A Mathematical Puzzle: The Ratio of Ages

In this article, we will explore a classic mathematical puzzle involving the ratio of ages among a father, a son, and a mother. The problem requires us to utilize the ratio of two individuals' ages and a future scenario to uncover the relationship between their present ages. Let's dive into the details and solve this intriguing problem step by step.

The Problem Statement

Currently, the ratio between the ages of a father and a son is given as 5:2. However, four years from now, the ratio between the son's age and his mother's age will be 1:2. Our goal is to determine what the ratio between the current ages of the father and the mother is.

Setting Up the Equations

Let's denote the present ages of the father, son, and mother as F, S, and M respectively. According to the problem, we can write the following equations:

The Current Ratio (1:2.5)

The ratio of the father's and the son's ages is given as F/S 5/2. This can be rewritten as:

F (5/2)S ...[1]

The Future Ratio (1:2)

Four years from now, the son's future age will be S 4 and the mother's future age will be M 4. The future ratio of the son's age to the mother's age is given as (S 4) / (M 4) 1/2. This can be rewritten as:

2(S 4) M 4

Expanding and simplifying the above equation, we get:

2S 8 M 4 ...[2]

Substituting and Solving for M

We can now substitute equation [1] into equation [2] to find the relationship between F and M.

Substitution Process

From equation [1], we know that:

S (2/5)F

Substituting this into equation [2] gives us:

M 2((2/5)F) 4 (4/5)F 4

Finding the Ratio Between Father and Mother's Ages

Now that we have expressions for S and M in terms of F, we can find the ratio between the present ages of the father and the mother:

F/M F/((4/5)F 4)

To simplify this equation, we can multiply the numerator and the denominator by 5:

F/M (5F) / (4F 20)

Determining the Value of F

To find the value of F that satisfies the ratio, let's test a value of F 20:

Verification

If F 20, then:

S (2/5) times; 20 8

M (4/5) times; 20 4 16 4 20

Substituting back into the ratio, we get:

F/M 20/20 1

Thus, the ratio of the present ages of the father and the mother is 5:4.

Conclusion

Therefore, the final answer is:

The ratio between the present ages of the father and the mother is 5:4.

Related Articles

1. How to Solve Ratio Problems in Mathematics

2. Understanding the Concept of Ratios and Proportions

3. Age Calculation and Mathematical Puzzles