Time & Work Set 6 : How many Days required ?
Work : Speed : Distance : Time
Main Formulas
- Speed = Distance ÷ Time (v = d/t)
- Distance = Speed × Time ( d = v × t)
- Time = Distance ÷ Speed ( t = d/v)
- Time & Work Formula = M × D × H ∝ Work
Where:
- M = Men
- D = Days
- H = Hours
- Work = Total work done / Output
Main Rule
If the work is constant → M × D × H = Constant
i. Men & Days Relation
If hours/day are the same: M1⋅D1=M2⋅D2
- More men → work finishes faster
- Fewer men → work takes longer
ii. Men & Hours Relation
If the number of days is the same: M1⋅H1=M2⋅H2
iii. Days & Hours Relation
If the number of men is the same: D1⋅H1=D2⋅H2
1. 40 workers can complete a house in 20 days. How many days will 8 workers take to complete the same house ?
Soln.
Use Men × Days = Constant
M1⋅D1 = M2⋅D2
40⋅20 = 8⋅D
800 = 8⋅D
D = 800/8 = 100 days
Ans: 100 days
2. Ravi can complete a work in 8 days and Shyam can complete the same work in 24 days. If they work alternately one day each, in how many days will the work be completed ?
Soln.
Work done per day:
- Ravi = 1/8 per day
- Shyam = 1/24 per day
Work done in 2 days (Ravi + Shyam): 1/8+1/24 = (3+1)/24 = 4/24 =1/6 work in 2 days
Total cycles needed:
- Total work = 1
- Each 2-day cycle completes 1/6 of work
- Number of 2-day cycles = 6 → 6 × 2 = 12 days
Ans: 12 days
3. 1234 × 9999 = ?
Trick:
- Write as many zeros as the number of 9s.
→ There are 4 nines → write 4 zeros after 1234 → 12340000 - Subtract the number itself (1234).
→ 12340000 − 1234 = 12338766
Ans: 12338766
4. A and B together can complete a work in 10 days, B and C together in 12 days, and C and A together in 15 days. How many days will it take if all three work together ?
Given:
- A + B = 10 days
- B + C = 12 days
- C + A = 15 days
Add all pair rates: (A+B) + (B+C) + (C+A) = 1/10 + 1/12 + 1/15 = 15/60
LCM of 10,12,15 = 60
Divide by 2 to get A + B + C: 15/60 ÷ 2 =1/8
Time for all three: 1÷(1/8) = 8 days
Ans: 8 days
5. 120 workers can complete a house in 60 days. How many days will 80 workers take to complete the same house ?
Soln
Use Men × Days = Constant
M1⋅D1 = M2⋅D2
120⋅60 = 80⋅D
7200 = 80⋅D
D = 7200 / 80 = 90 days
Ans: 90 days
6. Ram can complete a work in 15 days, and Shyam can complete the same work in 60 days. How many days will both working together take to finish it ?
Soln
Work done per day
Ram’s 1 day work = 1/15
Shyam’s 1 day work=1/60
Combined 1 day work
1/15 + 1/60 = 4/60 + 1/60 = 5/60 = 1/12
Total time to complete work
Time = 1/(1/12)=12 days
Ans: 12 days
7. A, B, and C can complete a work in 20, 40, and 60 days respectively. How many days will it take if all three work together ?
Soln
Take LCM as total work
LCM of 20, 40, 60 = 120 units
Find individual efficiency (work per day)
· A = 120 ÷ 20 = 6 units/day
· B = 120 ÷ 40 = 3 units/day
· C = 120 ÷ 60 = 2 units/day
Combined efficiency
6 + 3 + 2 = 11 units/day
Time required : Time = Total Work ÷ Combined Efficiency
120÷11=10 10/11 days
Ans: 10 10/11 days
8. A can complete a work in 24 days. A and B together can complete it in 8 days. How many days will B alone take to finish the work ?
Soln
Work per day
- A’s 1-day work = 1/24
- (A + B)’s 1-day work = 1/8
B’s 1 - day work
B = (A+B) - A
= 1/8 - 1/24
= 3/24 - 1/24
= 2/24
= 1/12
Time taken by B alone : Time = 1/(1/12) = 12 days
Ans: 12 days
9. Julie can cover 140 meters in 18 seconds. At the same speed, how much distance will she cover in 1 hour ?
Options: (a) 25.2 km (b) 31.5 km (c) 28 km (d) 29.4 km
Soln
Calculate Speed
Speed = Distance/Time = 140/18 m/s
Convert m/s to km/h
1 m/s = 18/5 km/h
140/18 × 18/5
Cancel 18:140 / 5 = 28 km/h
Ans = 28 km/h
Final Distance in 1 Hour
Since speed = 28 km/h, Distance in 1 hour = 28 km
Ans: (c) 28 km
Shortcut
Speed (km/h) = Distance in m / Time in sec × 18/5
=140/18 × 18/5 = 28km
Ans: 28km