In an examination, 80% of students passed in English, 70% of students passed in Hindi and
15% failed in both the subjects. What is the percentage of students who failed in only one
subject?
(a) 15%
(b) 20%
(c) 25%
(d) 35%
20%
Calculating Percentage of Students Failing Only One Subject
This problem involves using percentages and set theory concepts, often visualized with Venn diagrams, to find the number of students who failed in exactly one subject in an examination.
Let's break down the given information:
- Percentage of students who passed in English: ${P(E) = 80\%}$
- Percentage of students who passed in Hindi: ${P(H) = 70\%}$
- Percentage of students who failed in both subjects: ${P(\overline{E} \cap \overline{H}) = 15\%}$
The total percentage of students is ${100\%}$.
If ${15\%}$ of students failed in both subjects, then the percentage of students who passed in at least one subject is the complement of failing in both. This can be calculated as:
$${P(E \cup H) = 100\% - P(\overline{E} \cap \overline{H})}$$$${P(E \cup H) = 100\% - 15\% = 85\%}$$
The percentage of students who passed in at least one subject (${P(E \cup H)}$) is also related to the percentages who passed English, passed Hindi, and passed both (${P(E \cap H)}$) by the formula:
$${P(E \cup H) = P(E) + P(H) - P(E \cap H)}$$
We can use this formula to find the percentage of students who passed in both subjects:
$${85\% = 80\% + 70\% - P(E \cap H)}$$$${85\% = 150\% - P(E \cap H)}$$$${P(E \cap H) = 150\% - 85\% = 65\%}$$
So, ${65\%}$ of students passed in both English and Hindi.
Now, we want to find the percentage of students who failed in only one subject. This means students who passed English but failed Hindi, OR students who passed Hindi but failed English.
- Percentage passed English only (${P(E \cap \overline{H})}$): These are students who passed English but did not pass Hindi. This group includes those who passed English and failed Hindi. It is calculated as the percentage who passed English minus the percentage who passed both.
$${P(E \cap \overline{H}) = P(E) - P(E \cap H) = 80\% - 65\% = 15\%}$$
These students passed English but failed Hindi, meaning they failed only Hindi. - Percentage passed Hindi only (${P(H \cap \overline{E})}$): These are students who passed Hindi but did not pass English. This group includes those who passed Hindi and failed English. It is calculated as the percentage who passed Hindi minus the percentage who passed both.
$${P(H \cap \overline{E}) = P(H) - P(E \cap H) = 70\% - 65\% = 5\%}$$
These students passed Hindi but failed English, meaning they failed only English.
The percentage of students who failed in only one subject is the sum of those who failed only Hindi and those who failed only English:
Percentage failed in only one subject = (Percentage passed English only) + (Percentage passed Hindi only)
Percentage failed in only one subject = ${15\% + 5\% = 20\%}$
Alternatively, let's consider the failing percentages directly.
- Failed English (${P(\overline{E})}$) = ${100\% - P(E) = 100\% - 80\% = 20\%}$
- Failed Hindi (${P(\overline{H})}$) = ${100\% - P(H) = 100\% - 70\% = 30\%}$
- Failed Both (${P(\overline{E} \cap \overline{H})}$) = ${15\%}$
The percentage who failed at least one subject (${P(\overline{E} \cup \overline{H})}$) can be found using the formula:
$${P(\overline{E} \cup \overline{H}) = P(\overline{E}) + P(\overline{H}) - P(\overline{E} \cap \overline{H})}$$$${P(\overline{E} \cup \overline{H}) = 20\% + 30\% - 15\%}$$$${P(\overline{E} \cup \overline{H}) = 50\% - 15\% = 35\%}$$
Students who failed in only one subject are those who failed in at least one subject BUT did NOT fail in both subjects. This is the difference between the percentage who failed at least one and the percentage who failed both.
Percentage failed in only one subject = ${P(\overline{E} \cup \overline{H}) - P(\overline{E} \cap \overline{H})}$
Percentage failed in only one subject = ${35\% - 15\% = 20\%}$
Both methods yield the same result.
| Category | Percentage |
|---|---|
| Passed English (E) | ${80\%}$ |
| Passed Hindi (H) | ${70\%}$ |
| Failed Both ($\overline{E} \cap \overline{H}$) | ${15\%}$ |
| Passed At Least One (E $\cup$ H) | ${100\% - 15\% = 85\%}$ |
| Passed Both (E $\cap$ H) | ${80\% + 70\% - 85\% = 65\%}$ |
| Passed English Only (E $\cap$ $\overline{H}$) / Failed Only Hindi | ${80\% - 65\% = 15\%}$ |
| Passed Hindi Only (H $\cap$ $\overline{E}$) / Failed Only English | ${70\% - 65\% = 5\%}$ |
| Failed Only One Subject | ${15\% + 5\% = 20\%}$ |
Revision Table: Exam Percentage Concepts
| Term | Explanation | Calculation Example |
|---|---|---|
| Passed Subject A | Percentage of students who passed in Subject A. | Given as input (e.g., 80%). |
| Failed Subject A | Percentage of students who did not pass Subject A. | ${100\%}$ - Passed Subject A. |
| Passed Both Subjects | Percentage of students who passed in both Subject A and Subject B. | ${P(A \cap B) = P(A) + P(B) - P(A \cup B)}$ |
| Passed At Least One | Percentage of students who passed in Subject A or Subject B or both. | ${P(A \cup B) = 100\%}$ - Failed Both. |
| Failed Both Subjects | Percentage of students who failed in Subject A and Subject B. | Given as input (e.g., 15%). |
| Failed Only One | Percentage of students who failed in exactly one subject (failed A but not B, OR failed B but not A). | (Passed A Only) + (Passed B Only) OR (Failed At Least One) - (Failed Both). |
Additional Information: Using Venn Diagrams for Percentages
Problems like this can be easily visualized using a Venn diagram. Imagine two overlapping circles, one for English (E) and one for Hindi (H).
- The overlapping section represents students who passed both (${E \cap H}$). We found this to be ${65\%}$.
- The part of the English circle outside the overlap represents students who passed English only (${E \cap \overline{H}}$). This is ${80\% - 65\% = 15\%}$. These students failed Hindi.
- The part of the Hindi circle outside the overlap represents students who passed Hindi only (${H \cap \overline{E}}$). This is ${70\% - 65\% = 5\%}$. These students failed English.
- The area outside both circles represents students who failed both (${ \overline{E} \cap \overline{H}}$), which is given as ${15\%}$.
Let's check if the percentages add up to ${100\%}$:
- Passed English Only: ${15\%}$
- Passed Hindi Only: ${5\%}$
- Passed Both: ${65\%}$
- Failed Both: ${15\%}$
Total = ${15\% + 5\% + 65\% + 15\% = 100\%}$. The numbers are consistent.
Students who failed in only one subject are those who passed English only (failed only Hindi) plus those who passed Hindi only (failed only English). This is ${15\% + 5\% = 20\%}$.
The percentage of students who failed in only one subject is ${20\%}$.
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