Episode 8 — Aptitude and Reasoning / 8.9 — Work and Time

8.9 Quick Revision -- Work and Time

A one-stop reference for last-minute revision before exams.

1. Core Formulas

RATE = 1 / TIME

If A does a job in X days:
   A's daily rate = 1/X
   Work done in D days = D/X

TWO WORKERS TOGETHER:

   Time = (a x b) / (a + b)

   where a, b are individual completion times.

THREE WORKERS TOGETHER:

   Combined rate = 1/a + 1/b + 1/c
   Time = 1 / (combined rate)

WORK LEFT after D days by a worker who takes X days:

   Fraction left = (X - D) / X

2. The LCM Method (Step by Step)

This method eliminates all fractions and is the recommended approach for every problem.

GIVEN: A takes 12 days, B takes 18 days.

Step 1: Total Work = LCM(12, 18) = 36 units

Step 2: Rates
        A = 36/12 = 3 units/day
        B = 36/18 = 2 units/day

Step 3: Combined = 3 + 2 = 5 units/day

Step 4: Time together = 36/5 = 7.2 days

When to Use LCM Method

  • Three or more workers
  • Workers joining / leaving mid-way
  • Alternating schedules
  • Finding individuals from pair data
  • Wage calculations

3. Key Shortcut Formulas

Efficiency Shortcut

A is k times as efficient as B:
   Time(A) = Time(B) / k
   Together = Time(B) / (k + 1)

Percentage Efficiency

A is p% more efficient than B:
   Time(A) = Time(B) x 100 / (100 + p)

Finding Individuals from Pairs

Given: (A+B) in x days, (B+C) in y days, (A+C) in z days

   2(A+B+C) rate = 1/x + 1/y + 1/z

   A's rate = (A+B+C) rate - (B+C) rate
   B's rate = (A+B+C) rate - (A+C) rate
   C's rate = (A+B+C) rate - (A+B) rate

"A Alone Takes 'a' More Hours" Type

If A alone takes 'a' hours MORE than together,
and B alone takes 'b' hours MORE than together:

   Time together = sqrt(a x b)

B Joins After d Days

A starts alone. After d days, B joins.

   Total time T = a(b + d) / (a + b)

4. Man-Days Formula

M1 x D1 x H1 / W1 = M2 x D2 x H2 / W2

M = men, D = days, H = hours/day, W = work amount

Chain Rule Quick Method

To find unknown variable, multiply the known value by:
   - Direct ratio for directly proportional quantities
   - Inverse ratio for inversely proportional quantities

Example: Need more men? -> Days decreased (inverse), Work increased (direct)

5. Alternating Work Pattern

Step 1: Find work per cycle
        Cycle = 2 days if 2 workers alternate
        Cycle = 3 days if 3 workers rotate

Step 2: Complete cycles = Total Work / Work per cycle (integer part)

Step 3: Remaining work = Total - (complete cycles x work per cycle)

Step 4: Assign remaining to next worker in sequence.
        If remaining > that worker's daily output,
        continue to the next worker.

6. Wages Distribution

RULE: Wages split in ratio of WORK DONE (not rate, not time).

If all work same duration:
   Wage ratio = Rate ratio = inverse of Time ratio

If different durations:
   Work(A) = Rate(A) x Days(A)
   Wage ratio = Work(A) : Work(B) : Work(C)

Quick Wage Split for Two Workers

A takes 'a' days, B takes 'b' days, total wage = W:

   A's share = W x b / (a + b)
   B's share = W x a / (a + b)

Note: b in A's numerator (counterintuitive but correct --
      less time = more efficient = more wage)

7. Common Patterns and Traps

Pattern: "X Can Do in a Days, Y Destroys in b Days"

Net rate = 1/a - 1/b     (b > a means net negative -- work never finishes)
Net time = ab / (b - a)   (if b > a, work eventually finishes)

Pattern: Decreasing / Increasing Workforce

Day 1: N workers, Day 2: N+k workers, etc.
=> Arithmetic progression of work per day
=> Sum the AP to find when total work is reached

Pattern: Fractional Statements

"A does 2/5 of work in 8 days"  =>  Full job = 8 x 5/2 = 20 days
"A completes 40% in 6 days"     =>  Full job = 6 / 0.4 = 15 days
"A does as much in 3 days as B in 5 days"  =>  Efficiency A:B = 5:3

Traps to Watch

TRAP 1: "A and B together in 10 days" ≠ "A in 10, B in 10"
TRAP 2: "Twice as efficient" = half the time (not double the time)
TRAP 3: Who starts in alternating problems changes the answer
TRAP 4: Destroyer subtracts rate, does not add
TRAP 5: Remaining work ≠ remaining time (rate may change)
TRAP 6: Workers joining ≠ working from the start

8. Memorize These Common Pairs

Individual times -> Together

(n, n)    -> n/2
(n, 2n)   -> 2n/3
(n, 3n)   -> 3n/4
(a, b)    -> ab/(a+b)

Quick values:
(3, 6)   -> 2
(4, 12)  -> 3
(5, 20)  -> 4
(6, 30)  -> 5
(6, 12)  -> 4
(10, 15) -> 6
(12, 18) -> 36/5 = 7.2
(20, 30) -> 12
(36, 45) -> 20

9. Exam Tips

  1. Always try LCM method first. It converts every problem to integer arithmetic.

  2. Read the question twice. Distinguish between "A and B together take X days" vs "A takes X days."

  3. Track who is working during each phase. Draw a timeline if needed:

    Day 1----4: A+B work (Phase 1)
Day 5----end: B+C work (Phase 2)

  4. For alternating work, always track remaining work day by day near the end -- do not assume the last cycle is complete.

  5. Efficiency problems are best solved by converting to time first, then using LCM.

  6. Man-days problems are just multiplication. Write the equation, plug in, solve.

  7. Wage problems: find work done by each person (not just their rate), then split proportionally.

  8. When stuck, assign Total Work = LCM and convert everything to units. This almost always simplifies the problem.

  9. Verify your answer by checking that total work adds up to 100% (or the total units).

  10. Time-saving order in exams: Do basic formula problems first (< 30 seconds each), then medium LCM problems (1-2 minutes), save complex multi-phase problems for last.

10. Formula Summary Card

#ConceptFormula
1Rate1/Time
2Together (2 people)ab/(a+b)
3Together (3 people)1/(1/a + 1/b + 1/c)
4k times efficientTogether = B_time/(k+1)
5p% more efficientTime_A = Time_B x 100/(100+p)
6Man-daysM1.D1.H1/W1 = M2.D2.H2/W2
7WagesRatio of wages = Ratio of work done
8Alternating (2 ppl)Cycle = 2 days; Work/cycle = R_A + R_B
9Work remaining(X - D)/X after D days if total = X days
10sqrt shortcutTogether = sqrt(a x b) when extras given

Back to 8.9 Work and Time -- Overview