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Ages in Ratio Form

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Introduction

कई age problems में ages को exact numbers में नहीं, बल्कि एक ratio में दिया जाता है। जैसे: "A : B = 3 : 4 और उनका sum 56 है।" ऐसे questions aptitude tests में बहुत common होते हैं और इन्हें ratio को actual values में बदलकर solve किया जाता है।

तरीका बहुत simple है: ratio parts को एक variable (multiplier) के साथ लिखो, फिर sum/difference/future/past जैसी जानकारी से equation बनाओ, solve करो और verify करो।

Pattern: Ages in Ratio Form

Pattern: Ages in Ratio Form

Key idea:

अगर ages a : b ratio में हों, तो actual ages को a·k और b·k लिखते हैं (जहाँ k एक positive multiplier है)।

दी गई extra जानकारी (sum, difference, future/past condition) से equation बनाकर k निकालते हैं।

Step-by-Step Example

Question

A और B की उम्र का ratio 3 : 5 है। दोनों की उम्र का sum 64 years है। Present ages निकालो।

Solution

  1. Step 1: Ages को multiplier के साथ लिखो।

    Words: मान लो multiplier k है। Math: A = 3k, B = 5k.

  2. Step 2: Sum condition लगाओ।

    Words: A + B = 64 → (3k + 5k) = 64. Math: 8k = 64.

  3. Step 3: k निकालो।

    Math: k = 64 ÷ 8 = 8.

  4. Step 4: Actual ages निकालो।

    Math: A = 3 × 8 = 24, B = 5 × 8 = 40.

  5. Step 5: Final Answer.

    A = 24 years, B = 40 years

  6. Step 6: Quick Check.

    Sum check: 24 + 40 = 64 ✅ Ratio check: 24 : 40 → divide by 8 → 3 : 5 ✅ Answer correct है।

Question

X और Y की उम्र का ratio 4 : 7 है। 6 years बाद ratio 5 : 8 हो जाता है। Present ages निकालो।

Solution

  1. Step 1: Present ages को multiplier से लिखो।

    Words: Present ages = 4k और 7k. Math: X = 4k, Y = 7k.

  2. Step 2: 6 साल बाद की उम्र लिखो।

    Words: दोनों में 6 add करो। Math: X_after6 = 4k + 6, Y_after6 = 7k + 6.

  3. Step 3: Future ratio से equation बनाओ।

    Words: 6 साल बाद ratio = 5 : 8 → (4k + 6)/(7k + 6) = 5/8.

  4. Step 4: Cross-multiply करके k निकालो।

    Math: 8(4k + 6) = 5(7k + 6) → 32k + 48 = 35k + 30 → 48 - 30 = 35k - 32k → 18 = 3k → k = 6

  5. Step 5: Present ages निकालो।

    Math: X = 4 × 6 = 24, Y = 7 × 6 = 42

  6. Step 6: Quick Check.

    6 साल बाद: X = 30, Y = 48 Ratio = 30 : 48 → divide by 6 → 5 : 8 ✅ Present ages सही हैं।

Quick Variations

अगर age difference दिया हो: (a·k - b·k) = difference का formula use करके k निकालो। Example: A : B = 2 : 3 और A, B से 4 साल छोटा है → (3k - 2k) = 4 → k = 4.

अगर sum future/past में दिया हो: दोनों ages को उतने साल आगे/पीछे shift करके sum condition apply करो।

अगर दो से ज़्यादा लोग हों: Ratios जैसे a : b : c को भी उसी तरह k से multiply करके ages represent करो।

Trick to Always Use

  • Step 1: Ages = ratio × k (youngest या convenient term)।
  • Step 2: Sum/difference/future/past condition से k की equation बनाओ।
  • Step 3: k निकालकर actual ages compute करो।
  • Step 4: हमेशा numeric condition और ratio दोनों verify करो।

Summary

Ages in Ratio Form वाले questions ऐसे solve होते हैं:

  • Ages को a·k, b·k (या a·k : b·k : c·k) में लिखो।
  • दी गई condition (sum/difference/future/past) से k निकालो।
  • Actual ages compute करो।
  • Quick Check से verify करो।

यह method ratio-based age problems को fast और reliable बनाता है।

Practice

(1/5)
1. The ages of A and B are in the ratio 3 : 5. If A is 18 years old, how old is B?
easy
A. 30
B. 25
C. 28
D. 32

Solution

  1. Step 1: Identify ratio and known value

    Ratio A : B = 3 : 5, A = 18 → 3 parts = 18.

  2. Step 2: Find the value of one part

    1 part = 18 ÷ 3 = 6.

  3. Step 3: Compute the unknown age

    B = 5 × 6 = 30.

  4. Final Answer:

    30 years → Option A

  5. Quick Check:

    18 : 30 = 3 : 5 ✅
Hint: Divide the known age by its ratio part to find one unit, then multiply.
Common Mistakes: Taking direct difference instead of proportional calculation.
2. The ratio of the ages of a father and son is 7 : 2. If the father is 42 years old, what is the son’s age?
easy
A. 10
B. 12
C. 14
D. 16

Solution

  1. Step 1: Note the ratio and the given age

    Father : Son = 7 : 2, Father = 42 → 7 parts = 42.

  2. Step 2: Calculate one part

    1 part = 42 ÷ 7 = 6.

  3. Step 3: Find son's age

    Son = 2 × 6 = 14.

  4. Final Answer:

    14 years → Option C

  5. Quick Check:

    42 : 14 = 7 : 2 ✅
Hint: Find one ratio unit by dividing the known age by its part.
Common Mistakes: Using subtraction (42 - 2) instead of ratio scaling.
3. The present ages of A and B are in the ratio 5 : 7. If their total is 72 years, find their ages.
medium
A. 30 and 42
B. 28 and 44
C. 32 and 40
D. 35 and 37

Solution

  1. Step 1: Add ratio parts

    Ratio = 5 : 7 → total parts = 12.

  2. Step 2: Determine one part value

    1 part = 72 ÷ 12 = 6.

  3. Step 3: Compute each age

    A = 5 × 6 = 30, B = 7 × 6 = 42.

  4. Final Answer:

    A = 30, B = 42 → Option A

  5. Quick Check:

    30 + 42 = 72 and ratio = 5 : 7 ✅
Hint: Divide total by sum of ratio parts, then multiply by each part.
Common Mistakes: Multiplying directly without dividing total by total parts.
4. The ratio of ages of P and Q is 4 : 3. If Q is 21 years old, how old is P?
medium
A. 24
B. 27
C. 28
D. 36

Solution

  1. Step 1: Identify ratio and given age

    P : Q = 4 : 3, Q = 21 → 3 parts = 21.

  2. Step 2: Compute one part

    1 part = 21 ÷ 3 = 7.

  3. Step 3: Find P's age

    P = 4 × 7 = 28.

  4. Final Answer:

    28 years → Option C

  5. Quick Check:

    28 : 21 = 4 : 3 ✅
Hint: Divide given age by its ratio part to find one unit.
Common Mistakes: Taking 21 as 4 parts instead of 3.
5. The ages of A, B, and C are in the ratio 2 : 3 : 5. If their total age is 50 years, find C’s age.
medium
A. 20
B. 25
C. 30
D. 15

Solution

  1. Step 1: Sum the ratio parts

    Ratio = 2 : 3 : 5 → total parts = 10.

  2. Step 2: Get value of one part

    1 part = 50 ÷ 10 = 5.

  3. Step 3: Compute C's age

    C = 5 × 5 = 25.

  4. Final Answer:

    25 years → Option B

  5. Quick Check:

    A=10, B=15, C=25 → sum = 50 ✅
Hint: Split the total in ratio parts and multiply for each age.
Common Mistakes: Using difference instead of ratio multiplication.