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A Starts, B Joins Later

Introduction

कई time and work problems में ऐसा होता है कि एक व्यक्ति काम शुरू करता है और दूसरा बाद में join करता है। यह pattern यह पता लगाने में मदद करता है कि पूरा काम कितने समय में होगा, दोनों का योगदान कितना है, या समय पर काम पूरा करने के लिए दूसरे व्यक्ति को कब join करना चाहिए।

यह type का problem इसलिए important है क्योंकि इसमें sequential work contribution और joint efficiency concepts दोनों शामिल होते हैं।

Pattern: A Starts, B Joins Later

Pattern

Key idea: Total Work = (A द्वारा अकेले किया गया काम) + (A + B द्वारा मिलकर किया गया काम).

Work = Rate × Time. इसलिए equation बनेगा: (A की Rate × A का Time) + ((A + B) की Rate × Remaining Time) = 1 (total work).

Step-by-Step Example

Question

A कोई काम 12 days में और B वही काम 16 days में करता है। A पहले 4 days अकेले काम करता है और फिर B join करता है। कुल काम कितने दिनों में पूरा होगा?

Solution

  1. Step 1: One-day works निकालें

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

  2. Step 2: A द्वारा पहले 4 days में किया गया काम

    Work = 4 × (1/12) = 1/3.

  3. Step 3: Remaining work निकालें

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

  4. Step 4: A और B की combined rate

    Combined rate = 1/12 + 1/16 = (4 + 3)/48 = 7/48.

  5. Step 5: Remaining work पूरा करने का time

    Time = (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 का work = 1/3; Remaining = 2/3; Together 32/7 days में = (32/7) × 7/48 = 32/48 = 2/3 → total = 1 ✅

Quick Variations

1. A शुरू करता है और B कुछ समय बाद join करता है (जैसा example में)।

2. B शुरू करता है और A बाद में join करता है।

3. किसी given दिन में काम खत्म करना हो तो B को कब join करना चाहिए, यह निकालना।

4. A और B द्वारा किया गया individual work निकालना।

Trick to Always Use

  • Step 1: पहले व्यक्ति द्वारा अकेले किए गए काम का fraction निकालें।
  • Step 2: उसे total में से subtract करके remaining work पाएं।
  • Step 3: Remaining work को combined rate से divide करके joint time निकालें।
  • Step 4: दोनों time periods को जोड़कर total time पाएं।

Summary

Summary

A Starts, B Joins Later pattern में:

  • Total work = (A का solo work) + (A + B का joint work).
  • हर stage में किए गए काम का fraction अलग-अलग निकालें।
  • Final time = A द्वारा अकेले किया गया time + बाकी काम का joint time।

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.

Mock Test

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