Solving the Work and Time Problem

This problem involves the concept of work and time. The fundamental idea is that the rate at which work is done is inversely proportional to the time taken. If a person takes \( T \) hours to complete a piece of work, their work rate is \( \frac{1}{T} \) of the work per hour.

Understanding Work Rates

• If a man can do a work in 30 hours, his work rate is \( \frac{1}{30} \) work per hour.
• If the man and his son together can do the same work in 20 hours, their combined work rate is \( \frac{1}{20} \) work per hour.

When two people work together, their individual work rates add up to form the combined work rate. Let's denote the time taken by the son alone to be \( x \) hours. Therefore, the son's work rate is \( \frac{1}{x} \) work per hour.

Setting up the Equation

The combined work rate of the man and his son is the sum of the man's work rate and the son's work rate:

Man's Rate + Son's Rate = Combined Rate

\( \frac{1}{30} + \frac{1}{x} = \frac{1}{20} \)

Solving for the Son's Time

Now, we need to solve this equation for \( x \), which represents the time the son takes to complete the work alone.

Subtract \( \frac{1}{30} \) from both sides of the equation:

\( \frac{1}{x} = \frac{1}{20} - \frac{1}{30} \)

To subtract the fractions on the right side, we find a common denominator. The least common multiple (LCM) of 20 and 30 is 60.

\( \frac{1}{x} = \frac{3}{60} - \frac{2}{60} \)

Now, subtract the numerators:

\( \frac{1}{x} = \frac{3 - 2}{60} \)

\( \frac{1}{x} = \frac{1}{60} \)

To find \( x \), we take the reciprocal of both sides:

\( x = 60 \)

So, the son can do the work alone in 60 hours.

Summary of Calculations

Equation: \( \frac{1}{30} + \frac{1}{x} = \frac{1}{20} \)

Solving for \( x \): \( x = 60 \)

The son working alone can do the work in 60 hours.

Revision Table: Time and Work Concepts

Additional Information: Solving Time and Work Problems

Time and work problems often involve calculating how long it takes for individuals or groups to complete a task, either alone or together. The key is to convert the time taken into a work rate. The work rate represents the fraction of the total work completed in one unit of time (e.g., per hour, per day).

Here are some common variations and tips:

• Individual Times Given, Find Combined Time: If Person A takes \( T_A \) and Person B takes \( T_B \), their combined rate is \( \frac{1}{T_A} + \frac{1}{T_B} \). The time taken together \( T_{combined} \) is \( \frac{1}{\frac{1}{T_A} + \frac{1}{T_B}} \).
• Combined Time and One Individual Time Given, Find Other Individual Time: This is the type of problem we just solved. Use the formula \( R_{total} = R_1 + R_2 \) and solve for the unknown rate, then find the time.
• Work Done in Fractions: Sometimes problems involve completing only a fraction of the work. Calculate the time needed for the full work first, then adjust for the required fraction.
• Efficiency: Efficiency is often related to work rate. Higher efficiency means a higher work rate (less time taken).

Always ensure that the time units are consistent throughout the problem (e.g., all in hours or all in days).