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Product of Ages or Squared Ages

Introduction

Some age problems involve the product of ages or squared ages. Instead of differences or ratios, the question gives the multiplication of ages or a squared-age relationship. These problems can be solved using simple algebra.

Once you translate the statements into an equation, you can solve them step by step using factorization or quadratic formula.

Pattern: Product of Ages or Squared Ages

Pattern

The key idea:

If the product of two ages is given, represent the younger age as x and the older as (x + difference).

Form the equation: x × (x + difference) = product. Solve the quadratic equation to get the ages.

Step-by-Step Example

Question

The product of Neha’s and Rahul’s ages is 180. Rahul is 3 years older than Neha. Find their present ages.

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: Represent the ages.

    Let Neha’s present age be x.
    Rahul’s age = x + 3.

  2. Step 2: Write the product equation.

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

  3. Step 3: Solve the quadratic equation.

    Find two numbers whose product = -180 and sum = 3 → 15 and -12.
    • Write the quadratic equation: x² + 3x - 180 = 0.
    • Split the middle term: x² - 12x + 15x - 180 = 0.
    • Group terms: (x² - 12x) + (15x - 180) = 0.
    • Factor each group: x(x - 12) + 15(x - 12) = 0.
    • Factor common term: (x + 15)(x - 12) = 0.
    • Solve for x: x + 15 = 0 → x = -15 (ignore negative age); x - 12 = 0 → x = 12.
    Thus, Neha’s age = 12 years and Rahul’s age = 12 + 3 = 15 years.

  4. Final Answer:

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

  5. Quick Check:

    Product: 12 × 15 = 180 ✅ (matches the given condition).
    Difference: 15 - 12 = 3 ✅ (matches the given condition).

Quick Variations

Sometimes the problem involves squared ages such as "the square of the elder age minus the square of the younger age is given". Represent the ages similarly with variables and form a quadratic equation.

Trick to Always Use

  • Step 1: Represent the younger age as x and older as x + difference.
  • Step 2: Form the equation using product or squared relationship.
  • Step 3: Solve quadratic equation carefully, discard negative age.
  • Step 4: Find actual ages and verify by checking product or square.

Summary

Summary

  • Represent the younger age as a variable and the elder as an expression in that variable.
  • Form the product or squared-age equation and simplify to a standard quadratic.
  • Factor or use the quadratic formula to solve; discard non-physical (negative) roots.
  • Substitute back to get both ages and verify the original conditions (product/difference).

Example to remember:
With product problems, set younger = x, elder = x + d, form x(x + d) = P, solve for x and check the answers.

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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