How to Find the Sum of Two People's Present Ages Based on Past and Future Ratios

Mathematics often invites us to solve intriguing puzzles related to real-life scenarios. One such interesting puzzle involves understanding the relationship between the ages of two individuals based on their past and future age ratios. In this article, we will explore a specific example where the past and future age ratios of Samir and Ashok are given, and our goal is to find their current sum of ages.

The Problem

One year ago, the ratio of Samir's age to Ashok's age was 4:3. One year from now, the ratio of their ages will be 5:4. We need to calculate the sum of their present ages.

Solving the Equation

Let S be Samir's current age, and A be Ashok's current age.

Step 1: Setting up the equations based on past and future ratios.

Age one year ago:

(frac{S - 1}{A - 1} frac{4}{3})

Cross-multiplying gives:

3S - 3 4A - 4

Simplifying, we get:

Equation 1: (3S - 4A -1)

Age one year from now:

(frac{S 1}{A 1} frac{5}{4})

Cross-multiplying gives:

4S 4 5A 5

Simplifying, we get:

Equation 2: (4S - 5A 1)

Step 2: Solving the system of equations using the elimination method.

Multiplying Equation 1 by 4 and Equation 2 by 3 to align the coefficients of S:

Equation 3: (12S - 16A -4)

Equation 4: (12S - 15A 3)

Subtracting Equation 4 from Equation 3:

(-16A 15A -4 - 3)

(-A -7)

Thus, (A 7)

Step 3: Substituting A back into Equation 1 to find S.

3S - 4(7) -1

3S - 28 -1

3S 27

Therefore, (S 9)

Samir's current age is 9 years, and Ashok's current age is 7 years.

The sum of their present ages is:

(S A 9 7 16)

The sum of their present ages is 16 years.

Conclusion

By carefully setting up and solving equations based on the given ratios, we can accurately determine the present ages of both individuals. This method not only helps in solving mathematical puzzles but also enhances our problem-solving skills in real-life scenarios involving age and ratio relationships.