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Basic Work Formula (W = R × T)

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Introduction

Every time and work problem revolves around three core elements - Work (W), Rate or Efficiency (R), and Time (T). Understanding their relationship helps you easily calculate how long a task takes or how much work is completed.

This pattern forms the foundation for all types of time and work problems, including combined work, efficiency, and wages.

Pattern: Basic Work Formula (W = R × T)

Pattern: Basic Work Formula (W = R × T)

The key concept is: Work = Rate × Time (W = R × T).

From this, we can derive:

  • R = W ÷ T → Rate (work done per unit time)
  • T = W ÷ R → Time taken for given work

Step-by-Step Example

Question

A can complete a piece of work in 10 days. Find the fraction of work A does in one day.

Options:

  • A. 1/10
  • B. 1/5
  • C. 1/8
  • D. 1/20

Solution

  1. Step 1: Assume total work.

    Assume the whole work W = 1 unit (standard convention).

  2. Step 2: Identify time given.

    A completes the work in 10 days → T = 10.

  3. Step 3: Compute the rate (work per day).

    Using W = R × T → R = W ÷ T = 1 ÷ 10 = 1/10.

  4. Final Answer:

    1/10 of the work in one day → Option A

  5. Quick Check:

    In 10 days A does 10 × (1/10) = 1 unit → completes the whole work ✅

Quick Variations

1. Finding the time when rate and total work are given.

2. Finding the rate (efficiency) when total work and time are known.

3. Expressing rate as a fraction of work done per day or per hour.

Trick to Always Use

  • Step 1: Assume total work = 1 unit unless stated otherwise.
  • Step 2: Use W = R × T to find any missing variable.
  • Step 3: For daily efficiency, use R = 1 ÷ total days.

Summary

  • Identify the formula relation: Work = Rate × Time and use it to set up the equation.
  • Convert the given information into total work or rate as needed (e.g., assume W = 1 unit).
  • Solve for the missing variable (R or T) using algebraic substitution.
  • Verify the result by plugging it back into W = R × T to see if the original condition holds.

Example to remember:
Assume whole work = 1 unit → rate = 1 ÷ days → multiply back to check the total.

Practice

(1/5)
1. A can complete a piece of work in 8 days. What fraction of work does A complete in one day?
easy
A. 1/8
B. 1/6
C. 1/10
D. 1/12

Solution

  1. Step 1: Assume total work = 1 unit.

  2. Step 2: Compute daily rate.

    A's daily work = 1 ÷ 8 = 1/8.

  3. Final Answer:

    1/8 → Option A

  4. Quick Check:

    8 × (1/8) = 1 (whole work) ✅
Hint: Daily work = 1 ÷ total days.
Common Mistakes: Mixing up time taken with rate of work.
2. B completes a work in 12 days. How much work will B complete in 3 days?
easy
A. 1/2
B. 1/4
C. 1/3
D. 1/5

Solution

  1. Step 1: Compute daily rate.

    Daily work of B = 1/12.

  2. Step 2: Work in 3 days.

    3 × (1/12) = 1/4.

  3. Final Answer:

    1/4 → Option B

  4. Quick Check:

    12 × (1/12) = 1 (full work) ✅
Hint: Multiply rate × number of days.
Common Mistakes: Dividing instead of multiplying when finding work done.
3. C can do 1/5 of a work in one day. In how many days can C complete the whole work?
easy
A. 4
B. 6
C. 5
D. 8

Solution

  1. Step 1: Identify daily rate.

    R = 1/5.

  2. Step 2: Use T = 1 ÷ R.

    T = 1 ÷ (1/5) = 5 days.

  3. Final Answer:

    5 days → Option C

  4. Quick Check:

    5 × (1/5) = 1 (whole work) ✅
Hint: Total days = reciprocal of one-day work.
Common Mistakes: Multiplying instead of taking reciprocal.
4. D can finish a work in 15 days. E can finish the same work in 10 days. What fraction of work will D complete compared to E in one day?
medium
A. 2/3
B. 3/2
C. 1/2
D. 1/3

Solution

  1. Step 1: Find rates.

    D = 1/15, E = 1/10.

  2. Step 2: Compare rates.

    (1/15) ÷ (1/10) = 10/15 = 2/3.

  3. Final Answer:

    2/3 → Option A

  4. Quick Check:

    (2/3) × (1/10) = 1/15 → D’s rate matches correctly ✅
Hint: Compare using (1/T₁) ÷ (1/T₂).
Common Mistakes: Using total time instead of daily rates.
5. A and B take 6 days and 8 days respectively to complete a job individually. B works alone until he completes half the work, then A finishes the remaining half alone. How many days in total are required to finish the job?
medium
A. 3
B. 4
C. 5
D. 7

Solution

  1. Step 1: Time for B to complete half.

    B's rate = 1/8 → time for 1/2 = (1/2) ÷ (1/8) = 4 days.

  2. Step 2: Time for A to finish remaining half.

    A's rate = 1/6 → time for 1/2 = (1/2) ÷ (1/6) = 3 days.

  3. Final Answer:

    Total = 4 + 3 = 7 days → Option D

  4. Quick Check:

    B completes 1/2 in 4 days, A completes 1/2 in 3 days → total work done = 1 ✅
Hint: For fractional work, use: time = fraction ÷ rate.
Common Mistakes: Assuming both work together.