Solving for the Present Ages of Three Individuals Given Their Ratios and a_sum_of Their Ages Decades Ago

In this article, we will explore a problem where the present ages of three individuals are in a certain ratio. We will use algebraic methods to determine their current ages based on the sum of their ages many years ago. This involves setting up equations and solving for the unknowns. This type of problem is common in algebra and can help enhance problem-solving skills in mathematics.

Problem Statement

The present ages of three individuals are in the proportions 4:7:9. Ten years ago, the sum of their ages was 56. Find their present ages in years.

Step-by-Step Solution

Let the present ages of the three individuals be represented as (4x), (7x), and (9x) where (x) is a common multiplier.

Transforming the Ages Ten Years Ago

Ten years ago, their ages would have been:

The first person's age: (4x - 10) The second person's age: (7x - 10) The third person's age: (9x - 10)

Solving the Equation

According to the problem, the sum of their ages ten years ago was 56. Therefore, we can set up the following equation:

(4x - 10) (7x - 10) (9x - 10) 56

We can simplify this equation as follows:

4x 7x 9x - 30 56

2 - 30 56

Adding 30 to both sides:

2 86

Dividing by 20:

x frac{86}{20} 4.3

Determining the Present Ages

Substituting (x) back into the expressions for the present ages:

The first person's age: (4x 4 times 4.3 17.2) years The second person's age: (7x 7 times 4.3 30.1) years The third person's age: (9x 9 times 4.3 38.7) years

Thus, the present ages of the three individuals are approximately:

First person: 17 years Second person: 30 years Third person: 39 years

Conclusion

The present ages of the three individuals in years are approximately 17, 30, and 39. These calculated ages can be verified by checking that their sum is 86, which satisfies the problem's original condition that the total age 10 years ago was 56.

This problem demonstrates the use of algebraic equations and ratios to solve real-world problems and enhance analytical skills. Understanding such relationships can be valuable in many areas of study and practical applications.