Discussion

Explanation:

Let the number of students in sections A, B, C and D be a, b, c and d respectively.
Let the number of students passing in sections A, B, C and D be p, q, r and w respectively.

Total number of students passing from sections A, B and C = p + q + r = (a + b + c) × 0.4
Total number of students passing from sections A, B and D = p + q + s =  (a + b + d) × 0.45
Total number of students passing from sections A, C and D = p + r + s = (a + c + d) × 0.50
Total number of students passing from sections B, C and D = q + r + s = (b + c + d) × 0.55

Adding the 4 equation, we get

3 × (p + q + r + s) = 1.35a + 1.4b + 1.45c + 1.5d

This can be written as:
⇒ 3 × (p + q + r + s) = 1.35(a + b + c + d) + 0.05b + 0.1c + 0.15d
⇒ (p + q + r + s) = 0.45(a + b + c + d) + $\frac{(0.05\mathrm{b}+0.1\mathrm{c}+0.15\mathrm{d})}{3}$

⇒ $\frac{(\mathrm{p}+\mathrm{q}+\mathrm{r}+\mathrm{s})}{(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})}$ = 0.45 + $\frac{(0.05\mathrm{b}+0.1\mathrm{c}+0.15\mathrm{d})}{3(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})}$

⇒ $\frac{(\mathrm{p}+\mathrm{q}+\mathrm{r}+\mathrm{s})}{(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})}\times 100\%$ = 45% + $\frac{(0.05\mathrm{b}+0.1\mathrm{c}+0.15\mathrm{d})}{3(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})}\times 100$%
⇒ Overall pass percentage is definitely greater than 45%.

This can be also be written as:
⇒ 3 × (p + q + r + s) = 1.5(a + b + c + d) - (0.15a + 0.1b + 0.05c)
⇒ (p + q + r + s) = 0.5(a + b + c + d) - $\frac{(0.15\mathrm{a}+0.1\mathrm{b}+0.05\mathrm{c})}{3}$

⇒ $\frac{(\mathrm{p}+\mathrm{q}+\mathrm{r}+\mathrm{s})}{(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})}$ = 0.5 - $\frac{(0.15\mathrm{a}+0.1\mathrm{b}+0.05\mathrm{c})}{3(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})}$

⇒ $\frac{(\mathrm{p}+\mathrm{q}+\mathrm{r}+\mathrm{s})}{(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})}\times 100\%$ = 50% - $\frac{(0.15\mathrm{a}+0.1\mathrm{b}+0.05\mathrm{c})}{3(\mathrm{a}+\mathrm{b}+\mathrm{c}+\mathrm{d})}$%
⇒ Overall pass percentage is definitely less than 50%.

∴ Overall pass percentage off all 4 sections combined should be between 45% and 50% (excluding both) .

∴ Overall pass percentage cannot be option (c) 51.3%

Hence, option (c).

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