Solving Mathematical Puzzles: The Daughter, Mother, and Father's Age Problem

Mysteries and puzzles have always captivated the human mind, offering a fun challenge while exercising our analytical skills. One such classic problem involves the ages of a daughter, mother, and father. Let's dive into the solution of this intriguing puzzle and explore how to approach it step by step.

Introduction to the Problem

The problem states the following relationships:

The daughter's age is half of her mother's age. The father's age is 15 years more than the mother's age. The total age of the daughter, mother, and father is 100.

We are to determine each individual's age.

Setting Up the Equations

Let's denote the ages as follows:

Daughter's age: ( D ) Mother's age: ( M ) Father's age: ( F )

Based on the problem, we can set up the following equations:

( D frac{1}{2} M ) ( F M 15 ) ( D M F 100 )

Substitution and Solving the Equations

First, we'll substitute ( D ) and ( F ) in terms of ( M ) into the third equation.

From the first equation:

$$ D frac{1}{2} M $$

From the second equation:

$$ F M 15 $$

Substituting these into the third equation:

$$ frac{1}{2} M M (M 15) 100 $$

Combine like terms:

$$ frac{1}{2} M M M 15 100 $$ frac{5}{2} M 15 100 $$

Subtract 15 from both sides:

$$ frac{5}{2} M 85 $$

Multiply both sides by (frac{2}{5}) to solve for ( M ):

$$ M 85 times frac{2}{5} 34 $$

Now, using the value of ( M ), we can determine the ages of the daughter and the father:

The daughter's age (( D )): $$ D frac{1}{2} M frac{1}{2} times 34 17 $$ The father's age (( F )): $$ F M 15 34 15 49 $$

Verification

To verify, let's sum up the ages:

$$ D M F 17 34 49 100 $$

The calculations are correct, and the puzzle is solved.

Conclusion

The ages of the daughter, mother, and father are as follows:

Daughter: 17 years old Mother: 34 years old Father: 49 years old

Thus, the solution to the problem is:

( F 49 ), ( M 34 ), ( D 17 )

Mathematics is a powerful tool for solving such puzzles, and understanding the steps involved helps in both problem-solving and logical thinking.