Calculating Work Hours to Complete a Task Faster

This problem involves understanding the relationship between the number of days taken to complete a task and the number of hours worked per day. When the total amount of work remains constant, reducing the number of days requires increasing the effort per day, such as working more hours.

This is a classic example of inverse variation. If one quantity increases, the other decreases proportionally, such that their product remains constant. Here, the quantities are 'hours worked per day' and 'number of days to complete the work'.

Setting up the Work and Time Problem

Let's define the variables given in the problem:

• Let $H_1$ be the number of hours Vaibhav works per day in the first scenario. $H_1 = 15$ hours.
• Let $D_1$ be the number of days taken to complete the work in the first scenario. $D_1 = 35$ days.
• Let $H_2$ be the number of hours Vaibhav should work per day in the second scenario (which we need to find).
• Let $D_2$ be the number of days targeted to complete the work in the second scenario. $D_2 = 25$ days.

The total work done can be considered constant in both scenarios. The total work is proportional to the product of the hours worked per day and the number of days.

Total Work = (Hours per Day) × (Number of Days)

Since the total work is the same in both cases, we can write the equation:

$\text{Total Work}_1 = \text{Total Work}_2$

$H_1 \times D_1 = H_2 \times D_2$

Solving for the Required Hours per Day

Now, we substitute the given values into the equation:

$15 \text{ hours/day} \times 35 \text{ days} = H_2 \text{ hours/day} \times 25 \text{ days}$

We need to solve for $H_2$. To isolate $H_2$, we divide both sides of the equation by 25 days:

$H_2 = \frac{15 \times 35}{25}$

Let's calculate the value:

$H_2 = \frac{525}{25}$

Performing the division:

$H_2 = 21$

So, Vaibhav should work 21 hours a day to finish the work in 25 days.

Understanding the Result

Working fewer days (25 instead of 35) requires working more hours per day (21 instead of 15), which is consistent with the concept of inverse variation for a fixed amount of work.

For the total work to be the same:

$H_2 \times 25 = 525$

$H_2 = \frac{525}{25} = 21$

Revision Table: Work and Time Concepts

Here's a quick review of key concepts related to work and time problems:

• Inverse Variation: When the total work is constant, the number of workers (or hours per day) is inversely proportional to the time taken to complete the work. More workers (or hours per day) mean less time.
• Direct Variation: If the number of workers (or hours per day) is constant, the amount of work done is directly proportional to the time taken. More time means more work.
• Total Work: Often represented as the product of the number of workers (or efficiency) and the time taken. Total Work = Rate × Time. In this problem, Rate can be seen as hours per day.

Additional Information: Inverse Proportionality

Inverse proportionality describes a relationship between two variables where their product is a constant. If $x$ and $y$ are inversely proportional, then $x \times y = k$, where $k$ is a constant.

In our problem:

• $H$ (hours per day) is inversely proportional to $D$ (number of days) for a fixed amount of work.
• So, $H \times D = \text{Constant Work}$.

Using this principle, $H_1 D_1 = H_2 D_2$ is the direct application of inverse proportionality.

Understanding inverse variation helps solve many problems involving work, time, speed, and distance where a quantity is fixed (like total work or total distance).