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.
If all the sheets are numbered as 1,2,3 ,...... and so on and one sheets is picked up at random ,then find the probability of picking up a sheet which is numbered as a multiple of 13 or 17.

Options

A. A)`(1)/(8)`
B. B)`(1)/(7)`
C. C)`(5)/(8)`
D. D)`(9)/(10)`

Answer

To solve the problem step by step, we will first determine the total number of sheets in the stationary shop and then calculate the probability of picking a sheet that is numbered as a multiple of 13 or 17.

### Step 1: Determine the total number of sheets (T)

We know the probabilities of selecting each colored sheet:
- Probability of red sheet (P(R)) = 1/3
- Probability of blue sheet (P(B)) = 2/7
- Probability of white sheet (P(W)) = 1/4
- Number of green sheets = 22

Using the formula for probability:
\[ P(X) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]

We can express the probabilities in terms of the total number of sheets (T):
- For red sheets: \( \frac{R}{T} = \frac{1}{3} \) → \( R = \frac{T}{3} \)
- For blue sheets: \( \frac{B}{T} = \frac{2}{7} \) → \( B = \frac{2T}{7} \)
- For white sheets: \( \frac{W}{T} = \frac{1}{4} \) → \( W = \frac{T}{4} \)

### Step 2: Set up the equation

Since the total number of sheets is the sum of all colored sheets:
\[ R + B + W + G = T \]
Substituting the values we have:
\[ \frac{T}{3} + \frac{2T}{7} + \frac{T}{4} + 22 = T \]

### Step 3: Find a common denominator and solve for T

The least common multiple of 3, 7, and 4 is 84. We can rewrite the equation:
\[ \frac{28T}{84} + \frac{24T}{84} + \frac{21T}{84} + 22 = T \]
Combining the fractions:
\[ \frac{73T}{84} + 22 = T \]

Now, rearranging gives:
\[ 22 = T - \frac{73T}{84} \]
\[ 22 = \frac{84T - 73T}{84} \]
\[ 22 = \frac{11T}{84} \]

Multiplying both sides by 84:
\[ 1848 = 11T \]
\[ T = \frac{1848}{11} = 168 \]

### Step 4: Calculate the number of sheets of each color

Now that we have T = 168, we can find the number of sheets of each color:
- Red sheets: \( R = \frac{168}{3} = 56 \)
- Blue sheets: \( B = \frac{2 \times 168}{7} = 48 \)
- White sheets: \( W = \frac{168}{4} = 42 \)
- Green sheets: \( G = 22 \)

### Step 5: Find the number of favorable outcomes for multiples of 13 or 17

Next, we need to find how many sheets are numbered as multiples of 13 or 17.

**Multiples of 13:**
The multiples of 13 up to 168 are:
\[ 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156 \]
This gives us 12 multiples of 13.

**Multiples of 17:**
The multiples of 17 up to 168 are:
\[ 17, 34, 51, 68, 85, 102, 119, 136, 153 \]
This gives us 9 multiples of 17.

### Step 6: Calculate the total favorable outcomes

Now we add the favorable outcomes:
Total favorable outcomes = Multiples of 13 + Multiples of 17 = 12 + 9 = 21

### Step 7: Calculate the probability

The probability of picking a sheet that is a multiple of 13 or 17 is given by:
\[ P = \frac{\text{Total favorable outcomes}}{\text{Total outcomes}} = \frac{21}{168} \]

Simplifying this:
\[ P = \frac{1}{8} \]

### Conclusion

Thus, the probability of picking a sheet which is numbered as a multiple of 13 or 17 is \( \frac{1}{8} \).

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