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Fraction of Work Done / Remaining

Introduction

In time and work problems, it’s often useful to know what fraction of work has been completed or left after a certain time. This pattern helps you determine how much work is done (or still pending) when only partial time or partial workers are involved.

This concept is crucial for understanding progress tracking, efficiency analysis, and problems where workers leave or join midway.

Pattern: Fraction of Work Done / Remaining

Pattern

The key idea is: Work done = Time taken ÷ Total time.

Similarly, Fraction of work remaining = 1 - (Time taken ÷ Total time).

Step-by-Step Example

Question

A can complete a work in 15 days. What fraction of the work will he have done after 9 days? Also, what fraction remains?

Solution

  1. Step 1: Identify given values

    Time taken = 9 days; Total time to finish the work = 15 days.

  2. Step 2: Compute fraction of work done

    Fraction of work done = Time taken ÷ Total time = 9 ÷ 15 = 3/5.

  3. Step 3: Compute remaining fraction

    Work remaining = 1 - 3/5 = 2/5.

  4. Final Answer:

    Work done = 3/5; Work remaining = 2/5.

  5. Quick Check:

    After full 15 days: 15 ÷ 15 = 1 (complete work) ✅

Quick Variations

1. Finding how much work is left after some days.

2. Calculating time required for a given fraction of work.

3. Combined fraction problems when multiple workers join or leave.

4. Applied in Pipes & Cistern or Efficiency-based mixed work questions.

Trick to Always Use

  • Step 1: Always use the ratio of time taken to total time for fraction done.
  • Step 2: For remaining work, simply subtract from 1.
  • Step 3: If multiple workers are involved, use their combined rate to find total time first.

Summary

Summary

In the Fraction of Work Done / Remaining pattern:

  • Work done = Time taken ÷ Total time.
  • Work left = 1 - Work done.
  • Useful for progress tracking and partial completion problems.

Practice

(1/5)
1. A can complete a work in 12 days. What fraction of the work will be completed in 3 days?
easy
A. 1/3
B. 1/4
C. 1/2
D. 1/6

Solution

  1. Step 1: Identify given values

    Total time = 12 days; time taken = 3 days.

  2. Step 2: Compute fraction of work done

    Fraction of work done = 3 ÷ 12 = 1/4.

  3. Final Answer:

    Work done = 1/4 → Option B.

  4. Quick Check:

    3 × (1/12) = 3/12 = 1/4 ✅
Hint: Divide time taken by total time to get fraction done.
Common Mistakes: Using full-time instead of partial time or failing to simplify the fraction.
2. B can do a job in 10 days. What fraction of the work remains after 6 days of work?
easy
A. 2/5
B. 1/5
C. 3/5
D. 4/5

Solution

  1. Step 1: Identify given values

    Total time = 10 days; time worked = 6 days.

  2. Step 2: Compute work done

    Work done = 6 ÷ 10 = 3/5.

  3. Step 3: Compute remaining work

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

  4. Final Answer:

    Work left = 2/5 → Option A.

  5. Quick Check:

    60% done → 40% left = 2/5 ✅
Hint: Remaining work = 1 - (fraction completed).
Common Mistakes: Adding fractions instead of subtracting from 1.
3. A can finish a work in 15 days. What fraction of the work will A complete in 5 days?
easy
A. 1/2
B. 1/4
C. 1/3
D. 2/3

Solution

  1. Step 1: Identify given values

    Total time = 15 days; time taken = 5 days.

  2. Step 2: Compute fraction completed

    Fraction = 5 ÷ 15 = 1/3.

  3. Final Answer:

    Work completed = 1/3 → Option C.

  4. Quick Check:

    5 × (1/15) = 1/3, matches result ✅
Hint: Fraction = time taken ÷ total time.
Common Mistakes: Incorrect simplification of 5/15.
4. A and B together can finish a work in 8 days. They work together for 3 days. What fraction of the work remains?
medium
A. 5/8
B. 3/8
C. 1/2
D. 1/4

Solution

  1. Step 1: Identify time worked

    Total time = 8 days; worked time = 3 days.

  2. Step 2: Compute fraction completed

    Completed = 3 ÷ 8 = 3/8.

  3. Step 3: Compute fraction remaining

    Remaining = 1 - 3/8 = 5/8.

  4. Final Answer:

    Remaining work = 5/8 → Option A.

  5. Quick Check:

    Done 3/8; left 5/8 → total = 1 ✔
Hint: Fraction done = worked_days / total_days.
Common Mistakes: Subtracting wrongly or confusing done vs remaining.
5. A can complete a work in 18 days and B can complete the same work in 12 days. Working together, what fraction of the work will be completed in 6 days?
medium
A. 1/2
B. 3/5
C. 2/3
D. 5/6

Solution

  1. Step 1: Compute individual daily rates

    A's rate = 1/18, B's rate = 1/12.

  2. Step 2: Compute combined daily work

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

  3. Step 3: Work done in 6 days

    6 × (5/36) = 30/36 = 5/6.

  4. Final Answer:

    Work completed = 5/6 → Option D.

  5. Quick Check:

    Remaining = 1 - 5/6 = 1/6 ✔
Hint: Add one-day works, then multiply by days worked.
Common Mistakes: Adding days instead of rates.

Mock Test

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