In an exam, 40\% of the students passed only in Hindi and 50\% passed only in English. No student failed in both subjects. If the number of students who failed only in English is 75, find the number of students who failed only in Hindi.

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**Solution:**

1. Let the total number of students be *N*.

2. Given:
   - Passed only in Hindi = 40\% of *N* = 0.40 *N*
   - Passed only in English = 50\% of *N* = 0.50 *N*
   - No student failed in both subjects.
   - Number of students who failed only in English = 75

3. Since no student failed in both subjects, the students are divided into four groups:
   - Passed only Hindi
   - Passed only English
   - Failed only Hindi
   - Failed only English

4. The sum of all these groups equals total students:
   $$0.40N + 0.50N + 	ext{Failed only Hindi} + 75 = N$$

5. Simplify:
   $$0.90N + 	ext{Failed only Hindi} + 75 = N$$

6. Rearranging to find failed only Hindi:
   $$	ext{Failed only Hindi} = N - 0.90N - 75 = 0.10N - 75$$

7. The number of students who failed only Hindi cannot be negative, so:
   $$0.10N - 75 \geq 0 \implies 0.10N \geq 75 \implies N \geq 750$$

8. To find the exact number, assume *N* = 750:
   $$	ext{Failed only Hindi} = 0.10 	imes 750 - 75 = 75 - 75 = 0$$

But zero is not an option, so let's check if the problem implies *N* = 750.

9. Alternatively, since the problem gives the number of students who failed only English as 75, and the percentage passed only Hindi and English, the remaining students must be those who failed only Hindi.

10. Using the equation:
    $$	ext{Failed only Hindi} = N - (0.40N + 0.50N + 75) = N - 0.90N - 75 = 0.10N - 75$$

11. To get a positive integer value for failed only Hindi, let’s find *N* such that failed only Hindi is one of the options.

12. Try option (b) 60:
    $$60 = 0.10N - 75 \implies 0.10N = 135 \implies N = 1350$$

13. Check if this satisfies all conditions:
    - Passed only Hindi = 0.40 * 1350 = 540
    - Passed only English = 0.50 * 1350 = 675
    - Failed only English = 75 (given)
    - Failed only Hindi = 60 (calculated)

14. Sum:
    $$540 + 675 + 75 + 60 = 1350$$

15. All conditions are satisfied.

**Answer:** 60

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