0
0

Ratio of Time and Work

Introduction

कई time-and-work problems में दो या अधिक workers की तुलना days से नहीं, बल्कि ratios से की जाती है। Time, work और efficiency का ratio form में relation समझकर आप आसानी से इनके बीच convert कर सकते हैं और बिना heavy algebra के comparative questions solve कर सकते हैं।

यह pattern important है क्योंकि ratio reasoning competitive aptitude tests में बहुत आती है और इससे लंबे calculations को shortcut किया जा सकता है।

Pattern: Ratio of Time and Work

Pattern

Key concept: Efficiency ∝ 1/Time और Work ∝ Efficiency × Time. इन्हीं relations से ratios के बीच convert किया जाता है।

Core relations जो याद रखने ज़रूरी हैं:

  • Efficiency (E) वही काम करने के लिए लिए गए time का reciprocal है: E ∝ 1/T.
  • Work done = Efficiency × Time: W = E × T.
  • अगर A : B = a : b time ratio है, तो उनकी efficiencies 1/a : 1/b = b : a होंगी।
  • अगर efficiencies का ratio p : q है, तो times का ratio q : p होगा (inverse relation)।

Step-by-Step Example

Question

A और B किसी काम को ऐसे times में पूरा करते हैं जिनका ratio 2 : 3 है। अगर A अकेले वह काम 12 days में करता है, तो B कितने दिनों में करेगा? साथ ही उनकी efficiencies का ratio क्या होगा?

Solution

  1. Step 1: दिए गए ratio को पहचानें और समझें कि यह क्या बताता है।

    Times A : B = 2 : 3. इसका मतलब A का time 2k और B का time 3k है, जहाँ k कोई positive number है।

  2. Step 2: A के actual time से k निकालें।

    A का actual time = 12 days = 2k ⇒ k = 12 ÷ 2 = 6.

  3. Step 3: B का time निकालें।

    B का time = 3k = 3 × 6 = 18 days.

  4. Step 4: Efficiency ratio निकालें (time ratio का inverse)।

    Efficiency A : B = (1/Time_A) : (1/Time_B) = (1/2k) : (1/3k) = 3 : 2.

  5. Final Answer:

    B काम 18 days में करेगा। Efficiency ratio A : B = 3 : 2.

  6. Quick Check:

    अगर A रोज़ 1/12 और B रोज़ 1/18 काम करें, तो combined rate = (3 + 2)/36 = 5/36। इसका inverse = 36/5 = 7.2 days - ज़रूरी नहीं था पर consistency दिखाता है ✅

Quick Variations

1. Efficiency ratio दिया हो तो time ratio पाने के लिए उसे invert करें।

2. Time ratio और किसी एक का actual time दिया हो तो k निकालकर बाकी values निकालें।

3. Ratios और fractional work (जैसे A, B से double work करे same time में) हों तो efficiencies बनाकर फिर time निकालें।

4. Ratios को percentages से मिलाकर questions (जैसे A is 20% faster than B → 6:5) solve करें।

Trick to Always Use

  • Step 1 → किसी भी time ratio को efficiency ratio में बदलने के लिए numbers invert करें।
  • Step 2 → Actual time दिया हो तो ratio के part से divide करके k निकालें।
  • Step 3 → Fractional-work या combined-work checks के लिए W = E × T का उपयोग करें।

Summary

Summary

Key takeaways for the Ratio of Time and Work pattern:

  • Time और efficiency एक-दूसरे के inverse होते हैं - ratios convert करते समय numbers swap करें।
  • Actual value मिलने पर k निकालकर ratio वाले parts को actual numbers में बदलें।
  • Combined-work या fractional-work के लिए daily rates (reciprocals) का उपयोग करें और W = rate × time लगाएं।
  • Quick check ज़रूर करें: निकाले गए time × rate = 1 unit work आना चाहिए।

Practice

(1/5)
1. The ratio of time taken by A and B to complete a work is 3 : 4. If A can finish the work in 15 days, how many days will B take?
easy
A. 20 days
B. 18 days
C. 22 days
D. 24 days

Solution

  1. Step 1: Express times using ratio

    Time ratio A : B = 3 : 4 ⇒ A = 3k, B = 4k.

  2. Step 2: Find the scale factor k

    A's actual time = 15 days ⇒ 3k = 15 ⇒ k = 5.

  3. Step 3: Compute B's time

    B's time = 4k = 4 × 5 = 20 days.

  4. Final Answer:

    20 days → Option A

  5. Quick Check:

    15 : 20 = 3 : 4 ✅
Hint: Multiply the given time by (B_part ÷ A_part).
Common Mistakes: Swapping ratio parts or dividing when you should multiply.
2. The ratio of efficiencies of A and B is 5 : 3. If B can complete the job in 30 days, how long will A take to complete it alone?
easy
A. 20 days
B. 18 days
C. 15 days
D. 24 days

Solution

  1. Step 1: Convert efficiency ratio to time ratio

    Efficiency A : B = 5 : 3 ⇒ Time A : B = 3 : 5 (inverse relation).

  2. Step 2: Scale the time ratio using B's time

    B's time = 30 days corresponds to 5k = 30 ⇒ k = 6.

  3. Step 3: Compute A's time

    A's time = 3k = 3 × 6 = 18 days.

  4. Final Answer:

    18 days → Option B

  5. Quick Check:

    Time ratio 18 : 30 = 3 : 5 ⇒ efficiencies 5 : 3 ✅
Hint: Invert efficiency ratio to get time ratio, then scale with the given time.
Common Mistakes: Using the efficiency ratio directly as time instead of inverting it.
3. A and B take 10 and 25 days respectively to finish a work. What is the ratio of their efficiencies (A : B)?
easy
A. 2 : 5
B. 5 : 12
C. 5 : 2
D. 25 : 10

Solution

  1. Step 1: Use reciprocals of times for efficiencies

    Efficiency ∝ 1/Time. So efficiency ratio = 1/10 : 1/25.

  2. Step 2: Compute and simplify the ratio

    1/10 : 1/25 = 25 : 10 = simplify → 5 : 2.

  3. Final Answer:

    5 : 2 → Option C

  4. Quick Check:

    Time ratio 10 : 25 = 2 : 5 ⇒ inverse = 5 : 2 ✅
Hint: Take reciprocals of times, then simplify the ratio.
Common Mistakes: Forgetting to invert the times when computing efficiencies.
4. A can do a work in 12 days and B in 20 days. What is the ratio of work done by A and B in one day?
medium
A. 2 : 3
B. 4 : 3
C. 1 : 1
D. 5 : 3

Solution

  1. Step 1: Compute one-day work for each

    A's one-day work = 1/12, B's = 1/20.

  2. Step 2: Form and simplify the ratio

    1/12 : 1/20 = 20 : 12 = simplify → 5 : 3.

  3. Final Answer:

    5 : 3 → Option D

  4. Quick Check:

    Inverse of 12 : 20 = 5 : 3 ✅
Hint: Invert the time ratio and simplify to get one-day work ratio.
Common Mistakes: Confusing A:B order when inverting ratios.
5. A is twice as efficient as B. Together they finish a job in 9 days. How many days will A alone take to finish the work?
medium
A. 27/2 days
B. 18/5 days
C. 15/2 days
D. 12/5 days

Solution

  1. Step 1: Set up rate-parts

    Let B's one-day work = x ⇒ A's = 2x. Combined = 3x = 1/9 (since together they finish in 9 days).

  2. Step 2: Solve for x

    x = 1/(9 × 3) = 1/27. So A's one-day work = 2/27.

  3. Step 3: Invert to get A's time

    A's time = 1 ÷ (2/27) = 27/2 = 13.5 days.

  4. Final Answer:

    27/2 days (13.5 days) → Option A

  5. Quick Check:

    Combined rate = (2/27) + (1/27) = 3/27 = 1/9 ⇒ total time = 9 days ✅
Hint: Express rates as parts (x and 2x), set sum = 1/total days, solve for x and invert for A's time.
Common Mistakes: Confusing rate-parts with time-parts or arithmetic slips when inverting fractions.

Mock Test

Ready for a challenge?

Take a 10-minute AI-powered test with 10 questions (Easy-Medium-Hard mix) and get instant SWOT analysis of your performance!

10 Questions
5 Minutes