In a stationary shop there are four types of colored sheets of red , blue ,green and white colors . The probability of selecting one red sheet out of the total sheets is `(1)/(3)` , the probability of selecting one blue sheet out of the sheets is `(2)/(7)` and the probability of selecting one white sheet out of the total sheets is `(1)/(4)` .the number of green sheets in the bag is 22.
What is the total number of sheets in the bag ?

Options

A. A)117
B. B)168
C. C)154
D. D)120

Answer

To find the total number of sheets in the bag, we can follow these steps:

### Step 1: Define the Variables
Let the total number of sheets in the bag be \( T \). We know the number of green sheets is 22.

### Step 2: Write the Probabilities
We are given the probabilities of selecting each color sheet:
- Probability of selecting a red sheet: \( P(R) = \frac{1}{3} \)
- Probability of selecting a blue sheet: \( P(B) = \frac{2}{7} \)
- Probability of selecting a white sheet: \( P(W) = \frac{1}{4} \)
- Probability of selecting a green sheet: \( P(G) = \frac{22}{T} \)

### Step 3: Set Up the Equation
The sum of the probabilities of selecting each color sheet must equal 1:
\[ P(R) + P(B) + P(W) + P(G) = 1 \]
Substituting the known probabilities:
\[ \frac{1}{3} + \frac{2}{7} + \frac{1}{4} + \frac{22}{T} = 1 \]

### Step 4: Find a Common Denominator
To add the fractions, we need a common denominator. The least common multiple (LCM) of 3, 7, and 4 is 84. We convert each fraction:
- \( \frac{1}{3} = \frac{28}{84} \)
- \( \frac{2}{7} = \frac{24}{84} \)
- \( \frac{1}{4} = \frac{21}{84} \)

### Step 5: Substitute Back into the Equation
Now substitute these values back into the equation:
\[ \frac{28}{84} + \frac{24}{84} + \frac{21}{84} + \frac{22}{T} = 1 \]
Combine the fractions:
\[ \frac{28 + 24 + 21}{84} + \frac{22}{T} = 1 \]
This simplifies to:
\[ \frac{73}{84} + \frac{22}{T} = 1 \]

### Step 6: Isolate the Fraction
Rearranging the equation gives:
\[ \frac{22}{T} = 1 - \frac{73}{84} \]
Calculating the right side:
\[ 1 - \frac{73}{84} = \frac{84 - 73}{84} = \frac{11}{84} \]
Thus, we have:
\[ \frac{22}{T} = \frac{11}{84} \]

### Step 7: Cross-Multiply to Solve for T
Cross-multiplying gives:
\[ 22 \cdot 84 = 11 \cdot T \]
Calculating \( 22 \cdot 84 \):
\[ 1848 = 11T \]
Now, divide both sides by 11:
\[ T = \frac{1848}{11} = 168 \]

### Conclusion
The total number of sheets in the bag is \( T = 168 \).

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