Time and Work

Work refers to any job or task completed over a period of time. The Time & Work concept deals with calculating the time required to complete a job or the work done in a given time by one or more workers.

Basic Relation:

Work=Rate×Time\text{Work} = \text{Rate} \times \text{Time}

Where rate is the amount of work done per unit time (e.g., per day, per hour).

Alternatively,

Time=WorkRateandRate=WorkTime\text{Time} = \frac{\text{Work}}{\text{Rate}} \quad \text{and} \quad \text{Rate} = \frac{\text{Work}}{\text{Time}}

Key Idea:

If a person can complete a job in 10 days, then their work rate is 110\frac{1}{10} of the job per day.

2. Key Formulas & Shortcuts

One Person:

  • If A can finish a job in x days, A’s 1 day work is 1x\frac{1}{x}

Multiple People Working Together:

  • If A and B can do a work in x and y days respectively:

    Their combined 1 day work=1x+1y\text{Their combined 1 day work} = \frac{1}{x} + \frac{1}{y}

  • Time to finish the work together:

    Time=xyx+y\text{Time} = \frac{xy}{x + y}

Work & Efficiency:

If A is twice as efficient as B, then:

  • A’s rate = 2 units/day, B’s = 1 unit/day
  • A does the same work in half the time B takes

Alternate Work:

If A and B work on alternate days:

  • Total work is done based on summing work done on individual days
  • Use LCM of days taken to find total work when needed

Pipes & Cisterns Analogy (Preview):

  • Inlets: positive work rate
  • Outlets: negative work rate

3. Conceptual Tips & Common Mistakes

  • LCM Trick: When multiple people are involved, assume total work = LCM of their individual times. It simplifies calculations.
  • Efficiency vs Days: Efficiency ∝ 1 / Time. If A is 25% more efficient than B, A takes 20% less time.
  • Beware of “work left” phrasing: Don’t assume what “remaining work” means—clarify with fractions.
  • Alternating workdays: Start carefully with Day 1 and alternate manually if needed.

4. Visual Explanation

Example: Work Timeline

Let’s say:

  • A finishes a job in 5 days → 15\frac{1}{5} work/day
  • B in 10 days → 110\frac{1}{10}

Combined rate: 15+110=310\frac{1}{5} + \frac{1}{10} = \frac{3}{10}
→ They’ll finish 3/10 of the work per day.
So, entire work will take 13/10=103=3.33\frac{1}{3/10} = \frac{10}{3} = 3.33 days.

This can be visualized using a stacked bar showing how much each person contributes per day until 1 unit of work is complete.

5. Solved Examples

Example 1: Basic One Worker

Q: A can do a work in 12 days. How much work does A do in 4 days?

A:
A’s 1-day work = 112\frac{1}{12}
In 4 days:

412=13 of the work\frac{4}{12} = \frac{1}{3} \text{ of the work}

Example 2: Two Workers Together

Q: A can do a job in 20 days, B in 30 days. Working together, how many days will they take?

A:
LCM of 20 and 30 = 60 (Total work)
A’s 1 day work = 6020=3\frac{60}{20} = 3, B’s = 6030=2\frac{60}{30} = 2
Combined = 5 units/day

Time=605=12 days\text{Time} = \frac{60}{5} = 12 \text{ days}

Example 3: One Starts, Another Joins

Q: A can complete work in 10 days. B in 20 days. A works alone for 2 days, then B joins. How many days in total to finish?

A:
A’s 1 day work = 110\frac{1}{10}, B’s = 120\frac{1}{20}
Work in 2 days = 2×110=152 \times \frac{1}{10} = \frac{1}{5}
Remaining = 1−15=451 - \frac{1}{5} = \frac{4}{5}

Combined rate = 110+120=320\frac{1}{10} + \frac{1}{20} = \frac{3}{20}
Time = 4/53/20=163=5.33 days\frac{4/5}{3/20} = \frac{16}{3} = 5.33 \text{ days}

Total time = 2+5.33=7.332 + 5.33 = 7.33 days

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