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Basic Time & Work

ITBP Constable · Quantitative Aptitude

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Core Concept

Read this first — the foundation of the topic

→Core Concept

Work is always considered as 1 unit (complete job). If A can complete work in 10 days, A's rate of work is 1/10 per day. This means A completes 1/10th of the total work each day

💡Key Rules

Work = Rate × Time. If multiple people work together, their rates add up. Total work is always 1 unit. Rate of work = 1/Time taken to complete the job

🔑Basic Formulas

- If A completes work in 'a' days, A's one day work = 1/a - If A and B together complete work in 'd' days, (1/a + 1/b) = 1/d - Time taken by A and B together = (a × b)/(a + b) days - If A completes work in 'x' days and B in 'y' days, working together they complete in xy/(x+y) days

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Exam Patterns

What examiners ask — read before attempting PYQs

📋Common patterns include

two people working together, one person leaving midway, comparing efficiencies, and alternative work scenarios

⚡Shortcut Trick - LCM Method

Take LCM of all time periods given. This becomes total work. Then find individual rates by dividing LCM by individual time. This eliminates fractions and makes calculations faster

✏️Worked Example 1

1

A's one day work = 1/12

2

B's one day work = 1/18

3

Combined one day work = 1/12 + 1/18 = (3+2)/36 = 5/36

4

Time taken together = 36/5 = 7.2 days Alternative (Shortcut): Time = (12 × 18)/(12 + 18) = 216/30 = 7.2 days Worked Example 2: A and B together can complete work in 6 days. A alone can complete it in 10 days. In how many days can B alone complete the work?

1

Combined one day work = 1/6

2

A's one day work = 1/10

3

B's one day work = 1/6 - 1/10 = (5-3)/30 = 2/30 = 1/15

4

B alone can complete work in 15 days Shortcut Formula: If A and B together complete work in T days and A alone in A days, then B alone takes: B = (A × T)/(A - T) days Applying: B = (10 × 6)/(10 - 6) = 60/4 = 15 days Most Common Trap: Students often add time instead of adding rates. Remember - when people work together, their RATES add up, not their TIME. If A takes 4 days and B takes 6 days, together they DON'T take 4+6=10 days. Instead, calculate 1/4 + 1/6 to find combined rate

⚠️Another Major Mistake

Forgetting that work is always 1 unit. Students sometimes assume work quantity changes, leading to wrong calculations. Always treat complete work as 1 unit regardless of the scenario

⚡Efficiency Trick

If A is twice as efficient as B, then A takes half the time B takes. If A:B efficiency ratio is 2:3, then their time ratio is 3:2 (inverse relationship)

→Negative Work Concept

Sometimes one person does work while another undoes it (like filling and emptying a tank). In such cases, subtract the rates instead of adding them.

🔑 Key Points

Work is always considered as 1 complete unit in all problems
Rate of work = 1/Time taken to complete the job alone
When working together, rates add up: 1/a + 1/b = 1/combined time
Combined time formula: (a × b)/(a + b) where a, b are individual times
LCM method eliminates fractions - take LCM of all given times as total work
Efficiency and time have inverse relationship - higher efficiency means less time
If A is n times efficient than B, then A takes 1/n times the time B takes
In problems with destructive work, subtract rates instead of adding
Never add individual times to get combined time - this is the biggest trap
Work = Rate × Time is the fundamental equation for all variations

📌 Exam Facts

Combined work formula: Time = (a × b)/(a + b) days
If A and B work together for T days and A alone for remaining, use: 1 = T(1/a + 1/b) + remaining_time(1/a)
Efficiency ratio of 2:3 means time ratio of 3:2
If A completes work in 'a' days, A's rate = 1/a work per day
LCM method: Total work = LCM of all individual times
For pipes: Inlet rate positive, outlet rate negative
Work done = Rate × Time, where total work = 1 unit
If A can do work in x days and B in y days, together they need xy/(x+y) days

🚀 60-Second Revision

Formula: Combined time = (a × b)/(a + b) for two people working together
Remember: Add RATES, never add TIME when people work together
Trap: Don't assume 4 days + 6 days = 10 days together
Quick method: Use LCM of times as total work to avoid fractions
Efficiency rule: Higher efficiency = Less time (inverse relation)
Standard approach: Find individual rates, add them, then find combined time
Check: Combined time should always be less than individual times

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