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Ratio of Time and Work

Introduction

Many time-and-work problems ask you to compare two or more workers - not with absolute days, but using ratios. Understanding how time, work, and efficiency relate in ratio form lets you quickly convert between them and solve comparative questions without heavy algebra.

This pattern is important because ratio reasoning appears frequently in competitive aptitude tests and lets you shortcut lengthy calculations.

Pattern: Ratio of Time and Work

Pattern

Key concept: Efficiency ∝ 1/Time and Work ∝ Efficiency × Time. Use these relations to convert between ratios.

Core relations you must remember:

  • Efficiency (E) is the reciprocal of time for the same job: E ∝ 1/T.
  • Work done = Efficiency × Time: W = E × T.
  • If A : B = a : b are times, then their efficiencies are 1/a : 1/b = b : a.
  • If efficiencies are in ratio p : q, their times are in ratio q : p (inverse relation).

Step-by-Step Example

Question

A and B can finish the same job in times that are in the ratio 2 : 3. If A alone takes 12 days to finish the job, how many days will B take? Also, what is the ratio of their efficiencies?

Solution

  1. Step 1: Identify the given ratio and what it represents.

    Times A : B = 2 : 3. This means A's time is 2k and B's time is 3k for some positive k.

  2. Step 2: Use A's actual time to find k.

    We are told A's actual time = 12 days = 2k ⇒ k = 12 ÷ 2 = 6.

  3. Step 3: Compute B’s time.

    B's time = 3k = 3 × 6 = 18 days.

  4. Step 4: Find efficiency ratio (inverse of time ratio).

    Efficiency A : B = (1/Time_A) : (1/Time_B) = (1/2k) : (1/3k) = 3 : 2.

  5. Final Answer:

    B takes 18 days. Efficiency ratio A : B = 3 : 2.

  6. Quick Check:

    If A does 1/12 per day and B does 1/18 per day, combined per day = 1/12 + 1/18 = (3 + 2)/36 = 5/36. Inverse = 36/5 = 7.2 days for both together - not required but helps validate consistency ✅

Quick Variations

1. Given efficiency ratio, find times: invert the ratio.

2. Given time ratio and one worker’s actual time, scale to find others (use the multiplier k).

3. Given ratios and fractional work (e.g., A does twice the work of B in same time), convert to efficiencies then to times.

4. Mix ratios with percentages (e.g., A is 20% faster than B → use 6:5 or multiply factors).

Trick to Always Use

  • Step 1 → Convert any time-ratio to efficiency-ratio by inverting the numbers.
  • Step 2 → When an actual time is given for one person, find k by dividing actual time by that person’s ratio-part.
  • Step 3 → Use W = E × T to handle fractional-work questions or combined-work checks.

Summary

Summary

Key takeaways for the Ratio of Time and Work pattern:

  • Time and efficiency are inverses: swap ratio numbers to go between them.
  • Use a multiplier (k) to convert ratio parts to actual values when one actual figure is given.
  • For combined-work or fractional-work checks, convert to daily rates (reciprocals) and use W = rate × time.
  • Always perform a quick check: multiply computed time by rate to ensure total work equals 1 unit.

Practice

(1/5)
1. The ratio of time taken by A and B to complete a work is 3 : 4. If A can finish the work in 15 days, how many days will B take?
easy
A. 20 days
B. 18 days
C. 22 days
D. 24 days

Solution

  1. Step 1: Express times using ratio

    Time ratio A : B = 3 : 4 ⇒ A = 3k, B = 4k.

  2. Step 2: Find the scale factor k

    A's actual time = 15 days ⇒ 3k = 15 ⇒ k = 5.

  3. Step 3: Compute B's time

    B's time = 4k = 4 × 5 = 20 days.

  4. Final Answer:

    20 days → Option A

  5. Quick Check:

    15 : 20 = 3 : 4 ✅
Hint: Multiply the given time by (B_part ÷ A_part).
Common Mistakes: Swapping ratio parts or dividing when you should multiply.
2. The ratio of efficiencies of A and B is 5 : 3. If B can complete the job in 30 days, how long will A take to complete it alone?
easy
A. 20 days
B. 18 days
C. 15 days
D. 24 days

Solution

  1. Step 1: Convert efficiency ratio to time ratio

    Efficiency A : B = 5 : 3 ⇒ Time A : B = 3 : 5 (inverse relation).

  2. Step 2: Scale the time ratio using B's time

    B's time = 30 days corresponds to 5k = 30 ⇒ k = 6.

  3. Step 3: Compute A's time

    A's time = 3k = 3 × 6 = 18 days.

  4. Final Answer:

    18 days → Option B

  5. Quick Check:

    Time ratio 18 : 30 = 3 : 5 ⇒ efficiencies 5 : 3 ✅
Hint: Invert efficiency ratio to get time ratio, then scale with the given time.
Common Mistakes: Using the efficiency ratio directly as time instead of inverting it.
3. A and B take 10 and 25 days respectively to finish a work. What is the ratio of their efficiencies (A : B)?
easy
A. 2 : 5
B. 5 : 12
C. 5 : 2
D. 25 : 10

Solution

  1. Step 1: Use reciprocals of times for efficiencies

    Efficiency ∝ 1/Time. So efficiency ratio = 1/10 : 1/25.

  2. Step 2: Compute and simplify the ratio

    1/10 : 1/25 = 25 : 10 = simplify → 5 : 2.

  3. Final Answer:

    5 : 2 → Option C

  4. Quick Check:

    Time ratio 10 : 25 = 2 : 5 ⇒ inverse = 5 : 2 ✅
Hint: Take reciprocals of times, then simplify the ratio.
Common Mistakes: Forgetting to invert the times when computing efficiencies.
4. A can do a work in 12 days and B in 20 days. What is the ratio of work done by A and B in one day?
medium
A. 2 : 3
B. 4 : 3
C. 1 : 1
D. 5 : 3

Solution

  1. Step 1: Compute one-day work for each

    A's one-day work = 1/12, B's = 1/20.

  2. Step 2: Form and simplify the ratio

    1/12 : 1/20 = 20 : 12 = simplify → 5 : 3.

  3. Final Answer:

    5 : 3 → Option D

  4. Quick Check:

    Inverse of 12 : 20 = 5 : 3 ✅
Hint: Invert the time ratio and simplify to get one-day work ratio.
Common Mistakes: Confusing A:B order when inverting ratios.
5. A is twice as efficient as B. Together they finish a job in 9 days. How many days will A alone take to finish the work?
medium
A. 27/2 days
B. 18/5 days
C. 15/2 days
D. 12/5 days

Solution

  1. Step 1: Set up rate-parts

    Let B's one-day work = x ⇒ A's = 2x. Combined = 3x = 1/9 (since together they finish in 9 days).

  2. Step 2: Solve for x

    x = 1/(9 × 3) = 1/27. So A's one-day work = 2/27.

  3. Step 3: Invert to get A's time

    A's time = 1 ÷ (2/27) = 27/2 = 13.5 days.

  4. Final Answer:

    27/2 days (13.5 days) → Option A

  5. Quick Check:

    Combined rate = (2/27) + (1/27) = 3/27 = 1/9 ⇒ total time = 9 days ✅
Hint: Express rates as parts (x and 2x), set sum = 1/total days, solve for x and invert for A's time.
Common Mistakes: Confusing rate-parts with time-parts or arithmetic slips when inverting fractions.

Mock Test

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