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📖 Time & Work

What You’ll Learn

The core idea: Work = Rate × Time

How to think in terms of one day’s work (fraction method)

The powerful LCM–Total Work method

The concept of Efficiency and its shortcut

How to handle workers joining/leaving, alternate days, and wage problems

A weird-but-unforgettable memory trick

How to never confuse time and work again

The King’s Wall: A Hook

Imagine you are a master builder in an ancient kingdom. The King orders you to build a protective wall in 10 days. You have 20 workers. You plan, you sweat, and you finish exactly on time. The King is happy. But then he asks, “What if I gave you 30 workers tomorrow? How fast can you finish the same wall?” Your mind races... This is exactly what Time & Work is about — predicting the future when the number of hands changes. And you are about to become a prediction wizard.

🧠 Pause and close your eyes: If 20 workers take 10 days, how many days will 40 workers take for the identical wall? (Answer in mind)

I Know It Feels Tricky

You’re not alone. Most students look at “A does in 12 days, B does in 8 days” and their brain goes fuzzy. Fraction addition? Invert the answer? Multiply? Relax. The confusion is not you — it’s the way it’s usually taught. We are going to rebuild the concept from your own common sense.

Chunk 1: One Day’s Work – The Magic Ingredient

Think of the whole job as one juicy pizza. If you alone can finish that pizza in 6 days, how much do you eat in one day? Exactly 1/6 of the pizza. That’s your “one day’s work”. Everything in Time & Work orbits around this tiny number.

If your friend finishes the same pizza in 3 days, then her one day’s work is 1/3.

Together in one day you both eat 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 of the pizza. So together you will finish the whole pizza in 2 days (because 1 ÷ (1/2) = 2). See? No magic.

🧠 Active Recall: A alone can do a job in 5 days, B alone in 10 days. Without looking, find their one day’s work individually, then together, then total time. Do it mentally now.

Chunk 2: The LCM–Total Work Superpower

Fractions are great, but when numbers get messy (like 7 and 13 days), adding fractions feels like chewing glass. That’s why exam masters invented the Total Work (LCM) Method.

Step-by-step:

Take the LCM of the given days. That becomes our “Total Work” — imagine it as the number of bricks to lay.

Divide Total Work by each person’s days to get their per-day Efficiency (bricks per day).

Now you’re just playing with whole numbers, no fractions.

Example: A takes 12 days, B takes 18 days. LCM of 12 and 18 is 36. Let total work = 36 units.
A’s efficiency = 36/12 = 3 units/day.
B’s efficiency = 36/18 = 2 units/day.
Together = 5 units/day → time = 36/5 = 7.2 days. Easy numbers, happy brain.

🧠 *Try it: A = 15 days, B = 10 days. Find LCM, efficiencies, and total time. Answer: LCM 30, A eff=2, B eff=3, together 5, time 6 days.*

Chunk 3: Efficiency Ratio – The Silent Shortcut

Often data comes as “A is twice as efficient as B”. Immediately, efficiency ratio A : B = 2 : 1. And what is the relationship between efficiency and time? Efficiency is inversely proportional to time. So if A : B efficiency = 2 : 1, then time ratio A : B = 1 : 2. This one insight solves many ‘workers replaced’ problems in seconds.

🧠 Active Recall: If A is three times as efficient as B, and together they finish a job in 10 days, can you set up the ratio and find their individual days? Try before reading further.

A Weird Mnemonic That Sticks

Picture a grumpy chef named WERD. WERD stands for:

Work done = Efficiency × Rate of time? No — here’s the silly image: Work = Man × Days is the base. For multiple persons, we use M₁D₁H₁/W₁ = M₂D₂H₂/W₂ (the “man-days” equation). To never forget the formula, imagine a giant MAN-DAYS cookie (Man × Days). The cookie has layers: Hours (H) and Work (W) as the filling. When you bite, you must keep the ratio equal on both sides.

Mantra: “Men and Days are best friends; Work and Hours complete the recipe. Keep them equal on both plates, or the cookie crumbles!”

That’s the Cookie Equation: M₁D₁H₁ / W₁ = M₂D₂H₂ / W₂. Whenever a problem talks about men, days, hours, and amount of work, just set up the cookie and solve.

Chunk 4: Alternate Days and Leaving/Joining

These are just logical extensions. If A and B work on alternate days, calculate work done in a cycle (usually 2 days) and then see how many full cycles fit. Remainder work is then done by the person whose turn it is. When a worker leaves before completion, simply add the work done together until the exit, then finish with the remaining person’s work. Always fall back on the LCM-Total Work method; it makes remainders child’s play.

🧠 *Active Recall: A does a job in 20 days. He works for 5 days and leaves. B finishes the remaining in 12 days. What is B’s individual time? (Hint: Find total work = 20, A’s eff=1, work left=15, B’s eff=15/12=1.25, so B’s time=20/1.25=16 days.) Do it now.*

Chunk 5: Wages and Work-Share

Wages are divided in the ratio of work done, not time spent. Since work done = efficiency × time spent, the ratio of wages equals the ratio of (efficiency × days) for each worker. If efficiencies are given, you can directly find each person’s share of money.

🧠 *A and B work together with efficiencies 3 and 2, finish a job in 5 days. Total wage is ₹500. What is A’s share? (Work ratio = 3:2, A gets 3/5 × 500 = ₹300.) Simple!*

Summary Table: Time & Work Toolkit

🧠 2-Day Review Prompt:
Take a blank paper tomorrow and again after two days. Write down from memory:

The three core steps of the LCM method.

The crazy Cookie mnemonic and its equation.

Solve one problem each of basic, efficiency ratio, alternate days, and wages without looking at the chapter.
Your brain will thank you during the exam. You are not just learning Time & Work — you are building a problem-solving reflex. Now go own it!