📘 Time and Work – Complete Notes
1. Basic Concept
- Work = Task to be completed
- Time = Duration required to finish work
- Efficiency = Ability to complete work in given time
Formula: \[ Work = Efficiency \times Time \]
If a person completes a work in \(N\) days → One day’s work = \[ \frac{1}{N} \]
2. Important Formulas
-
Individual Work
A finishes in \(x\) days → One day’s work = \(\frac{1}{x}\)
B finishes in \(y\) days → One day’s work = \(\frac{1}{y}\) -
Combined Work (Two Persons)
A and B together = \(\frac{1}{x} + \frac{1}{y}\)
Time = \(\frac{xy}{x+y}\) -
Three Persons
A, B, C take \(x, y, z\) days → Time = \[ \frac{xyz}{xy + yz + zx} \] -
Efficiency Relation
If A is \(n\) times efficient as B → Time ratio = \(1:n\) -
LCM Method (Shortcut)
Assume Total Work = LCM of given days. Efficiency = \(\frac{\text{Total Work}}{\text{Time}}\)
3. Solved Examples
Example 1
A = 12 days, B = 18 days. Find the number of days in which together they will completes in ?
LCM = 36 (work = 36 units).
A = \( \frac{36}{12} = 3 \), B = \( \frac{36}{18} = 2\)
Together = 5 units/day → Time = \( \frac{36}{5} = 7.2 \) days
Example 2
A is twice efficient as B. Together = 18 days then find A alone finish the Work in ?
Let B = 1 unit/day, A = 2 units/day.
Together = 3 units/day → Work = \(18 \times 3 = 54\).
A alone = \( \frac{54}{2} = 27 \) days
Example 3
A = 15 days, B = 20 days. They work 4 days together, then A leaves. find the total work completes in how many days ?
LCM = 60 → A = \( \frac{60}{15} = 4 \), B = \( \frac{60}{20} = 3\).
4 days work = \( (4+3) \times 4 = 28\).
Remaining = \(60 - 28 = 32\).
B alone = \( \frac{32}{3} = 10 \tfrac{2}{3}\).
Total = \(4 + 10 \tfrac{2}{3} = 14 \tfrac{2}{3}\) days
4. Work and Wages
Wages ∝ Work done.
If A twice as efficient as B → A:B = 2:1.
Example: A & B earn ₹1200. A:B = 2:1 → A=₹800, B=₹400
5. Pipes and Cisterns
- Inlet = Positive work
- Outlet = Negative work
Example: Pipe A fills in 12h, Pipe B empties in 18h.
Rate = \(\frac{1}{12} - \frac{1}{18} = \frac{1}{36}\)
So tank fills in 36 hours
6. Practice Questions
- A = 15 days, B = 20 days. Time together?
- A is thrice efficient as B. Together = 12 days. Find B’s alone time.
- Pipes: 24h, 36h fill; 12h empty. Time together?
- A, B, C finish in 12, 18, 36 days. Together?
- 16 women complete in 25 days. Find days for 20 women.