20 men and 35 women can complete a piece of work in $y$ and $y - 3$ days respectively, while 15 men and 20 women can complete the same piece of work in $x - 4$ and $x$ days respectively. If each man is 60\% more efficient than each woman, find the difference between $x$ and $y$.

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Accepted Answer

**Solution:**

1. Let the efficiency of each woman be $w$. Then the efficiency of each man is $1.6w$ (60\% more efficient).

2. Total work can be expressed in terms of efficiency and time. Since work = efficiency $	imes$ time,

- For 20 men and 35 women working together for $y$ days:
     $$	ext{Work} = (20 	imes 1.6w + 35 	imes w) 	imes y = (32w + 35w) y = 67w y$$

- For 20 men and 35 women working together for $y - 3$ days (women only):
     $$	ext{Work} = 35w (y - 3)$$

Since both represent the same work, equate:
   $$67w y = 35w (y - 3)$$

3. Simplify the above equation:
   $$67 y = 35 y - 105$$
   $$67 y - 35 y = -105$$
   $$32 y = -105$$

This leads to a contradiction (negative value), so the interpretation needs correction. Actually, the problem states 20 men and 35 women can complete the work in $y$ and $y - 3$ days respectively, meaning:

- 20 men alone complete the work in $y$ days
   - 35 women alone complete the work in $y - 3$ days

4. Calculate work done per day by men and women:

- Work done per day by 20 men = $\frac{1}{y}$
   - Work done per day by 35 women = $\frac{1}{y - 3}$

5. Express these in terms of $w$:

- 20 men efficiency = $20 	imes 1.6w = 32w$
   - 35 women efficiency = $35w$

6. Since work done per day is efficiency, equate:

$$32w = \frac{1}{y} \quad \Rightarrow \quad w = \frac{1}{32 y}$$

$$35w = \frac{1}{y - 3} \quad \Rightarrow \quad w = \frac{1}{35 (y - 3)}$$

7. Equate the two expressions for $w$:

$$\frac{1}{32 y} = \frac{1}{35 (y - 3)}$$

Cross multiply:

$$35 (y - 3) = 32 y$$

$$35 y - 105 = 32 y$$

$$35 y - 32 y = 105$$

$$3 y = 105$$

$$y = 35$$

8. Similarly, for 15 men and 20 women:

- 15 men complete work in $x - 4$ days
   - 20 women complete work in $x$ days

9. Work done per day:

- 15 men efficiency = $15 	imes 1.6w = 24w$
   - 20 women efficiency = $20w$

So,

$$24 w = \frac{1}{x - 4}$$

$$20 w = \frac{1}{x}$$

10. Express $w$ from both:

$$w = \frac{1}{24 (x - 4)}$$

$$w = \frac{1}{20 x}$$

11. Equate:

$$\frac{1}{24 (x - 4)} = \frac{1}{20 x}$$

Cross multiply:

$$20 x = 24 (x - 4)$$

$$20 x = 24 x - 96$$

$$24 x - 20 x = 96$$

$$4 x = 96$$

$$x = 24$$

12. Finally, find the difference:

$$x - y = 24 - 35 = -11$$

The difference in absolute value is $11$.

**Answer:** $11$

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