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Combined Work (A + B Together)

Introduction

Combined work problems ask how long two or more workers (or machines) take when they work together. These problems are common because many real-world tasks are completed by teams - knowing how to add efficiencies correctly lets you estimate total time or shared contributions.

This pattern is important because it generalizes the basic formula W = R × T to multiple agents and forms the basis for more advanced mixture and collaborative work problems.

Pattern: Combined Work (A + B Together)

Pattern

Key concept: Add individual one-day works (rates) to get the combined one-day work; then take reciprocal to find total time.

If A takes TA days and B takes TB days to finish 1 work individually:
One-day work of A = 1/TA, One-day work of B = 1/TB.
Combined one-day work = 1/TA + 1/TB.
Total time when both work together = 1 ÷ (1/TA + 1/TB).

Step-by-Step Example

Question

A can finish a job in 12 days. B can finish the same job in 8 days. If both work together, how many days will they take to complete the job?

Options:

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

Solution

  1. Step 1: Identify individual times and convert to one-day works.

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

  2. Step 2: Add the one-day works to get the combined rate.

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

  3. Step 3: Invert the combined rate to find total time.

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

  4. Final Answer:

    4.8 days → Option A

  5. Quick Check:

    Check by multiplying time × combined rate: (24/5) × (5/24) = 1 (complete work) ✅

Quick Variations

1. More than two workers: add all individual one-day works (e.g., A + B + C = 1/TA + 1/TB + 1/TC).

2. One worker does part of the job first, then both work together - compute work done by the first part, subtract from 1, then use combined rate for remaining work.

3. Workers with different units (hours vs days): convert to the same time unit before adding rates.

4. When efficiencies given (e.g., A is k times as efficient as B), convert to rates using ratios and then compute combined time.

Trick to Always Use

  • Step 1 → Convert each person's time to one-day work (reciprocal).
  • Step 2 → Add all one-day works to get combined rate.
  • Step 3 → Invert the combined rate to get total time (Time = 1 ÷ combined rate).

Summary

Summary

For combined work problems:

  • Always convert times to one-day works first (use reciprocals).
  • Add rates (do not add times) to get the combined rate.
  • Take the reciprocal of the combined rate to find the time required when working together.
  • Use the same unit (days/hours) across all workers and include a quick check by multiplying time × combined rate to ensure the result equals 1.

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.

Mock Test

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