Solving Age Ratio Problems in Algebra: A Comprehensive Guide

Age ratio problems, such as those involving the current and future ages of individuals, are a fundamental aspect of algebra. These problems often require the application of basic algebraic principles and can provide insight into the relationships between variables. In this article, we will explore various methods to solve these problems, using detailed examples to illustrate the steps involved.

Problem 1: The Ratio of A and B's Ages

Given that the ages of A and B are in the ratio 8:3, six years from now, their ages will be in the ratio 9:4. Determine the present ages of A and B.

Solution:

Let the current ages of A and B be 8x and 3x, respectively.

Current ages: A 8x, B 3x Six years from now: A 6 9x, B 6 4x

Substitute the values:

$$8x 6 9x$$ $$3x 6 4x$$

Both equations simplify to:

$$x 6$$

Therefore:

Current age of A: A 8x 8 * 6 48 years Current age of B: B 3x 3 * 6 18 years

The correct solution is A 48 years and B 18 years.

Problem 2: Another Approach to Age Ratio Problems

This problem involves a different set of ratios and variables. Let's solve it step by step.

Solution:

Problem: A:B current age ratio 5:7. After 6 years, their ages will be 6 more and the ratio of their ages will be 7:9.

New ratio after 6 years: A 6 7, B 6 9.

Express the new ratios in terms of x:

$$frac{A 6}{B 6} frac{7}{9}$$

Let A 5x and B 7x, where x is a common factor:

$$frac{5x 6}{7x 6} frac{7}{9}$$

Cross multiply to solve for x:

$$9(5x 6) 7(7x 6)$$ $$45x 54 49x 42$$ $$45x - 49x 42 - 54$$ $$-4x -12$$ $$x 3$$

Substitute x back into the original ratios to find the current ages:

Current age of A: A 5x 5 * 3 15 years Current age of B: B 7x 7 * 3 21 years

Problem 3: Another Example

Another problem involves the current ages of A and B, where A is 35 years old and B is 49 years old. The ratio of their ages is 5:7. After 7 years, the ratio of their ages will be 4:5.

Solution:

Current ages: A 35 years, B 49 years

Ratio after 7 years: (frac{A 7}{B 7} frac{4}{5})

Let A 5x and B 7x, where x is a common factor:

$$frac{A 7}{B 7} frac{4}{5}$$

Substitute the values:

$$frac{5x 7}{7x 7} frac{4}{5}$$

Cross multiply to solve for x:

$$5(5x 7) 4(7x 7)$$ $$25x 35 28x 28$$ $$25x - 28x 28 - 35$$ $$-3x -7$$ $$x frac{7}{3}$$

Substitute x back into the original ratios to find the current ages:

Current age of A: A 5x 5 * (frac{7}{3}) (frac{35}{3}) ≈ 11.67 years (rounded to two decimal points) Current age of B: B 7x 7 * (frac{7}{3}) (frac{49}{3}) ≈ 16.33 years (rounded to two decimal points)

Conclusion

In conclusion, solving age ratio problems in algebra requires a systematic approach. By setting up the correct equations and solving them step by step, we can determine the present ages of individuals or variables. Understanding these methods is essential for solving a wide range of algebraic problems involving ratios and proportions.