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Product of Ages or Squared Ages

Introduction

कुछ age problems में product of ages या squared ages की जानकारी दी जाती है। Difference या ratio देने के बजाय, question ages के multiplication या squared-age relationship देता है। ऐसे questions simple algebra से आसानी से solve हो जाते हैं।

Statements को equation में translate करने के बाद, आप factorization या quadratic formula से step-by-step solve कर सकते हैं।

Pattern: Product of Ages or Squared Ages

Pattern

मुख्य idea:

अगर दो ages का product दिया है, तो younger age को x मानें और elder age को (x + difference) लिखें।

Equation बनाएँ: x × (x + difference) = product. इसे solve करने पर quadratic equation मिलती है जिसे factor करके solve किया जाता है।

Step-by-Step Example

Question

Neha’s और Rahul’s ages का product 180 है। Rahul, Neha से 3 years older है। उनकी present ages निकालें।

Options:
  • A: Neha = 12 years, Rahul = 15 years
  • B: Neha = 10 years, Rahul = 13 years
  • C: Neha = 9 years, Rahul = 12 years
  • D: Neha = 15 years, Rahul = 18 years

Solution

  1. Step 1: Ages को represent करें।

    Neha की present age = x
    Rahul की age = x + 3

  2. Step 2: Product equation लिखें।

    The product of their ages is 180.
    x × (x + 3) = 180 → x² + 3x - 180 = 0

  3. Step 3: Quadratic equation solve करें।

    ऐसे दो numbers खोजें जिनका product = -180 और sum = 3 हो → 15 और -12
    • Equation: x² + 3x - 180 = 0
    • Middle term split: x² - 12x + 15x - 180 = 0
    • Group: (x² - 12x) + (15x - 180) = 0
    • Factorize: x(x - 12) + 15(x - 12) = 0
    • Common factor: (x + 15)(x - 12) = 0
    • Solve: x + 15 = 0 → x = -15 (ignore) ; x - 12 = 0 → x = 12
    इसलिए, Neha = 12 years और Rahul = 12 + 3 = 15 years.

  4. Final Answer:

    Neha = 12 years; Rahul = 15 years → Option A

  5. Quick Check:

    Product: 12 × 15 = 180 ✅
    Difference: 15 - 12 = 3 ✅

Quick Variations

कभी-कभी questions में squared ages होते हैं, जैसे - “elder age का square minus younger age का square given है।” ऐसे cases में भी उम्रों को variable से represent करके quadratic equation बनाई जाती है।

Trick to Always Use

  • Step 1: Younger age = x, elder age = x + difference.
  • Step 2: Product या squared relation से equation बनाएँ।
  • Step 3: Quadratic equation carefully solve करें और negative age discard करें।
  • Step 4: Actual ages निकालकर product या square relation verify करें।

Summary

Summary

  • आप younger age को variable मानें और elder age को उसी variable के expression के रूप में लिखें।
  • Product या squared-age relation से equation बनाएँ और उसे quadratic form में simplify करें।
  • Factorization या quadratic formula से solve करें; negative roots ignore करें।
  • Values वापस रखकर original condition verify करें।

याद रखने लायक example:
Product problems में younger = x, elder = x + d, equation x(x + d) = P बनाएँ और solve करें।

Practice

(1/5)
1. The product of ages of two friends is 96. If one is 12 years old, find the other.
easy
A. 6
B. 7
C. 8
D. 9

Solution

  1. Step 1: Use the product relation.

    The product of their ages is 96 and one friend is 12.

  2. Step 2: Divide to find the other age.

    Other friend = 96 ÷ 12 = 8.

  3. Final Answer:

    8 → Option C

  4. Quick Check:

    12 × 8 = 96 ✅
Hint: Divide the product by the known age.
Common Mistakes: Multiplying instead of dividing.
2. The square of Ramesh’s age is 144. What is his age?
easy
A. 10
B. 11
C. 12
D. 14

Solution

  1. Step 1: Use the squared-age relation.

    Age² = 144.

  2. Step 2: Take square root.

    Age = √144 = 12.

  3. Final Answer:

    12 → Option C

  4. Quick Check:

    12² = 144 ✅
Hint: Take square root directly.
Common Mistakes: Forgetting that only the positive root applies.
3. The product of two siblings’ ages is 180. If one is 12, what is the other?
easy
A. 12
B. 13
C. 14
D. 15

Solution

  1. Step 1: Use the product relation.

    Product = 180 and one sibling is 12.

  2. Step 2: Divide to find other age.

    Other = 180 ÷ 12 = 15.

  3. Final Answer:

    15 → Option D

  4. Quick Check:

    12 × 15 = 180 ✅
Hint: Divide product by known age.
Common Mistakes: Division mistakes.
4. The product of ages of a father and son is 480. If father is 40, find the son’s age.
medium
A. 10
B. 11
C. 12
D. 13

Solution

  1. Step 1: Write the product relation.

    Product = 480, father = 40.

  2. Step 2: Divide to get son's age.

    Son = 480 ÷ 40 = 10.

  3. Final Answer:

    10 → Option A

  4. Quick Check:

    40 × 10 = 480 ✅
Hint: Divide product by elder’s age.
Common Mistakes: Multiplying instead of dividing.
5. If the square of a boy’s age is 225, find his age.
medium
A. 13
B. 14
C. 15
D. 16

Solution

  1. Step 1: Apply the squared-age relation.

    Age² = 225.

  2. Step 2: Take square root.

    Age = √225 = 15.

  3. Final Answer:

    15 → Option C

  4. Quick Check:

    15² = 225 ✅
Hint: Use square root for squared age equations.
Common Mistakes: Choosing incorrect root.

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