Time and Work - Mathematics (SSC CGL CHSL)

When two or more people work together on a job, the total work completed per day is the sum of their individual work rates.

Let’s understand this through structured explanation and examples.

1. Combined Work Rate

If person A can complete a work in \( x \) days and person B in \( y \) days, then:

\[ \text{A's 1-day work} = \frac{1}{x}, \quad \text{B's 1-day work} = \frac{1}{y} \]

When A and B work together, their 1-day work is:

\[ \frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy} \]

Total time to finish the work together:

\[ \text{Time} = \frac{1}{\left(\frac{1}{x} + \frac{1}{y}\right)} \]

2. Example 1: Two People Working Together

Question: A can do a work in 12 days, and B can do the same work in 18 days. In how many days will they complete it together?

Step 1: A’s 1-day work = \( \frac{1}{12} \)

Step 2: B’s 1-day work = \( \frac{1}{18} \)

Step 3: Combined 1-day work =

\[ \frac{1}{12} + \frac{1}{18} = \frac{3 + 2}{36} = \frac{5}{36} \]

Step 4: Time to complete work together =

\[ \frac{1}{\left(\frac{5}{36}\right)} = \frac{36}{5} = 7.2 \text{ days} \]

Answer: They will finish the work in 7.2 days.

3. Using the LCM Method

The LCM method is useful for avoiding fractions and making calculations faster.

Let us understand this method step-by-step:

1. Assume the total work = LCM of the number of days each person takes.
2. Calculate individual work per day.
3. Add the rates to get combined work per day.
4. Divide total work by combined rate to find total days.

Example 2: Using the LCM Method

Question: A can do a work in 10 days, B in 15 days. How many days will they take to complete it together?

Step 1: LCM of 10 and 15 = 30 (Assume total work = 30 units)

Step 2:

• A’s work rate = \( \frac{30}{10} = 3 \) units/day
• B’s work rate = \( \frac{30}{15} = 2 \) units/day

Step 3: Combined rate = \( 3 + 2 = 5 \) units/day

Step 4: Time = \( \frac{30}{5} = 6 \) days

Answer: They will finish the work together in 6 days.

Example 2 (Quick-Solve with Diagram Style)

Question: A can do a work in 10 days, and B in 15 days. How many days will they take together?

       A          B
      10         15
       \         /
        \       /
         \     /
        LCM = 30
         /     \
        /       \
     3 units   2 units

Step 1: LCM of 10 and 15 = 30 (We assume total work = 30 units)

Step 2: A’s efficiency = \( \frac{30}{10} = 3 \) units/day

Step 3: B’s efficiency = \( \frac{30}{15} = 2 \) units/day

Step 4: Combined efficiency = \( 3 + 2 = 5 \) units/day

Step 5: Time to complete = \( \frac{30}{5} = 6 \) days

Answer: A and B together will finish the work in 6 days.

4. Key Takeaways

• Use the formula \( \frac{1}{x} + \frac{1}{y} \) for combined 1-day work.
• Use LCM method to simplify fractions and speed up calculations.
• For 3 or more persons, keep adding their 1-day work rates.

In the next part, we will learn how to handle situations where people join or leave work midway, which involves carefully tracking time and partial work.