Time and Work
Time and Work - Work efficiency, working together, alternate work, pipes and cistern problems
Learning Content
⏱️ What is period and work?
The relationship between the amount of time it takes to complete a job and the amount of that workPeriod and work(Time and Work) accounts explain.
📌 Basic concepts
| Comment | Explanation |
|---|---|
| Work | Total work done. Generally let's assume 1 unit |
| Time | Time required to complete the work (days/hours) |
| Efficiency | Amount of work done in a day/hour |
| Men | Number of people employed |
🔗 Basic relationships
W = E × T
| relationship | Explanation |
|---|---|
| If one completes the work in n days | 1 day's work = 1/n share |
| If 1 day's work = 1/n | n days to complete the entire work |
| Capacity of A : Capacity of B = m : n | Period of A : Period of B = n : m |
📊 Ratio relationships
| factor | Relationship with work | ratio |
|---|---|---|
| People (M) | More → more work | direct ratio |
| Days (D) | More → more work | direct ratio |
| Hour (H) | More → more work | direct ratio |
| Efficiency (E) | More → Less time | The opposite ratio |
🔢 LCM Method (Best Method)
It is easy to calculate the LCM of the period as total work.
1. LCM of all periods = total work
2. 1 day's work of each = total work / period
3. 1 day's work done together = addition of separate works
4. Total period = Total work / Cumulative work
Example:A - 10 days, B - 15 days
Total work = LCM(10,15) = 30 units
1 day of A = 30/10 = 3 units
1 day of B = 30/15 = 2 units
Together 1 day = 3+2 = 5 units
Total period = 30/5 =6 days
📚 Key Codes
| Code | Meaning | English |
|---|---|---|
| W | Work | Work |
| T/D | Duration / Days | Time / Days |
| E | ability | Efficiency |
| M | People | Men |
| H | hours | Hours |
🚰 Pipes and Cistern
| Type | Explanation | Code |
|---|---|---|
| Inlet | Filling the tank | + (addition) |
| Outlet | Emptying the tank | - (Subtraction) |
| Leak | Water comes out | - (Subtraction) |
💡 Things to remember
- If one finishes in n days:1 day's work = 1/n
- Capacity ↑ → Duration ↓(counter ratio)
- People ↑ → Period ↓(counter ratio)
- Working together:1/A + 1/B = 1/T
- Working against:1/A - 1/B = 1/T
- LCM method:Very simple & fast!
📐 Period and Work Formulas
All necessary formulas for TNPSC Exam!
📌 Basic formulas
| Work | W = E × T | Capacity × Duration |
| 1 day work | = 1/n | If completed in n days |
| Full working days | = 1 ÷ (1 day's work) |
👥 Working Together
1/T = 1/a + 1/b
T = (a × b) / (a + b)
| persons | formula |
|---|---|
| A + B | T = ab/(a+b) |
| A + B + C | 1/T = 1/a + 1/b + 1/c |
| A – B (opposite) | T = ab/(a-b) [a > b] |
🔄 Alternate Days
2 day's work = 1/a + 1/b
Total Days = 2 × (Total Work / 2 Days Work)
Note:Watch out who starts first!
👷 Men and Days
M₁ × D₁ × H₁ = M₂ × D₂ × H₂(including hours)
| situation | formula |
|---|---|
| Work is equal | M₁D₁ = M₂D₂ |
| Work varies | M₁D₁/W₁ = M₂D₂/W₂ |
| Including hours | M₁D₁H₁/W₁ = M₂D₂H₂/W₂ |
📈 Efficiency Formulas
If A:B capacity = m:n, then A:B duration = n:m
| A is x% more efficient than B | Duration of A = Duration of B × 100/(100+x) |
| A is x% less efficient than B | Duration of A = Duration of B × 100/(100-x) |
💰 Wages
A:B wage = A:B job = A:B skill
Example:A - 10 days, B - 15 days, total pay ₹5000
Capacity ratio = 1/10 : 1/15 = 3:2
Salary of A = 5000 × 3/5 = ₹3000
Salary of B = 5000 × 2/5 = ₹2000
🚰 Pipes & Cistern
| situation | formula |
|---|---|
| 2 filling tubes | 1/T = 1/a + 1/b |
| 1 fill + 1 empty | 1/T = 1/a - 1/b (full > empty) |
| If there is leakage | 1/T = 1/Fill - 1/Leak |
Drain pipe / leakage → - (Negative)
🔢 LCM method formulas
Step 2:1 day's work of each = total work / period
Step 3:Cumulative 1 day's work = sum/subtraction of separate works
Step 4:Duration = Total Work / Cumulative 1 Day Work
⚡ Special formulas
| situation | formula |
|---|---|
| A+B = T₁, A+C = T₂, B+C = T₃ | A+B+C = 2/(1/T₁ + 1/T₂ + 1/T₃) |
| A alone | 1/A = (1/T₁ + 1/T₂ - 1/T₃)/2 |
| If A works for n days | Remaining work = 1 - n/a |
| Duration of x stock job | T = x × total duration |
📊 Major relationships table
| A days | B days | together (ab/a+b) |
|---|---|---|
| 2 | 2 | 1 |
| 3 | 6 | 2 |
| 4 | 4 | 2 |
| 5 | 10 | 10/3 = 3.33 |
| 6 | 12 | 4 |
| 10 | 15 | 6 |
| 12 | 18 | 36/5 = 7.2 |
| 15 | 20 | 60/7 ≈ 8.57 |
📝 Tenses and Work - Solved Examples
10 Important Questions for TNPSC Exam
Question 1: Basic – Two together
Question:A completes a piece of work in 12 days and B in 18 days. If both of them work together, in how many days will they finish?
Solution (LCM method):
Total work = LCM(12, 18) = 36 units
1 day's work of A = 36/12 = 3 units
1 day's work of B = 36/18 = 2 units
Together 1 day = 3 + 2 = 5 units
Period = 36/5 =7.2 days = 7 days 4 hours 48 minutes
Formula Method:T = ab/(a+b) = (12×18)/(12+18) = 216/30 =36/5 days
Question 2: Three together
Question:A - 10 days, B - 12 days, C - 15 days. If all three join?
Solution (LCM method):
Total work = LCM(10, 12, 15) = 60 units
1 day of A = 60/10 = 6 units
1 day of B = 60/12 = 5 units
1 day of C = 60/15 = 4 units
Together 1 day = 6 + 5 + 4 = 15 units
Period = 60/15 =4 days
Question 3: If one leaves
Question:A and B together finish in 6 days. A alone finishes in 10 days. A leaves after working together for 4 days. In how many days will B complete the remaining work?
Solution:
A+B together 6 days, A alone 10 days
B alone: 1/B = 1/6 - 1/10 = (5-3)/30 = 2/30 = 1/15
∴ B alone will finish in 15 days
Total work = LCM(6, 10, 15) = 30 units
A+B 1 day = 30/6 = 5 units
In 4 days = 4 × 5 = 20 units (completed)
Remainder = 30 - 20 = 10 units
1 day of B = 30/15 = 2 units
Required for B = 10/2 =5 days
Question 4: Shift work
Question:A - 20 days, B - 30 days. In how many days will it be completed if A starts from the first day and works in shifts?
Solution:
Total work = LCM(20, 30) = 60 units
1 day of A = 60/20 = 3 units
1 day of B = 60/30 = 2 units
In 2 days (A+B) = 3 + 2 = 5 units
60 units to complete = 60/5 × 2 = 24 days
But let's check:
In 22 days = 11 rounds = 55 units
23rd day A work = 3 units, total = 58
24th Day B Work = 2 units, Total = 60 ✓
24 days
Question 5: Persons and Days
Question:15 men complete a piece of work in 20 days. How many people are needed to finish in 10 days?
Solution:
M₁ × D₁ = M₂ × D₂
15 × 20 = M₂ × 10
M₂ = (15 × 20) / 10 = 300/10 =30 people
Question 6: Skill difference
Question:A is 50% more efficient than B. If B completes a work in 18 days, in how many days will A complete it?
Solution:
Efficiency of A = Efficiency of B + 50% = 150%
Capacity ratio A:B = 150:100 = 3:2
Period ratio A:B = 2:3 (inverse ratio)
If B = 18 days, A:B = 2:3
A = 18 × 2/3 =12 days
Question 7: Pipe and tank
Question:A pipe fills the tank at 12 o'clock. B tube empties at 18 hours. If both are open, at what time will it be full?
Solution:
Total capacity = LCM(12, 18) = 36 units
1 hour of A = +36/12 = +3 units (complements)
1 hour of B = -36/18 = -2 units (empties)
Together 1 hour = 3 - 2 = +1 unit
Period = 36/1 =36 hours
Question 8: Leak
Question:The tap fills the tank at 6 o'clock. If there is a leak it fills up at 8 o'clock. At what time will the leak clear itself?
Solution:
Pipe 1 hour = 1/6
Pipe + leak = 1/8
Leakage = 1/6 - 1/8 = (4-3)/24 = 1/24
To empty the leak =24 hours
Question 9: Pay distribution
Question:A will complete the work in 6 days, B in 8 days. How much for A if they work together and earn ₹2800?
Solution:
Capacity of A : Capacity of B = 1/6 : 1/8
= 8 : 6 = 4 : 3
Total share = 4 + 3 = 7
Wages of A = 2800 × 4/7 =₹1600
Salary of B = 2800 × 3/7 = ₹1200
Question 10: Additional persons
Question:20 men will complete the work in 30 days. If 5 men leave after 10 days, in how many days will the remaining work be completed?
Solution:
Total work = 20 × 30 = 600 man-days
Completed in 10 days = 20 × 10 = 200 man-days
Remaining work = 600 - 200 = 400 man-days
Remaining persons = 20 - 5 = 15 persons
Days required = 400/15 =26⅔ days ≈ 27 days
📚 Additional practice questions
- If A - 15 days, B - 20 days, together =60/7 ≈ 8.57 days
- 12 persons 18 day → 9 persons =24 days
- Pipe 4 o'clock fill, 6 o'clock empty → together =12 o'clock
- A:B capacity = 5:3, B = 15 day → A =9 days
- A 10 days, B 15 days, ₹3000 → A wages =₹1800
⚡ Period and Work - Crossroads
Super Tricks to Save Time in TNPSC Exam!
🚀 Trick 1: LCM method - exponential solution
1. Total work = LCM (periods)
2. 1 day's work = total work / period
3. Together = joint, against = waste
Example:A - 6 days, B - 8 days
Work = LCM(6,8) = 24
A = 4, B = 3, together = 7
Period = 24/7 =3 3/7 days
🚀 Trick 2: Two together - direct formula
"Multiplication Division Addition"
| a | b | Together with T |
|---|---|---|
| 2 | 3 | 6/5 = 1.2 |
| 3 | 6 | 18/9 = 2 |
| 4 | 6 | 24/10 = 2.4 |
| 5 | 10 | 50/15 = 10/3 |
| 6 | 12 | 72/18 = 4 |
| 10 | 15 | 150/25 = 6 |
🚀 Trick 3: Find B alone
B alone = (a × T) / (a – T)
Example:A+B = 6 days, A = 10 days
B = (10 × 6) / (10 - 6) = 60/4 =15 days
🚀 Trick 4: Converter works - quick method
Calculate 2 day's work → Calculate complete rounds → Check balance
Quick Method:
Work = LCM, 2 days work = a + b
Rounds = Work / (a+b) × 2
🚀 Trick 5: The man-day formula
"Total work = persons × days" (constant)
Example:10 people 12 days → 15 people how many days?
10 × 12 = 15 × D₂
D₂ = 120/15 =8 days
🚀 Trick 6: Efficiency Ratio = Inverse of Time
Example:
A = 10 days, B = 15 days
Capacity A:B = 15:10 = 3:2
Wages A:B = 3:2
🚀 Trick 7: % Capacity → Duration
Duration of A = Duration of B × 100/(100+x)
A is x% less efficient than B:
Duration of A = Duration of B × 100/(100-x)
| Efficiency % | Period ratio |
|---|---|
| 25% more | 4/5 = 0.8 times |
| 50% more | 2/3 times |
| 100% more (double) | 1/2 times |
🚀 Trick 8: Pipe Account Shortcut
Vacancy / leakage → - (Negative)
Do it like a working account!
Example:Fill 4 hours, empty 6 hours
Work = LCM(4,6) = 12
Fill = +3, Empty = -2
Together = 3-2 = +1
Period = 12/1 =12 o'clock
🚀 Trick 9: Rest work account
Completed work = n/a
Remaining work = 1 - n/a = (a-n)/a
Example:A completes in 12 days and works in 4 days
Done = 4/12 = 1/3
Remainder = 1 - 1/3 = 2/3
🚀 Trick 10: Duration for x stock job
Example:A 15 day full work, 2/5 part work?
Period = 15 × 2/5 =6 days
📊 TNPSC Frequently Asked Categories
| Question type | Crossroads |
|---|---|
| Two together | T = ab/(a+b) |
| Three together | LCM method |
| If one leaves | Calculate the remaining work |
| man-day | M₁D₁ = M₂D₂ |
| Tubular tank | Fill +, Empty - |
| Wages | Efficiency ratio |
💡 Important memory tips
- 1 day's work = 1/n(if completed in n days)
- Together: 1/a + 1/b
- Against: 1/a - 1/b
- T = ab/(a+b)- Two together
- M₁D₁ = M₂D₂- man-day
- Capacity ↑ → Duration ↓
- LCM method = fast solution!
- Pipe: Fill +, Empty/Drain -
🎯 Quick reference table
| A day | B day | along with | LCM |
|---|---|---|---|
| 2 | 4 | 4/3 | 4 |
| 3 | 6 | 2 | 6 |
| 4 | 12 | 3 | 12 |
| 5 | 20 | 4 | 20 |
| 6 | 12 | 4 | 12 |
| 8 | 12 | 24/5 | 24 |
| 10 | 15 | 6 | 30 |
| 12 | 18 | 36/5 | 36 |
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