A can complete a piece of work in *a* days by working *m* hours per day, and B can complete the same work in *b* days by working *n* hours per day. If each of them reduces their working hours per day by 4 hours, each would take 5 days more to finish the work. Given that *m, n

Question

A can complete a piece of work in *a* days by working *m* hours per day, and B can complete the same work in *b* days by working *n* hours per day. If each of them reduces their working hours per day by 4 hours, each would take 5 days more to finish the work. Given that *m, n < 16* and *m, n, a, b* are all integers, what is the time taken if A and B start working together such that A works for *m* hours per day and B works for *n* hours per day?

Choose the correct option(s) from the following:

Accepted Answer

**Solution:**

1. Let the total work be *W* units.

2. For A:
   - Work done per day = *m* hours
   - Days taken = *a*
   - So, total work: $W = a 	imes m 	imes 	ext{rate of work per hour for A}$
   - Rate of work per hour for A = $\frac{W}{a m}$

3. For B:
   - Work done per day = *n* hours
   - Days taken = *b*
   - Rate of work per hour for B = $\frac{W}{b n}$

4. When both reduce their working hours by 4 hours:
   - A works $(m - 4)$ hours per day and takes $(a + 5)$ days
   - B works $(n - 4)$ hours per day and takes $(b + 5)$ days

5. Using the work formula for A:
   $$W = (a + 5)(m - 4) 	imes 	ext{rate of work per hour for A}$$

Since rate of work per hour for A is $\frac{W}{a m}$, equate:
   $$a m = (a + 5)(m - 4)$$

6. Similarly for B:
   $$b n = (b + 5)(n - 4)$$

7. Expanding and simplifying for A:
   $$a m = (a + 5)(m - 4) = a m - 4 a + 5 m - 20$$
   $$a m = a m - 4 a + 5 m - 20$$
   $$0 = -4 a + 5 m - 20$$
   $$5 m = 4 a + 20$$
   $$m = \frac{4 a + 20}{5}$$

8. Similarly for B:
   $$b n = (b + 5)(n - 4) = b n - 4 b + 5 n - 20$$
   $$0 = -4 b + 5 n - 20$$
   $$5 n = 4 b + 20$$
   $$n = \frac{4 b + 20}{5}$$

9. Since *m* and *n* are integers less than 16, try integer values of *a* and *b* such that *m* and *n* are integers and less than 16.

10. For example, if *a = 5*:
    $$m = \frac{4 	imes 5 + 20}{5} = \frac{40}{5} = 8$$

For *b = 5*:
    $$n = \frac{4 	imes 5 + 20}{5} = 8$$

11. Now, work done by A in one day:
    $$	ext{Work per hour} 	imes m = \frac{W}{a m} 	imes m = \frac{W}{a}$$

Similarly, work done by B in one day:
    $$\frac{W}{b}$$

12. So, combined work per day when working together:
    $$\frac{W}{a} + \frac{W}{b} = W \left( \frac{1}{a} + \frac{1}{b} ight)$$

13. Time taken to complete work together:
    $$= \frac{W}{	ext{combined work per day}} = \frac{W}{W \left( \frac{1}{a} + \frac{1}{b} ight)} = \frac{1}{\frac{1}{a} + \frac{1}{b}} = \frac{a b}{a + b}$$

14. Using the example values *a = 5* and *b = 5*:
    $$	ext{Time} = \frac{5 	imes 5}{5 + 5} = \frac{25}{10} = 2.5 	ext{ days}$$

15. Testing other values of *a* and *b* that satisfy the integer condition for *m* and *n* will give other possible times such as 5 days and $\frac{10}{3}$ days.

**Answer:** The correct options are I (5 days), III ($10/3$ days), and IV (2.5 days).

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