Read the following information carefully and answer the questions given below-
1000 students appeared in an examination Out of which 275 students passed in physics, 320 students passed in chemistry and 350 students passed in mathematics. 84 students passed in both chemistry and mathematics 60 students passed in both physics and mathematics, 50 students passed in both physics and chemistry and 34 students passed in all the three subjects.
What is the percentage of students who passed in only one subject ?

Options

A. 52
B. 66
C. 69.3
D. 65.9

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion to find the number of students who passed in only one subject.

### Step 1: Define Variables
Let:
- A = Number of students who passed only Mathematics
- B = Number of students who passed only Chemistry
- C = Number of students who passed only Physics
- D = Number of students who passed in both Chemistry and Mathematics but not Physics
- E = Number of students who passed in both Physics and Mathematics but not Chemistry
- F = Number of students who passed in both Physics and Chemistry but not Mathematics
- G = Number of students who passed in all three subjects

### Step 2: Given Data
From the problem, we have:
- Total students = 1000
- Students passed in Physics (P) = 275
- Students passed in Chemistry (C) = 320
- Students passed in Mathematics (M) = 350
- Students passed in both Chemistry and Mathematics (D + G) = 84
- Students passed in both Physics and Mathematics (E + G) = 60
- Students passed in both Physics and Chemistry (F + G) = 50
- Students passed in all three subjects (G) = 34

### Step 3: Find Values of D, E, F, and G
Using the values of G:
- D = 84 - G = 84 - 34 = 50
- E = 60 - G = 60 - 34 = 26
- F = 50 - G = 50 - 34 = 16

### Step 4: Substitute Values into the Total Passes
Now, we can substitute D, E, F, and G into the equations for A, B, and C.

1. For Physics:
\[ A + E + F + G = 275 \]
\[ A + 26 + 16 + 34 = 275 \]
\[ A + 76 = 275 \implies A = 275 - 76 = 199 \]

2. For Chemistry:
\[ B + D + F + G = 320 \]
\[ B + 50 + 16 + 34 = 320 \]
\[ B + 100 = 320 \implies B = 320 - 100 = 220 \]

3. For Mathematics:
\[ C + D + E + G = 350 \]
\[ C + 50 + 26 + 34 = 350 \]
\[ C + 110 = 350 \implies C = 350 - 110 = 240 \]

### Step 5: Calculate Students Passing Only One Subject
Now we have:
- A = 199 (only Mathematics)
- B = 220 (only Chemistry)
- C = 240 (only Physics)

Total students passing only one subject:
\[ Total = A + B + C = 199 + 220 + 240 = 659 \]

### Step 6: Calculate the Percentage
To find the percentage of students who passed in only one subject:
\[ \text{Percentage} = \left( \frac{Total}{1000} \right) \times 100 = \left( \frac{659}{1000} \right) \times 100 = 65.9\% \]

### Final Answer
The percentage of students who passed in only one subject is **65.9%**.

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