selecting a red ball at random from a jar that contains only red, blue and orange balls is `(1)/(4)` The probability of selecting a blue ball at random from the same jar is `(1)/(3)` If the jar contains 10 orange balls, find the total number of balls in the jar.

Answer

To solve the problem step by step, we will use the information given about the probabilities of selecting red and blue balls, along with the number of orange balls.

### Step 1: Understand the probabilities given
We know:
- Probability of selecting a red ball, \( P(R) = \frac{1}{4} \)
- Probability of selecting a blue ball, \( P(B) = \frac{1}{3} \)

### Step 2: Set up the equation for total probability
Since the jar contains only red, blue, and orange balls, the total probability must equal 1:
\[ P(R) + P(B) + P(O) = 1 \]
where \( P(O) \) is the probability of selecting an orange ball.

### Step 3: Substitute the known probabilities into the equation
Substituting the known probabilities into the equation:
\[ \frac{1}{4} + \frac{1}{3} + P(O) = 1 \]

### Step 4: Find a common denominator
To combine the fractions, we need a common denominator. The least common multiple of 4 and 3 is 12. We convert each fraction:
\[ \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{3} = \frac{4}{12} \]
Now substitute these into the equation:
\[ \frac{3}{12} + \frac{4}{12} + P(O) = 1 \]

### Step 5: Simplify the equation
Combine the fractions:
\[ \frac{3 + 4}{12} + P(O) = 1 \]
This simplifies to:
\[ \frac{7}{12} + P(O) = 1 \]

### Step 6: Solve for \( P(O) \)
Now, isolate \( P(O) \):
\[ P(O) = 1 - \frac{7}{12} = \frac{5}{12} \]

### Step 7: Relate the probability of orange balls to the total number of balls
We know that the probability of selecting an orange ball is given by:
\[ P(O) = \frac{\text{Number of orange balls}}{\text{Total number of balls}} \]
Let \( N \) be the total number of balls in the jar. We have:
\[ \frac{10}{N} = \frac{5}{12} \]

### Step 8: Cross-multiply to find \( N \)
Cross-multiplying gives:
\[ 10 \cdot 12 = 5 \cdot N \]
This simplifies to:
\[ 120 = 5N \]

### Step 9: Solve for \( N \)
Now, divide both sides by 5:
\[ N = \frac{120}{5} = 24 \]

### Final Answer
The total number of balls in the jar is \( \boxed{24} \).

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