In an examination `53%` students passed in mathematics, `61%` students passed in physics `60%` students passed in chemistry `24%` students passed in mathematics and physics `35%` students passed in physics and chemistry, `27%` students passed in mathematics and chemistry and `5%` in none. The ratio of `%` of passes in mathematics and chemistry but not is physics in relation to the `%` of passes in physics and chemistry but not in mathematics is ?

Options

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

Answer

To solve the problem step by step, we will use the principle of set theory and Venn diagrams to find the required percentages.

### Step 1: Define the Variables
Let:
- M = Percentage of students who passed in Mathematics = 53%
- P = Percentage of students who passed in Physics = 61%
- C = Percentage of students who passed in Chemistry = 60%
- MP = Percentage of students who passed in both Mathematics and Physics = 24%
- PC = Percentage of students who passed in both Physics and Chemistry = 35%
- MC = Percentage of students who passed in both Mathematics and Chemistry = 27%
- N = Percentage of students who passed in none = 5%

### Step 2: Calculate the Percentage of Students Who Passed in All Three Subjects
Let X be the percentage of students who passed in all three subjects (Mathematics, Physics, and Chemistry).

Using the principle of inclusion-exclusion for three sets, we have:

\[ M + P + C - (MP + PC + MC) + X = 100 - N \]

Substituting the known values:

\[ 53 + 61 + 60 - (24 + 35 + 27) + X = 100 - 5 \]

Calculating the left side:

\[ 53 + 61 + 60 = 174 \]
\[ 24 + 35 + 27 = 86 \]
So, substituting these values in:

\[ 174 - 86 + X = 95 \]
\[ 88 + X = 95 \]

Thus, solving for X:

\[ X = 95 - 88 = 7 \]

### Step 3: Calculate the Percentage of Students Who Passed in Each Subject Alone
Now we can find the percentages of students who passed in each subject but not in others.

1. **Mathematics only (M only)**:
\[ M_{only} = M - (MP + MC - X) = 53 - (24 + 27 - 7) = 53 - 44 = 9 \]

2. **Physics only (P only)**:
\[ P_{only} = P - (MP + PC - X) = 61 - (24 + 35 - 7) = 61 - 52 = 9 \]

3. **Chemistry only (C only)**:
\[ C_{only} = C - (MC + PC - X) = 60 - (27 + 35 - 7) = 60 - 55 = 5 \]

### Step 4: Calculate the Required Percentages
1. **Passed in Mathematics and Chemistry but not Physics**:
\[ MC_{only} = MC - X = 27 - 7 = 20 \]

2. **Passed in Physics and Chemistry but not Mathematics**:
\[ PC_{only} = PC - X = 35 - 7 = 28 \]

### Step 5: Calculate the Ratio
Now we need to find the ratio of the percentage of students who passed in Mathematics and Chemistry but not Physics to those who passed in Physics and Chemistry but not Mathematics:

\[ \text{Ratio} = \frac{MC_{only}}{PC_{only}} = \frac{20}{28} = \frac{5}{7} \]

### Final Answer
The ratio of the percentage of passes in Mathematics and Chemistry but not in Physics to the percentage of passes in Physics and Chemistry but not in Mathematics is **5:7**.

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