1. **Stating the problem:** We have three tests: Written (W), Practical (P), and Oral (O). Candidates pass different combinations of these tests. Given: - 5 candidates passed only one test. - 27 candidates passed only two tests. - 4 candidates passed all three tests. - 8 candidates didn't pass any test. - 16 candidates passed the practical test. We need to: (i) Complete the Venn diagram with the given data. (ii) Find the total number of candidates who sat for the test. 2. **Understanding the sets and notation:** Let: - $a$ = number of candidates who passed only Written test. - $b$ = number of candidates who passed only Practical test. - $c$ = number of candidates who passed only Oral test. - $d$ = number who passed exactly Written and Practical but not Oral. - $e$ = number who passed exactly Practical and Oral but not Written. - $f$ = number who passed exactly Written and Oral but not Practical. - $g$ = number who passed all three tests. - $h$ = number who passed none. 3. **Given values:** - $a + b + c = 5$ (only one test) - $d + e + f = 27$ (only two tests) - $g = 4$ (all three tests) - $h = 8$ (none) - $P = b + d + e + g = 16$ (passed Practical test) 4. **Expressing the practical test passers:** $$b + d + e + g = 16$$ Substitute $g=4$: $$b + d + e + 4 = 16$$ $$b + d + e = 12$$ 5. **Using the sums for one and two tests:** From one test: $$a + b + c = 5$$ From two tests: $$d + e + f = 27$$ 6. **Sum of all who passed at least one test:** $$a + b + c + d + e + f + g = ?$$ We know $a + b + c = 5$, $d + e + f = 27$, and $g = 4$, so: $$5 + 27 + 4 = 36$$ 7. **Find $f$ using $b + d + e = 12$:** We know $d + e + f = 27$, so: $$f = 27 - (d + e)$$ But $b + d + e = 12$, so: $$d + e = 12 - b$$ Therefore: $$f = 27 - (12 - b) = 15 + b$$ 8. **Recall $a + b + c = 5$, so $a + c = 5 - b$.** 9. **Total candidates who sat for the test:** Sum of all who passed at least one test plus those who didn't pass any: $$N = (a + b + c + d + e + f + g) + h = 36 + 8 = 44$$ 10. **Find values of $a, b, c, d, e, f$ using $b$ as variable:** From step 7, $f = 15 + b$. From step 5, $d + e + f = 27$, so: $$d + e = 27 - f = 27 - (15 + b) = 12 - b$$ From step 4, $b + d + e = 12$, so: $$d + e = 12 - b$$ Both expressions for $d + e$ match, so consistent. 11. **Assign values to $a, c$ to satisfy $a + c = 5 - b$.** Since no further info, we can distribute $a$ and $c$ arbitrarily as long as sum is $5 - b$. 12. **Summary of Venn diagram values:** - Only Written: $a$ - Only Practical: $b$ - Only Oral: $c$ - Written & Practical only: $d$ - Practical & Oral only: $e$ - Written & Oral only: $f = 15 + b$ - All three: $g = 4$ - None: $h = 8$ 13. **Final answer:** (i) The Venn diagram is completed with the above values. (ii) Total candidates who sat for the test is: $$\boxed{44}$$