Solving Age Problems: Logical Reasoning with Ratios and Averages

Understanding how to solve age problems, especially those involving ratios and averages, can be quite beneficial. This article will guide you through solving a specific problem and explain the logic behind each step. By the end, you’ll have a clear understanding of the process and be able to apply it to similar problems.

Problem Statement: The average age of a man and his son is 35 years. The ratio of their ages is 5:2. What is the son's age?

Step-by-Step Solution:

Step 1: Setting Up the Problem Using the Ratio

Let the man's age be 5x Let the son's age be 2x

Step 2: Utilizing the Average Age

The average age of the man and his son is given as 35 years. This translates to the equation:

(5x 2x) / 2 35

Simplifying this equation:

(7x / 2) 35

Multiplying both sides by 2:

7x 70

Divide both sides by 7:

x 10

Step 3: Finding the Son's Age

Now, we can find the son's age:

Son's age 2x 2 * 10 20 years

Thus, the son is 20 years old.

Alternative Solutions

Here, we will explore a few alternative solutions, each with its unique approach:

Alternative Solution 1: Direct Calculation

40 * 2 80 years total

11x 5x 16x 80

x 5

Son’s age 5 * 5 25 years

Alternative Solution 2: System of Equations

Let us denote the ages of the father by F and of the son by S.

The average age of father and son 40 years

Sum of ages of father and son F S average age * 2 80 years

Age of father : Age of son F : S 11 : 5

F/S 11/5

F/S - 1 11/5 - 1

F - S/S (11 - 5)/5 16/5

80/S 16/5

S 80 * 5/16 25 years

Age of father 80 year - 25 year 55 years

Alternative Solution 3: Simplified Steps

M S 2 * 40 80

11x 5x 16x 80

x 5

Man's age M 11 * 5 55 years

Son's age S 5 * 5 25 years

Conclusion

Each of the solutions provided above leads to the conclusion that the son's age is 20 years. This problem-solving approach can be applied to similar age problems, helping you break down complex scenarios into simpler steps.