Understanding Time and Work Problems: Women vs Boys

This problem involves the concept of Time and Work, specifically dealing with different types of workers (women and boys) and their respective work rates. The key is to establish a relationship between the work rates of women and boys and then use the given information to find the total work required.

Calculating Total Work Capacity

We are given that a boy can finish the work in 60 days. If one boy works for 60 days, the total amount of work done can be considered as 60 'boy-days'.

Total Work $= 1 \text{ boy} \times 60 \text{ days} = 60 \text{ boy-days}$

Relating Work Rates of Women and Boys

The problem states that 4 women or 6 boys can finish the work in the same number of days. Let this number of days be $D$.

Work done by 4 women in $D$ days = Total Work

Work done by 6 boys in $D$ days = Total Work

Since both groups finish the same work in the same number of days:

Work done by 4 women = Work done by 6 boys (in the same time period)

Let $W_w$ be the work rate of one woman (work done per day) and $W_b$ be the work rate of one boy (work done per day).

$4 \times W_w = 6 \times W_b$

This equation shows the relationship between the collective work rate of 4 women and 6 boys. We can simplify this to find the relationship between individual work rates:

$W_w = \frac{6}{4} W_b = \frac{3}{2} W_b$

This means one woman works $\frac{3}{2}$ times as fast as one boy. Equivalently, 1 woman's work rate is equal to the work rate of 1.5 boys.

Expressing Total Work in Woman-Days

We know the total work is 60 boy-days. We can convert this to an equivalent amount of work in 'woman-days'.

Since 1 boy's work rate is $W_b$ and 1 woman's work rate is $W_w = \frac{3}{2} W_b$, 1 boy-day is equivalent to the amount of work a boy does in one day. To find its equivalent in woman-days, we can use the relationship $W_b = \frac{2}{3} W_w$.

Work done by 1 boy in 1 day = $1 \text{ boy} \times 1 \text{ day} = 1 \text{ boy-day}$.

Work done by 1 woman in 1 day = $1 \text{ woman} \times 1 \text{ day} = 1 \text{ woman-day}$.

From $4 \times W_w = 6 \times W_b$, we can say that the work done by 4 women in a day is equal to the work done by 6 boys in a day.

So, 4 woman-days are equivalent to 6 boy-days.

1 woman-day $= \frac{6}{4}$ boy-days $= \frac{3}{2}$ boy-days.

Alternatively, 1 boy-day $= \frac{4}{6}$ woman-days $= \frac{2}{3}$ woman-days.

Total Work $= 60 \text{ boy-days}$.

Let's convert this total work into woman-days:

Total Work $= 60 \times (1 \text{ boy-day}) = 60 \times \left(\frac{2}{3} \text{ woman-day}\right) = \frac{60 \times 2}{3} \text{ woman-days} = \frac{120}{3} \text{ woman-days} = 40 \text{ woman-days}$.

The total work is 40 woman-days.

Finding Days for 5 Women

We need to find the number of days it takes for 5 women to finish the total work (40 woman-days), working together every day.

Let the number of days required be $D_5$.

Work done by 5 women in $D_5$ days = Total Work

Number of women $\times$ Number of days = Total Work (in woman-days)

$5 \text{ women} \times D_5 \text{ days} = 40 \text{ woman-days}$

$5 \times D_5 = 40$

To find $D_5$, we divide the total work by the number of women:

$D_5 = \frac{40}{5}$

$D_5 = 8$ days.

Therefore, 5 women can finish the work in 8 days.

Summary of Steps:

1. Calculated the total work based on the boy's time: 60 boy-days.
2. Established the equivalence between women's and boys' work rates: 4 women = 6 boys.
3. Used the equivalence to convert total work into woman-days: 40 woman-days.
4. Calculated the time taken by 5 women using the total work in woman-days.

Revision Table: Time and Work Concepts

Additional Information: Work and Time Efficiency

Problems involving work and time often assume that all workers of the same type have equal efficiency and that efficiency remains constant throughout the work. The total work is considered a fixed quantity.

• Efficiency: Efficiency is inversely proportional to the time taken to complete a certain amount of work. If A is twice as efficient as B, A takes half the time B takes to do the same work. In our problem, since 1 woman works as fast as 1.5 boys, a woman is 1.5 times more efficient than a boy.
• Multiple Workers: If $M$ workers of the same efficiency can do a job in $D$ days, then $M \times D$ is the total work unit (worker-days). If you change the number of workers to $M'$, the new time $D'$ will be such that $M' \times D' = M \times D$. So, $D' = (M \times D) / M'$. This assumes the work rate per worker is constant.
• Combining Workers: When different types of workers are involved, like women and boys here, their work capacities are made equivalent. We found that the work rate of 4 women equals the work rate of 6 boys. This means 4 women are equivalent to 6 boys in terms of productivity over any given period. This equivalence is crucial for solving such problems.