Solving the Age Ratio Puzzle for Parents and Son

Mathematical problems involving age ratios can be intriguing and challenging. This article demonstrates how to solve a particular problem where the age ratio between a father and his son is given. We'll walk through a detailed step-by-step solution, leveraging algebraic equations to uncover the ages and the historical ratio between the two.

Problem Statement

The problem sets a current age ratio between a father and his son as 17:5. After a span of nine years, the age ratio will be 5:2. Additionally, we need to find the ratio between the current age of the son to the father's age six years ago. The given answer states that the ratio six years ago was 1:3. We will validate this step-by-step and solve the problem accurately.

Step-by-Step Solution

Step 1: Define Variables
Let the current age of the father be ( F ) and the son be ( S ).

Step 2: Express Given Ratios as Equations
a. The current ratio is 17:5, so:

[ frac{F}{S} frac{17}{5} ]

From this equation, we can express ( F ) in terms of ( S ):
[ F frac{17}{5}S quad text{(1)} ]

b. After nine years, the ratio will be 5:2, so:

[ frac{F 9}{S 9} frac{5}{2} ]

From this, we derive:

[ 2(F 9) 5(S 9) ]

[ 2F 18 5S 45 quad text{(2)} ]

Step 3: Substitute Equation (1) into Equation (2)

[ 2left(frac{17}{5}Sright) 18 5S 45 ]

[ frac{34}{5}S 18 5S 45 ]

Step 4: Simplify the Equation

Eliminate the fraction by multiplying the entire equation by 5:

[ 34S 90 25S 225 ]

Rearrange to isolate ( S ):
[ 34S - 25S 225 - 90 ]

[ 9S 135 ]

[ S 15 ]

Step 5: Find ( F )

Substitute ( S ) back into equation (1):

[ F frac{17}{5} times 15 51 ]

Step 6: Calculate the Ratio Six Years Ago

Six years ago, the son's age was ( 15 - 6 9 )

And the father's age was ( 51 - 6 45 )

The ratio at that time is:

[ frac{9}{45} frac{1}{5} ]

Conclusion

The solution shows that the ratio between the age of the son and the age of the father six years ago is 1:5, not 1:3 as stated in the given answer. This ensures accuracy and understanding of the age ratio problem.

Additional Examples

Let's verify the provided solutions with similar examples:

Example 1: If we set the son's age as 7x and the father's age as 17x, and the current ratio to be 7:17:

[ frac{17x}{7x} frac{17}{7} ]

Similarly, the ages six years ago are:

[ 17x - 6 quad text{and} quad 7x - 6 ]

The ratio given is 3:1:

[ frac{17x - 6}{7x - 6} frac{3}{1} ]

Cross multiplication gives:

[ 17x - 6 3(7x - 6) ]

[ 17x - 6 21x - 18 ]

[ 21x - 17x -6 18 ]

[ 4x 12 ]

[ x 3 ]

So, the son's age is ( 7x 21 ) and the father's age is ( 17x 51 ).

Final Answer

[ text{Son's age} 21 text{ years}text{, Father's age} 51 text{ years} ]

The ratio six years ago was indeed 1:5 as calculated in the detailed step-by-step solution.