Solving Age Problems Using Algebraic Equations

Age problems are a common type of word problem that frequently appear in mathematics. In this article, we will explore how to solve these problems using algebraic equations, focusing on a specific example and providing step-by-step solutions.

Problem Statement

What is Aman's present age if after 20 years, his age will be 10 times his age 10 years back?

Solution

Let Aman’s present age be (x). According to the problem, in 20 years his age will be (x 20) and his age 10 years back was (x - 10). The equation based on the given condition is:

(x 20 10(x - 10))

Step-by-Step Solution

Expand the right side: (x 20 1 - 100) Rearrange the equation to bring all terms involving (x) to one side and constant terms to the other side: (20 100 1 - x) (120 9x) Divide both sides by 9 to solve for (x): (x frac{120}{9} frac{40}{3} approx 13.33)

Therefore, Aman’s present age is approximately 13 years and 4 months.

Additional Examples

Example 1

A is 30 years old. As present age is thirty years old. This bodes well with A’s present age being 150 of what it was ten years ago.

Let's solve the problem:

Based on the problem, A's age 10 years ago was ( frac{30}{1.5} 20 ). The equation is ( A frac{150}{100}A - 10 ). Solving the equation: ( A 30 ) years old.

Example 2

According to the given information: (A{20} 10A - 10). Solve for (A):

Using the equation:

(A{20} 10A - 100) Solving the equation: ( 9A 120 ). ( A frac{120}{9} 13.33 )

Therefore, Aman’s present age is approximately 13 years and 4 months, with decimals being rounded as necessary.

Example 3

Let Aman’s present age (A).

The equation is:

(A{20} 10A - 10) Solving the equation: ( 9A 120 ). ( A frac{120}{9} 13.33 ) Aman’s present age 13 y 4 m.

Conclusion

Solving age problems using algebraic equations involves setting up equations based on the given information and solving them step by step. By following these methods, you can easily solve a wide range of age problems. Whether it's the initial Aman problem or the variations presented here, understanding how to set up and solve these equations is key to tackling age problems successfully.

Key Takeaways

Use algebraic equations to represent the information given in the problem statement. Rearrange and simplify the equations to solve for the unknown variable. Check your solution by plugging it back into the original equation or problem context.

Practice Problems

If someone’s age 15 years ago was one-third of their age 10 years ago, what is their current age? Find the current age of a person if in 5 years, their age will be twice what it was 10 years ago. Solve the problem where someone’s age in 10 years will be 120 less than three times their current age.