Time and Work Principles

• Work = Efficiency * Time.
• Efficiency represents the amount of work done by an individual in a single day.
• Efficiency is determined by the formula: Work / Time.
• Time is calculated as: Work / Efficiency.
• Efficiency and Time are inversely proportional.
• For inversely proportional quantities, use: E1/E2 = T2/T1.
• Focus on finding efficiency to solve time and work problems.

Approach to solving problems:

• Prioritize finding the efficiency of each worker or group.
• If A takes 4 days and B takes 3 days to complete a task:
  • Determine the total work by finding the least common multiple (LCM) of 4 and 3.
  • Calculate A's one-day work by dividing the total work by A's time.
  • Calculate B's one-day work by dividing the total work by B's time.

Example problem 1

• A can complete work in 6 days, B can complete it in 9 days.
• If A and B work together, how long will it take to complete the work?
  • A's time = 6 days.
  • B's time = 9 days.
  • Total work = LCM(6, 9) = 18 units.
  • A's one-day work (efficiency) = 18/6 = 3 units.
  • B's one-day work (efficiency) = 18/9 = 2 units.
  • A + B's combined one-day work = 3 + 2 = 5 units.
  • Time for A & B together = Total work / combined efficiency = 18/5 = 3.6 days.

Example problem 2

• A, B, and C can complete a piece of work in 20, 30, and 60 days respectively.
• How long will A and C take to complete the work together?
  • A's time = 20 days, B's time = 30 days, C's time = 60 days.
  • Total work = LCM(20, 30, 60) = 60 units.
  • A's efficiency = 60/20 = 3 units/day.
  • B's efficiency = 60/30 = 2 units/day.
  • C's efficiency = 60/60 = 1 unit/day.
  • A and C working together:
    • A+C efficiency = 3+1 = 4 units/day.
    • Time to complete = 60/4 = 15 days.

Example problem 3

• A and B can complete a piece of work in 15 and 10 days respectively.
• They work together for 4 days. What fraction of the work is left incomplete?
  • A's time = 15 days, B's time = 10 days.
  • Total work = LCM(15, 10) = 30 units.
  • A's efficiency = 30/15 = 2 units/day.
  • B's efficiency = 30/10 = 3 units/day.
  • A+B efficiency = 2+3 = 5 units/day.
  • Work done in 4 days = 5 * 4 = 20 units.
  • Work left = 30 - 20 = 10 units.
  • Fraction of work left = 10/30 = 1/3.

Example problem 4

• A can finish a piece of work in 18 days, and B can also do it in 18 days. How much of the work can they finish together in one day?
  • A's time = 18 days, B's time = 18 days.
  • Total work = 18 units (LCM).
  • A's efficiency = 18/18 = 1 unit/day.
  • B's efficiency = 18/18 = 1 unit/day.
  • Combined efficiency = 1+1 = 2 units/day.
  • Work done together in one day is 2 out of 18 = 2/18 = 1/9.

Example problem 5

• A and B can do a piece of work in 12 days, B and C in 16 days, and A and C in 24 days. How much time will A, B, and C take to do the work together?
  • A+B time = 12, B+C time = 16, C+A time = 24.
  • Total work = LCM(12, 16, 24) = 48 units.
  • Efficiency: A+B=4, B+C=3, C+A=2.
  • Adding all efficiencies: 2(A+B+C)=9, so A+B+C=4.5.
  • Time for A, B, C working together = 48/4.5 = 10.67 days.
  • To find the time for C alone:
    • C = (A+B+C) - (A+B) = 4.5 - 4 = 0.5 Efficiency
    • C Time = 48 / 0.5 = 96 days

Example problem 6

• A can complete a piece of work in 4 hours. B and C can do it in 3 hours, and A and C can do it in 2 hours. How long will it take for B alone to complete the work?
  • A: 4 hours, B+C: 3 hours, A+C: 2 hours
  • Total Work = LCM (4, 3, 2) = 12
  • Efficiencies: A = 3, B+C = 4, A+C = 6
  • C = (A+C) - A = 6 - 3 = 3
  • B = (B+C) - C = 4 - 3 = 1
  • B Time = 12 / 1 = 12 hours

Example problem 7

• A can complete a piece of work in 6 days, while B takes 12 days to complete the same work. If they work together, what portion of the work has A completed.
  • A finishes in 6 days, B finishes in 12 days.
  • Total work = LCM(6, 12) = 12.
  • A's efficiency = 2, B's efficiency = 1.
  • A and B working together finish in 4 days (12 / (2 + 1)).
  • Portion of work done by A = 8/12 = 2/3.

Example problem 8

• A is 150% more efficient than B, and B alone completes the work in 20 days. How long will it take A to do the same work?
  • A = 2.5B
  • Work = Efficiency * Time
  • 20 = B Time
  • T = W / E
  • 100 / 250 = 0.4 * 20 = 8
  • Time that A takes = 8 days.

Example problem 9

• A can complete work in 15 days, and B can complete it in 10 days. They have a contract worth RS 30,000. What is A's share. Note that the share ratio is dependent on efficiency share.
  • A can complete work in 15 days, B can complete in 10 days.
  • Total Work = LCM(15, 10) = 30.
  • A's efficiency = 2, B's efficiency = 3.
  • A/(A + B) = 2 / 5 = 0.4
  • Share of A = 0.4 * 30000 = 12000

Example problem 10

• A takes 6 days less than B to complete a work. A and B together take 4 days. How many days does B alone take?
  • Let B alone = X, A = X-6.
  • A and B together take 4 days
  • Eq is X^2-14x+24=0
  • Need Quad Eq
  • B == 12

Example problem 11

• The efficiencies of A, B, and C are in the ratio 2:3:5. A alone takes 50 days to complete the work. If A works for 15 days, how long will A and B take to complete the remaining work?
  • Total Work -- -A = 2. 50 = 1--Share
  • A, B, C: 2:3:5
  • Efficiency = 10
  • Rem = 75 left. and A+B=5 --Rem
  • A and together ==15

Example problem 12

• Six Times BE
• More BE
• B: 32 A: time
• BE = 1
• M = 32 A Share Total