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Basic Work Formula (W = R × T)

Introduction

हर time and work problem तीन core elements पर आधारित होता है - Work (W), Rate या Efficiency (R), और Time (T)। इनका relationship समझने से यह आसानी से पता लगाया जा सकता है कि कोई काम कितने समय में पूरा होगा या दिए गए समय में कितना काम हुआ है।

यह pattern सभी प्रकार के time and work problems-combined work, efficiency, wages-का foundation बनाता है।

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

Pattern

Key concept: Work = Rate × Time (W = R × T).

इससे हम निकाल सकते हैं:

  • R = W ÷ T → Rate (प्रति unit समय में किया गया काम)
  • T = W ÷ R → दिए गए काम को पूरा करने में लगने वाला समय

Step-by-Step Example

Question

A किसी काम को 10 दिनों में पूरा कर सकता है। एक दिन में A कितना हिस्सा पूरा करेगा?

Options:

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

Solution

  1. Step 1: Total work assume करें।

    पूरा काम W = 1 unit मान लें (standard convention)।

  2. Step 2: दिया गया समय समझें।

    A काम को 10 दिनों में पूरा करता है → T = 10.

  3. Step 3: Rate निकालें (per day work).

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

  4. Final Answer:

    एक दिन में 1/10 काम → Option A

  5. Quick Check:

    10 दिनों में A करता है: 10 × (1/10) = 1 unit → पूरा काम पूरा हुआ ✅

Quick Variations

1. Rate और total work दिए हों तो time निकालना।

2. Total work और time दिए हों तो efficiency (rate) निकालना।

3. Rate को per day या per hour fraction के रूप में व्यक्त करना।

Trick to Always Use

  • Step 1: Total work = 1 unit मान लें (जब तक कुछ और न दिया हो)।
  • Step 2: Missing variable निकालने के लिए W = R × T इस्तेमाल करें।
  • Step 3: Daily efficiency के लिए R = 1 ÷ total days।

Summary

Summary

  • Formula relation पहचानें: Work = Rate × Time, और इसी से equation सेट करें।
  • दिए गए data को total work या rate में convert करें (जैसे W = 1 unit लेना)।
  • Missing variable (R या T) को algebraic substitution से निकालें।
  • Result को वापस W = R × T में डालकर verify करें कि condition match करती है।

याद रखने वाला example:
पूरा काम = 1 unit मानें → rate = 1 ÷ days → multiply करके check करें कि काम पूरा हो रहा है या नहीं।

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.

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