Solving Age Problems Using Algebra: A Man and His Daughter

Understanding and solving age-related problems is a fundamental skill in algebra. Let's explore a classic puzzle involving a man and his daughter, as well as various approaches to derive their ages in the present and in the future. This article will explore the process step-by-step using both direct and simultaneous equation methods.

Problem Statement

Problem: A man is 12 years older than his daughter. Five years ago, their ages were in the ratio of 5:3. What are their ages in 2 years time?

Algebraic Solution

Step 1: Define Variables
Let ( x ) be the current age of the daughter. Then, the current age of the man (father) would be ( x 12 ).

Step 2: Past Ages
Five years ago, the daughter's age was ( x - 5 ), and the father's age was ( (x 12) - 5 x 7 ).

Step 3: Set Up the Equation
According to the problem, their ages 5 years ago were in the ratio 5:3. Therefore, we can write the equation:

(frac{x 7}{x - 5} frac{5}{3})

Step 4: Cross Multiply and Solve
Cross-multiplying gives:

(3(x 7) 5(x - 5))

Expanding and rearranging the equation:

(3x 21 5x - 25)

(21 25 5x - 3x)

(46 2x)

(x 23)

Step 5: Determine Current Ages
The current age of the daughter is ( 23 ), and the current age of the man (father) is ( 23 12 35 ).

Step 6: Future Ages
In 2 years, the daughter will be ( 23 2 25 ) years old, and the father will be ( 35 2 37 ) years old.

Alternative Solution: Simplified Approach

A second approach is to directly use the given ratios and simplify the process:

Step 1: Use Given Ratios
Five years ago, the ratio of their ages was 5:3. If we let the daughter's age be ( 3x ) and the father's age be ( 5x ), then five years ago the daughter was ( 3x - 5 ) and the father was ( 5x - 5 ).

Step 2: Solve for ( x )
If the father is currently 12 years older than the daughter, then:

(5x - (3x 5) 12)

Simplifying:

(2x - 5 12)

(2x 17)

(x 8.5)

However, because the problem involves whole years, this step appears to suggest an error. Recheck the initial algebraic method, which provides correct ages.

Step 3: Calculate Current and Future Ages
From the algebraic method, the correct current ages are 23 for the daughter and 35 for the father. In 2 years, the daughter will be 25 years old, and the father will be 37 years old.

Simultaneous Equation Solution

Step 1: Define the Premises
M - D 12, and (M - 5) / (D - 5) 5/3.

Step 2: Simultaneous Equation Setup
Lets set up the equations:

M - D 12 and (M - 5) / (D - 5) 5/3.

From the first equation, ( M D 12 ).

Substituting into the second equation:

(frac{D 12 - 5}{D - 5} frac{5}{3})

(frac{D 7}{D - 5} frac{5}{3})

Cross-multiplying and solving:

(3(D 7) 5(D - 5))

(3D 21 5D - 25)

(3D - 5D -25 - 21)

(-2D -46)

(D 23)

Thus, ( M 23 12 35).

Step 3: Future Ages
In 2 years, the daughter will be ( 23 2 25 ) and the father will be ( 35 2 37 ).

Conclusion

The accurate ages in the present and in 2 years can be derived using algebraic methods, such as simulating and solving simultaneous equations. These methods provide a structured and reliable approach to solving age problems in algebra.