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Combined Work (A + B Together)

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Introduction

Combined work problems पूछते हैं कि दो या ज़्यादा workers (या machines) मिलकर कोई काम कितने समय में पूरा करेंगे। ये problems common हैं क्योंकि real-world में कई काम teams द्वारा पूरे होते हैं - efficiencies को सही तरीके से जोड़ना जानकर आप total time या shared contribution को सही तरह estimate कर सकते हैं।

यह pattern important है क्योंकि यह basic formula W = R × T को multiple agents तक generalize करता है और आगे आने वाले mixture और collaborative work problems की foundation बनता है।

Pattern: Combined Work (A + B Together)

Pattern: Combined Work (A + B Together)

Key concept: Individual one-day works (rates) को जोड़कर combined one-day work मिलता है; फिर उसका reciprocal लेकर total time मिलता है।

अगर A कोई काम अकेले TA days में और B वही काम TB days में पूरा करता है:
One-day work of A = 1/TA, One-day work of B = 1/TB.
Combined one-day work = 1/TA + 1/TB.
जब दोनों साथ काम करें तो total time = 1 ÷ (1/TA + 1/TB).

Step-by-Step Example

Question

A एक काम 12 days में पूरा करता है। B वही काम 8 days में करता है। अगर दोनों साथ काम करें, तो उन्हें काम पूरा करने में कितने दिन लगेंगे?

Options:

  • A. 4.8 days
  • B. 4 days
  • C. 4.5 days
  • D. 5 days

Solution

  1. Step 1: Individual times को one-day works में बदलें।

    A का one-day work = 1/12.
    B का one-day work = 1/8.

  2. Step 2: One-day works को जोड़कर combined rate निकालें।

    Combined one-day work = 1/12 + 1/8 = (2 + 3) / 24 = 5/24.

  3. Step 3: Combined rate का reciprocal लेकर total time निकालें।

    Time = 1 ÷ (5/24) = 24/5 days = 4.8 days.

  4. Final Answer:

    4.8 days → Option A

  5. Quick Check:

    Time × combined rate चेक करें: (24/5) × (5/24) = 1 (complete work) ✅

Quick Variations

1. दो से ज़्यादा workers हों: सभी के one-day works जोड़ें (जैसे A + B + C = 1/TA + 1/TB + 1/TC).

2. एक worker पहले कुछ काम करे, फिर दोनों मिलकर काम करें - पहले part का work निकालें, उसे 1 से subtract करें, फिर remaining work के लिए combined rate लगाएं।

3. अगर workers के units अलग हों (hours vs days): जोड़ने से पहले same unit में convert करें।

4. Efficiencies दी हों (जैसे A is k times as efficient as B), तो ratios से rates बनाएं और फिर combined time निकालें।

Trick to Always Use

  • Step 1 → हर व्यक्ति के time को one-day work (reciprocal) में बदलें।
  • Step 2 → सभी one-day works जोड़ें और combined rate पाएं।
  • Step 3 → Combined rate का reciprocal लेकर total time निकालें (Time = 1 ÷ combined rate).

Summary

Combined work problems में:

  • हमेशा times को one-day works में बदलें (reciprocal का इस्तेमाल करें)।
  • Times को नहीं, rates को जोड़ें - इसी से combined rate मिलता है।
  • Combined rate का reciprocal लेकर total time निकालें।
  • सभी workers के लिए same unit (days/hours) रखें और अंत में time × combined rate करके quick check करें कि result 1 आ रहा है या नहीं।

Practice

(1/5)
1. A can finish a job in 6 days and B can finish it in 12 days. How long will they take to complete the work together?
easy
A. 4 days
B. 3 days
C. 5 days
D. 6 days

Solution

  1. Step 1: Identify the values.

    A's one-day work = 1/6; B's one-day work = 1/12.

  2. Step 2: Add rates.

    Combined one-day work = 1/6 + 1/12 = (2 + 1)/12 = 3/12 = 1/4.

  3. Step 3: Invert to get time.

    Total time = 1 ÷ (1/4) = 4 days.

  4. Final Answer:

    They finish the work together in 4 days → Option A.

  5. Quick Check:

    4 × (1/4) = 1 (complete work) ✅
Hint: Add individual one-day works and take reciprocal for total time.
Common Mistakes: Adding times directly instead of adding rates.
2. A can do a work in 10 days and B can do it in 20 days. Working together, how much work will they complete in one day?
easy
A. 1/10
B. 3/20
C. 1/15
D. 1/8

Solution

  1. Step 1: Identify the values.

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

  2. Step 2: Add rates.

    Combined one-day work = 1/10 + 1/20 = (2 + 1)/20 = 3/20.

  3. Final Answer:

    They complete 3/20 of the work in one day → Option B.

  4. Quick Check:

    Reciprocal time = 20/3 ≈ 6.67 days; (3/20) × (20/3) = 1 ✅
Hint: Add the reciprocals (1/T) to get combined one-day work.
Common Mistakes: Averaging times or using wrong LCM when adding fractions.
3. A and B can complete a work together in 8 days. If A alone can do it in 12 days, in how many days can B alone finish it?
easy
A. 18 days
B. 20 days
C. 24 days
D. 30 days

Solution

  1. Step 1: Identify the values.

    Combined one-day work = 1/8; A's one-day work = 1/12.

  2. Step 2: Subtract to find B's rate.

    B's one-day work = 1/8 - 1/12 = (3 - 2)/24 = 1/24.

  3. Step 3: Invert to get time for B.

    Time for B = 1 ÷ (1/24) = 24 days.

  4. Final Answer:

    B alone can finish the work in 24 days → Option C.

  5. Quick Check:

    1/12 + 1/24 = (2 + 1)/24 = 3/24 = 1/8 ✅
Hint: Find missing rate by subtracting known rate from combined rate, then invert.
Common Mistakes: Subtracting times instead of rates or mixing denominators incorrectly.
4. A and B together can finish a work in 9 days. A alone can finish it in 18 days. How long will B alone take to finish the same work?
medium
A. 15 days
B. 20 days
C. 12 days
D. 18 days

Solution

  1. Step 1: Identify the values.

    Combined one-day work = 1/9; A's one-day work = 1/18.

  2. Step 2: Subtract to find B's rate.

    B's one-day work = 1/9 - 1/18 = (2 - 1)/18 = 1/18.

  3. Step 3: Invert to get time for B.

    Time for B = 1 ÷ (1/18) = 18 days.

  4. Final Answer:

    B alone can finish the work in 18 days → Option D.

  5. Quick Check:

    1/18 + 1/18 = 2/18 = 1/9 ✅
Hint: Use 1/Tb = 1/Tcombined - 1/Ta, then take reciprocal.
Common Mistakes: Using subtraction on times rather than on their reciprocals (rates).
5. A and B can complete a piece of work in 5 days. If A alone can complete it in 15 days, how long will B alone take to complete it?
medium
A. 7.5 days
B. 6 days
C. 8 days
D. 10 days

Solution

  1. Step 1: Identify the values.

    Combined one-day work = 1/5; A's one-day work = 1/15.

  2. Step 2: Subtract to find B's rate.

    B's one-day work = 1/5 - 1/15 = (3 - 1)/15 = 2/15.

  3. Step 3: Invert to get time for B.

    Time for B = 1 ÷ (2/15) = 15/2 = 7.5 days.

  4. Final Answer:

    B alone can finish the work in 7.5 days → Option A.

  5. Quick Check:

    1/15 + 1/7.5 = (1 + 2)/15 = 3/15 = 1/5 ✅
Hint: Compute B's rate = combined rate - A's rate, then invert to get time.
Common Mistakes: Multiplying times directly or forgetting to use reciprocals when combining.