In an examination `50%` of the students passed in maths, `55%` of the students passed in english `40%` of the students passed in Hindi. If `15%` of the students passed in maths and english , `20%` of the students passed in english and hindi and `25%` of the students passed in maths and hindi and `10%` of the students passed in all the three subjects then find the `%` of students who failed in all three subjects ?

Options

A. `11%`
B. `7%`
C. `9%`
D. `5%`

Answer

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

### Step 1: Understand the given data
- Let the total number of students be represented as 100% for simplicity.
- The percentages of students passing in each subject are:
- Maths (M) = 50%
- English (E) = 55%
- Hindi (H) = 40%
- The percentages of students passing in combinations of subjects are:
- Maths and English (M ∩ E) = 15%
- English and Hindi (E ∩ H) = 20%
- Maths and Hindi (M ∩ H) = 25%
- All three subjects (M ∩ E ∩ H) = 10%

### Step 2: Set up the Venn diagram
- We will denote the sections of the Venn diagram as follows:
- Let \( x \) be the percentage of students who passed in only Maths.
- Let \( y \) be the percentage of students who passed in only English.
- Let \( z \) be the percentage of students who passed in only Hindi.
- Let \( a \) be the percentage of students who passed in both Maths and English but not Hindi.
- Let \( b \) be the percentage of students who passed in both English and Hindi but not Maths.
- Let \( c \) be the percentage of students who passed in both Maths and Hindi but not English.
- The percentage of students who passed in all three subjects is given as 10%.

### Step 3: Fill in the Venn diagram
From the given data:
- For Maths and English (M ∩ E):
- \( a + 10\% = 15\% \) → \( a = 5\% \)
- For English and Hindi (E ∩ H):
- \( b + 10\% = 20\% \) → \( b = 10\% \)
- For Maths and Hindi (M ∩ H):
- \( c + 10\% = 25\% \) → \( c = 15\% \)

### Step 4: Calculate the remaining percentages
Now we can express the total percentages for each subject:
- For Maths:
\[ x + a + c + 10\% = 50\% \]
Substituting \( a \) and \( c \):
\[ x + 5\% + 15\% + 10\% = 50\% \]
\[ x + 30\% = 50\% \]
\[ x = 20\% \]

- For English:
\[ y + a + b + 10\% = 55\% \]
Substituting \( a \) and \( b \):
\[ y + 5\% + 10\% + 10\% = 55\% \]
\[ y + 25\% = 55\% \]
\[ y = 30\% \]

- For Hindi:
\[ z + b + c + 10\% = 40\% \]
Substituting \( b \) and \( c \):
\[ z + 10\% + 15\% + 10\% = 40\% \]
\[ z + 35\% = 40\% \]
\[ z = 5\% \]

### Step 5: Calculate the total percentage of students who passed at least one subject
Now we can sum up all the percentages:
\[ \text{Total passing} = x + y + z + a + b + c + (M \cap E \cap H) \]
Substituting the values:
\[ \text{Total passing} = 20\% + 30\% + 5\% + 5\% + 10\% + 15\% + 10\% = 95\% \]

### Step 6: Calculate the percentage of students who failed in all subjects
To find the percentage of students who failed in all three subjects:
\[ \text{Percentage failed} = 100\% - \text{Total passing} \]
\[ \text{Percentage failed} = 100\% - 95\% = 5\% \]

### Final Answer
The percentage of students who failed in all three subjects is **5%**.

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