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A Starts, B Joins Later

Introduction

In many time and work problems, one person starts the work, and another joins later. This pattern helps determine how long it takes to complete the job, how much each person contributes, or when the second person should join for timely completion.

This type of problem is important because it combines sequential work contribution with joint efficiency concepts.

Pattern: A Starts, B Joins Later

Pattern

The key idea is: Total Work = (Work done by A alone) + (Work done by A + B together).

Work = Rate × Time. So, the equation becomes: (A’s Rate × A’s Time) + ((A + B)’s Rate × Remaining Time) = 1 (total work).

Step-by-Step Example

Question

A can do a piece of work in 12 days and B in 16 days. A works alone for 4 days and then B joins. In how many total days will the work be completed?

Solution

  1. Step 1: Compute one-day works

    A’s one-day work = 1/12; B’s one-day work = 1/16.

  2. Step 2: Find A’s work in the first 4 days

    Work done by A in first 4 days = 4 × (1/12) = 1/3.

  3. Step 3: Compute remaining work

    Remaining work = 1 - 1/3 = 2/3.

  4. Step 4: Compute combined rate of A and B

    Combined rate of A and B = 1/12 + 1/16 = (4 + 3)/48 = 7/48.

  5. Step 5: Find time to finish remaining work

    Time required to complete remaining 2/3 work = (2/3) ÷ (7/48) = (2/3) × (48/7) = 32/7 days ≈ 4.57 days.

  6. Final Answer:

    60/7 days ≈ 8.57 days

  7. Quick Check:

    A’s work = 1/3; Remaining = 2/3; Together for 32/7 days = (32/7) × 7/48 = 32/48 = 2/3 → sum = 1 ✅

Quick Variations

1. A starts and B joins after some time (as above).

2. B starts and A joins later (reverse case).

3. Find when B should join for the work to finish in a given number of days.

4. Find individual shares of work done by A and B.

Trick to Always Use

  • Step 1: Compute work done by the person who starts first.
  • Step 2: Subtract from total to get remaining work.
  • Step 3: Divide remaining work by combined rate to get remaining time.
  • Step 4: Add both time periods for total time.

Summary

Summary

In the A Starts, B Joins Later pattern:

  • Total work = (A’s solo work) + (A + B’s joint work).
  • Always compute fraction of work done separately for each stage.
  • Final time = A’s solo time + time taken jointly to complete the rest.

Practice

(1/5)
1. A can do a piece of work in 10 days and B in 20 days. A works alone for 4 days and then B joins. In how many total days will the work be completed?
easy
A. 8 days
B. 6 days
C. 7 days
D. 9 days

Solution

  1. Step 1: Compute one-day works

    A's one-day work = 1/10; B's one-day work = 1/20.

  2. Step 2: Find A's solo work

    Work done by A in first 4 days = 4 × (1/10) = 2/5.

  3. Step 3: Compute remaining work

    Remaining work = 1 - 2/5 = 3/5.

  4. Step 4: Compute combined rate

    Combined rate = 1/10 + 1/20 = 3/20.

  5. Step 5: Compute time for remaining work

    Time = (3/5) ÷ (3/20) = (3/5) × (20/3) = 4 days.

  6. Final Answer:

    8 days → Option A

  7. Quick Check:

    A does 2/5, both next 4 days do 3/5 → total = 1 ✅
Hint: Compute solo work first, subtract from 1, then divide remaining by combined rate.
Common Mistakes: Adding times instead of adding rates or forgetting that B joins later.
2. A can finish a work in 12 days and B in 24 days. A works alone for 6 days and then B joins. In how many total days will the work be completed?
easy
A. 10 days
B. 9 days
C. 8 days
D. 11 days

Solution

  1. Step 1: Compute daily rates

    A = 1/12; B = 1/24.

  2. Step 2: A's work in 6 days

    A completes 6 × (1/12) = 1/2.

  3. Step 3: Remaining work

    1 - 1/2 = 1/2.

  4. Step 4: Combined rate

    1/12 + 1/24 = 1/8.

  5. Step 5: Time for remaining work

    (1/2) ÷ (1/8) = 4 days.

  6. Final Answer:

    10 days → Option A

  7. Quick Check:

    A does 1/2; both do remaining 1/2 in 4 days → total = 1 ✅
Hint: Work done alone + joint work must add up to 1.
Common Mistakes: Forgetting to add solo and joint parts correctly.
3. A can do a work in 15 days and B in 20 days. A works alone for 5 days, then B joins. How many more days will they take to finish the work?
easy
A. 5 days
B. 40/7 days
C. 6 days
D. 8/3 days

Solution

  1. Step 1: Compute one-day works

    A = 1/15; B = 1/20.

  2. Step 2: A's work in 5 days

    5 × (1/15) = 1/3.

  3. Step 3: Remaining work

    1 - 1/3 = 2/3.

  4. Step 4: Combined rate

    1/15 + 1/20 = 7/60.

  5. Step 5: Compute remaining time

    (2/3) ÷ (7/60) = (2/3) × (60/7) = 40/7 days.

  6. Final Answer:

    40/7 days → Option B

  7. Quick Check:

    (40/7) × (7/60) = 2/3 → plus 1/3 = 1 ✅
Hint: Keep fractions exact, convert remaining work to days using combined rate.
Common Mistakes: Rounding too early, miscomputing combined rate.
4. A can do a work in 18 days and B in 12 days. A works alone for 6 days and then B joins. How many total days will it take to finish the work?
medium
A. 10 days
B. 11/3 days
C. 54/5 days
D. 12 days

Solution

  1. Step 1: Compute one-day works

    A's one-day work = 1/18; B's one-day work = 1/12.

  2. Step 2: A's solo work

    Work done by A in 6 days = 6 × (1/18) = 1/3.

  3. Step 3: Remaining work

    Remaining work = 1 - 1/3 = 2/3.

  4. Step 4: Combined rate

    Combined rate = 1/18 + 1/12 = (2 + 3)/36 = 5/36.

  5. Step 5: Time to finish remaining work

    Time for remaining 2/3 = (2/3) ÷ (5/36) = (2/3) × (36/5) = 24/5 days.

  6. Step 6: Total time

    Total time = A's solo 6 days + 24/5 days = 6 + 24/5 = 54/5 days.

  7. Final Answer:

    54/5 days → Option C

  8. Quick Check:

    1/3 + (24/5 × 5/36) = 1/3 + 24/36 = 1/3 + 2/3 = 1 ✅
Hint: Compute A's solo work first, then finish the remaining work using the combined rate, and finally add both times.
Common Mistakes: Forgetting to add the initial solo 6 days to the remaining time.
5. A can do a work in 16 days and B in 8 days. B joins A after 4 days. Find the total time taken to finish the work.
medium
A. 7 days
B. 6 days
C. 9 days
D. 8 days

Solution

  1. Step 1: Compute one-day works

    A = 1/16; B = 1/8.

  2. Step 2: A's solo work

    4 × (1/16) = 1/4.

  3. Step 3: Remaining work

    1 - 1/4 = 3/4.

  4. Step 4: Combined rate

    1/16 + 1/8 = 3/16.

  5. Step 5: Time to finish remaining

    (3/4) ÷ (3/16) = (3/4) × (16/3) = 4 days.

  6. Final Answer:

    8 days → Option D

  7. Quick Check:

    A alone: 1/4; both 4 days: 4×3/16 = 3/4; sum = 1 ✅
Hint: Always convert remaining work to days using combined rate.
Common Mistakes: Forgetting that A works alone initially.

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