Time & work

Time & Work – Quick Study Notes

1. What is “Time & Work”?

When people (or machines) finish a job, we can talk about

• How long it takes (time) and
• How much of the job is done (work).

The basic idea is that more time usually means more work done, and faster people finish a job in less time.

2. Everyday Analogy

• Think of a pizza that needs to be cut.
• One person can cut the whole pizza in 4 minutes.
• Another can cut it in 6 minutes.
• If they cut together, they finish the pizza faster than either alone.

3. Core Concepts

3.1 Rate of Work (Simplified)

• If a person finishes a job in T days, his rate is 1/T of the job each day.
• Example: 4 days → rate = 1/4 of the job per day.

3.2 Work Done

• Work done = Rate × Time.
• Example: a rate of 1/4 job/day for 2 days → 1/4 × 2 = 1/2 of the job.

3.3 Adding Rates

• When two people work together, their rates add.
• Example: 1/4 + 1/6 = 5/12 of the job per day.

4. Simple Rules to Remember

5. Tiny Example

Problem:
Alice finishes a task in 4 days. Bob finishes the same task in 6 days. How long will they take if they work together?

Step‑by‑step

1. Find each rate

   • Alice: 1/4 of the job per day.
   • Bob: 1/6 of the job per day.

2. Add the rates
   1/4 + 1/6 = 3/12 + 2/12 = 5/12 of the job per day.

3. Find the time together
   Time = 1 ÷ (5/12) = 12/5 = 2.4 days.

Answer: They finish the job together in 2.4 days (about 2 days and 10 hours).

6. Another Quick Example

Problem:
A works alone for 3 days. After that, B joins and together they finish the job in 5 more days. How long would B take alone?

Solution

1. Rate of A

   • Assume the whole job = 1.
   • Work done by A in 3 days = Rate_A × 3.

2. Work left after A

   • 1 – (Rate_A × 3).

3. Together Rate

   • Let Rate_B be B’s rate.
   • Together rate = Rate_A + Rate_B.

4. Time for the remaining work

   • Time_left = 5 days →
     5 × (Rate_A + Rate_B) = Work_left.