Given that $x$ women can complete a work in $2y$ days, $1.5x$ men can complete the same work in $y$ days, and $2x$ children can complete the same work in $3y$ days. Also, 8 women, 8 men, and 8 children together can complete the work in $22 \frac{1}{2}$ days, and 9 men can complete the work in $(y + 20)$ days.

If $(x - 6)$ women worked for $(y - 6)$ days and $(x - 6)$ men worked for $(y - 10)$ days, then in how many days will the remaining work be completed by $(x - 6)$ children?

Choose the correct option.

Question

Given that $x$ women can complete a work in $2y$ days, $1.5x$ men can complete the same work in $y$ days, and $2x$ children can complete the same work in $3y$ days. Also, 8 women, 8 men, and 8 children together can complete the work in $22 \frac{1}{2}$ days, and 9 men can complete the work in $(y + 20)$ days.

If $(x - 6)$ women worked for $(y - 6)$ days and $(x - 6)$ men worked for $(y - 10)$ days, then in how many days will the remaining work be completed by $(x - 6)$ children?

Choose the correct option.

Accepted Answer

**Solution:**

1. Let the total work be $W = 1440a$ units (where $a$ is a unit of work rate).

2. Work done by $(x - 6)$ women in $(y - 6)$ days:
   - Number of women = 6 (since $x - 6 = 6$)
   - Number of days = 14 (since $y - 6 = 14$)
   - Work done = $6 	imes 3a 	imes 14 = 252a$ units

3. Work done by $(x - 6)$ men in $(y - 10)$ days:
   - Number of men = 6
   - Number of days = 10
   - Work done = $6 	imes 4a 	imes 10 = 240a$ units

4. Total work completed by women and men:
   $$252a + 240a = 492a 	ext{ units}$$

5. Remaining work:
   $$1440a - 492a = 948a 	ext{ units}$$

6. Work rate of $(x - 6)$ children:
   - Number of children = 6
   - Rate per child = $a$ units/day
   - Total rate = $6a$ units/day

7. Time required to finish remaining work by children:
   $$\frac{948a}{6a} = 158 	ext{ days}$$

**Answer:** $158$ days

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