Given two statements:

I. Solve the quadratic equation $x^2 - 9x + 20 = 0$.

II. A can complete a work in 12 days. B is 50\% more efficient than A. Find the total time taken by A and B working together to finish the work.

Compare the values obtained from I and II and choose the correct relation:

Question

Given two statements:

I. Solve the quadratic equation $x^2 - 9x + 20 = 0$.

II. A can complete a work in 12 days. B is 50\% more efficient than A. Find the total time taken by A and B working together to finish the work.

Compare the values obtained from I and II and choose the correct relation:

Accepted Answer

**Solution:**

1. Solve statement I: $x^2 - 9x + 20 = 0$

- Factorize the quadratic:
     $$x^2 - 9x + 20 = (x - 4)(x - 5) = 0$$

- Roots are $x = 4$ and $x = 5$.

- For comparison, consider the larger root $x = 5$.

2. Solve statement II:

- A completes the work in 12 days, so A's work rate is:
     $$\frac{1}{12} 	ext{ work/day}$$

- B is 50\% more efficient than A, so B's rate:
     $$1.5 	imes \frac{1}{12} = \frac{1.5}{12} = \frac{1}{8} 	ext{ work/day}$$

- Combined rate of A and B:
     $$\frac{1}{12} + \frac{1}{8} = \frac{2}{24} + \frac{3}{24} = \frac{5}{24} 	ext{ work/day}$$

- Total time taken together:
     $$\frac{1}{\frac{5}{24}} = \frac{24}{5} = 4.8 	ext{ days}$$

3. Compare the values:

- From I, the larger root is 5.

- From II, time taken is 4.8 days.

- Since $5 > 4.8$, we have $I > II$.

**Answer:** (b) I > II

**Note:** The correct answer is option (b), but the question asks to select the correct relation. Since the roots are 4 and 5, and the time from II is 4.8, the relation is $I > II$ if we consider the larger root. If the question intends to compare the roots individually, the answer could vary, but typically the larger root is considered for such comparisons.

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