In a bag there are three types of colored boxes of black ,blue and white colors .the probability of selecting one black box out of the total boxes is `(2)/(5)` and the probability of selecting one blue box out of the total boxes is `(3)/(7)` .The number of white boxes in the bag is 18 .
What is the total number of boxes in the bag ?

Options

A. A)117
B. B)105
C. C)114
D. D)120

Answer

To solve the problem step by step, we need to find the total number of boxes in the bag based on the given probabilities and the number of white boxes.

### Step 1: Define Variables
Let:
- \( B \) = number of black boxes
- \( U \) = number of blue boxes
- \( W \) = number of white boxes = 18
- \( T \) = total number of boxes = \( B + U + W \)

### Step 2: Use Given Probabilities
The probability of selecting one black box is given as:
\[ P(B) = \frac{B}{T} = \frac{2}{5} \]
From this, we can express \( B \) in terms of \( T \):
\[ B = \frac{2}{5} T \]

The probability of selecting one blue box is given as:
\[ P(U) = \frac{U}{T} = \frac{3}{7} \]
From this, we can express \( U \) in terms of \( T \):
\[ U = \frac{3}{7} T \]

### Step 3: Substitute Known Values
Now, we can substitute \( B \) and \( U \) into the total box equation:
\[ T = B + U + W \]
Substituting the expressions we found:
\[ T = \frac{2}{5} T + \frac{3}{7} T + 18 \]

### Step 4: Find a Common Denominator
To solve for \( T \), we need to combine the fractions. The least common multiple of 5 and 7 is 35. Thus, we rewrite the fractions:
\[ \frac{2}{5} T = \frac{14}{35} T \quad \text{and} \quad \frac{3}{7} T = \frac{15}{35} T \]
Now, substituting back into the equation:
\[ T = \frac{14}{35} T + \frac{15}{35} T + 18 \]
Combining the fractions:
\[ T = \frac{29}{35} T + 18 \]

### Step 5: Isolate \( T \)
To isolate \( T \), we can subtract \( \frac{29}{35} T \) from both sides:
\[ T - \frac{29}{35} T = 18 \]
This simplifies to:
\[ \frac{6}{35} T = 18 \]

### Step 6: Solve for \( T \)
Now, we can solve for \( T \):
\[ T = 18 \times \frac{35}{6} = 105 \]

### Conclusion
The total number of boxes in the bag is \( T = 105 \).