In an examination `60%` of the students passed in maths, `40%` of the students passed in english `30%` of the students passed in reasoning. If `10%` of the students passed in maths and reasoning `25%` of the students passed in english and maths and `15%` of the students passed in reasoning and english then at least how many `%` students passed in all subjects in none of the students failed in any subjects ?

Options

A. `28%`
B. `22%`
C. `24%`
D. `20%`

Answer

To solve the problem step by step, we will use the principle of inclusion-exclusion and a Venn diagram approach.

### Step 1: Define Variables
Let:
- \( X \) = percentage of students who passed in all three subjects (Maths, English, and Reasoning).

### Step 2: Set Up the Venn Diagram
From the problem, we have the following information:
- 60% passed in Maths.
- 40% passed in English.
- 30% passed in Reasoning.
- 10% passed in both Maths and Reasoning.
- 25% passed in both English and Maths.
- 15% passed in both Reasoning and English.

### Step 3: Express Overlaps
Using the variable \( X \):
- Students who passed in both Maths and Reasoning but not English: \( 10\% - X \)
- Students who passed in both English and Maths but not Reasoning: \( 25\% - X \)
- Students who passed in both Reasoning and English but not Maths: \( 15\% - X \)

### Step 4: Write Equations Based on Total Percentages
Now we can set up equations based on the total percentages for each subject:
1. For Maths:
\[ (25\% - X) + (10\% - X) + X + \text{(only Maths)} = 60\% \]
This simplifies to:
\[ 35\% - X + \text{(only Maths)} = 60\% \]
Therefore,
\[ \text{(only Maths)} = 60\% - 35\% + X = 25\% + X \]

2. For English:
\[ (25\% - X) + X + (15\% - X) + \text{(only English)} = 40\% \]
This simplifies to:
\[ 25\% + 15\% - X + \text{(only English)} = 40\% \]
Therefore,
\[ \text{(only English)} = 40\% - 40\% + X = X \]

3. For Reasoning:
\[ (10\% - X) + (15\% - X) + X + \text{(only Reasoning)} = 30\% \]
This simplifies to:
\[ 25\% - X + \text{(only Reasoning)} = 30\% \]
Therefore,
\[ \text{(only Reasoning)} = 30\% - 25\% + X = 5\% + X \]

### Step 5: Total Students
Now, we know that the total percentage of students who passed in at least one subject must equal 100%:
\[ \text{(only Maths)} + \text{(only English)} + \text{(only Reasoning)} + (10\% - X) + (25\% - X) + (15\% - X) + X = 100\% \]
Substituting the expressions we found:
\[ (25\% + X) + X + (5\% + X) + (10\% - X) + (25\% - X) + (15\% - X) + X = 100\% \]
Combining like terms:
\[ 25\% + 5\% + 10\% + 25\% + 15\% + 3X = 100\% \]
This simplifies to:
\[ 85\% + 3X = 100\% \]
Thus,
\[ 3X = 15\% \]
So,
\[ X = 5\% \]

### Step 6: Conclusion
The percentage of students who passed in all subjects is \( 5\% \).

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