How to Solve: Time and Work

How to Solve: Time and Work

(For Students Who Want to Ace Their Exam & Teachers Who Need a Ready-to-Record Script)

Introduction

"Imagine you’re given 3 workers and a deadline—how do you figure out if they’ll finish the job on time? This is exactly what ‘Time and Work’ problems test, and mastering them can save you 5+ marks on your exam!

What You Need To Know First

Before diving into Time and Work, ensure you understand: 1. Unitary Method – Finding the value of one unit (e.g., if 5 workers finish a job in 10 days, how much does 1 worker do in 1 day?). 2. Fractions & LCM – Adding/subtracting fractions and finding the Least Common Multiple (LCM) for efficiency. 3. Basic Algebra – Solving simple equations (e.g., if A’s work rate + B’s work rate = combined rate).

Key Vocabulary

Formulas To Know

Step-by-Step Method

Step 1: Identify the Total Work

• Assume the total work is 1 unit (e.g., "painting a house" = 1 job).
• If the problem gives a specific quantity (e.g., "digging 100 meters"), treat that as the total work.

Step 2: Find Individual Work Rates

• Work Rate = Work / Time
• If Worker A takes 5 days to finish 1 job, their rate = 1/5 per day.
• If Worker B takes 10 days, their rate = 1/10 per day.

Step 3: Calculate Combined Work Rate (If Working Together)

• Combined Rate = Sum of Individual Rates
• A + B together: 1/5 + 1/10 = 3/10 per day.

Step 4: Find Time Taken Together

• Time = Work / Rate
• If combined rate = 3/10 per day, time to finish 1 job = 1 / (3/10) = 10/3 days ≈ 3.33 days.

Step 5: Adjust for Efficiency (If Given)

• If Worker A is twice as efficient as Worker B:
• Let B’s rate = x, then A’s rate = 2x.
• Combined rate = x + 2x = 3x.
• Time = 1 / 3x.

Step 6: Use LCM for Complex Problems (Optional Shortcut)

• If A takes 6 days, B takes 4 days, find LCM of 6 and 4 = 12.
• Assume total work = 12 units (LCM).
• A’s rate = 12/6 = 2 units/day.
• B’s rate = 12/4 = 3 units/day.
• Combined rate = 2 + 3 = 5 units/day.
• Time = 12 / 5 = 2.4 days.

Step 7: Solve for Unknowns (If Needed)

• If the problem asks for partial work (e.g., "How much work is done in 2 days?"), multiply rate by time.
• Combined rate = 3/10 per day.
• Work in 2 days = (3/10) × 2 = 6/10 = 3/5 of the job.

Worked Examples

Example 1 – Basic (Single Worker)

Problem: If Ramesh can complete a job in 8 days, how much work does he do in 3 days?

Solution: 1. Total work = 1 job. 2. Ramesh’s rate = 1/8 per day. 3. Work in 3 days = Rate × Time = (1/8) × 3 = 3/8 of the job.

What we did and why: - We used Work = Rate × Time to find partial work. - Since Ramesh does 1/8 per day, in 3 days he does 3/8.

Example 2 – Medium (Two Workers, Different Rates)

Problem: A can finish a job in 12 days, B in 6 days. How long will it take if they work together?

Solution: 1. Total work = 1 job. 2. A’s rate = 1/12 per day. 3. B’s rate = 1/6 per day. 4. Combined rate = 1/12 + 1/6 = 1/12 + 2/12 = 3/12 = 1/4 per day. 5. Time = Work / Rate = 1 / (1/4) = 4 days.

What we did and why: - We added individual rates to get the combined rate. - Then used Time = Work / Rate to find the total time.

Example 3 – Exam Style (Efficiency & Partial Work)

Problem: P is twice as efficient as Q. Together, they finish a job in 6 days. How long will Q take alone?

Solution: 1. Let Q’s rate = x per day. 2. P’s rate = 2x per day (since P is twice as efficient). 3. Combined rate = x + 2x = 3x per day. 4. Total work = Rate × Time = 3x × 6 = 18x. 5. Q alone: Time = Work / Rate = 18x / x = 18 days.

What we did and why: - We used efficiency ratios to express P’s rate in terms of Q’s. - Combined rate gave us the total work (18x). - Finally, we found Q’s individual time by dividing work by Q’s rate.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in—here’s what you need to remember for Time and Work problems:

1. Assume total work = 1 unit (unless given otherwise).
2. Find individual rates (Work / Time).
3. Add rates for combined work (never add times!).
4. Time = Work / Rate—this is your go-to formula.
5. Efficiency? Express rates in ratios (e.g., if A is twice as fast as B, A’s rate = 2 × B’s rate).
6. LCM shortcut—if times are messy, assume total work = LCM of individual times.

Common traps? Watch for workers leaving early, negative work (like leaks), and fractional workers. Always double-check: Did I add rates or times? Did I account for efficiency?

Now go crush those problems—you’ve got this!