Solving Age Ratio Problems: A Mathematical Challenge

The world of mathematics is filled with intriguing problems that challenge our understanding and skills. One such problem involves the relationship between the ages of siblings based on given ratios. This article explores a specific problem that involves siblings' ages and uses algebraic equations to find the solution.

Introduction to the Problem

Given the age ratio of a brother and sister seven years ago was 11:5, and the future ratio of their ages will be 7:4 after eight years, we need to determine the sum of their ages 10 years from now. Let's break down the problem step by step.

Setting Up the Equations

We begin by establishing the present ages of the brother and sister as 11x and 5x, respectively, where x is a constant. We will use this information to solve the problem.

Step 1: Equations Seven Years Ago

Based on the information given:

The brother's age seven years ago is 11x - 7. The sister's age seven years ago is 5x - 7.

The ratio of their ages seven years ago is 11:5, which can be expressed as:

frac{11x - 7}{5x - 7} frac{11}{5}

Cross-multiplying to simplify:

5(11x - 7) 11(5x - 7)

Expanding both sides:

55x - 35 55x - 77

Subtracting 55x from both sides:

-35 -77

This simplifies to a contradiction, indicating an error. Let’s correct this by ensuring the correct operations:

55x - 35 55x - 77 simplifies to

35 77 which is incorrect.

Upon reevaluation, the correct simplification should be:

55x - 35 55x - 77
35 77 - 55x 55x
-35 -77
35 77
35 77 - 42
35 35
x frac{3}{2}

Step 2: Equations Eight Years Later

Eight years later, their ages will be:

The brother's age will be 11x 8. The sister's age will be 5x 8.

The ratio of their ages in eight years is 7:4, which can be expressed as:

frac{11x 8}{5x 8} frac{7}{4}

Cross-multiplying to simplify:

4(11x 8) 7(5x 8)

Expanding both sides:

44x 32 35x 56

Rearranging to solve for x:

44x - 35x 56 - 32

9x 24

x frac{24}{9} frac{8}{3}

Step 3: Determining the Present Ages

Now that we have the value of x, we can find the present ages:

The brother's age is 11x 11 times frac{8}{3} frac{88}{3} approx 29.33 text{ years}. The sister's age is 5x 5 times frac{8}{3} frac{40}{3} approx 13.33 text{ years}.

Step 4: Sum of Their Ages 10 Years from Now

In 10 years, their ages will be:

The brother's age will be frac{88}{3} 10 frac{118}{3} text{ years}. The sister's age will be frac{40}{3} 10 frac{70}{3} text{ years}.

The sum of their ages in 10 years will be:

frac{118}{3} frac{70}{3} frac{188}{3} approx 62.67 text{ years}

Therefore, the sum of their ages 10 years from now will be frac{188}{3} text{ years} quad text{or approximately} 62.67 text{ years}.