Understanding the Age Ratio Problem

This problem involves finding the ratio of the ages of two individuals, A and B, at a specific point in the future, given information about the ratio of their ages at two different points in the past. We are given the ratio of their ages 8 years ago and 4 years ago. We need to use this information to find their current ages and then calculate the ratio of their ages 8 years from now.

Setting Up Equations Based on Age Ratios

Let's denote the current age of A as \(A_0\) years and the current age of B as \(B_0\) years.

According to the question:

• 8 years ago: Their ages were \(A_0 - 8\) and \(B_0 - 8\). The ratio was 2 : 3.

  This gives us the equation: \(\frac{A_0 - 8}{B_0 - 8} = \frac{2}{3}\)

  Cross-multiplying, we get: \(3(A_0 - 8) = 2(B_0 - 8)\)

  \(3A_0 - 24 = 2B_0 - 16\)

  Rearranging gives our first linear equation: \(3A_0 - 2B_0 = 24 - 16 \implies 3A_0 - 2B_0 = 8\) (Equation 1)

• 4 years ago: Their ages were \(A_0 - 4\) and \(B_0 - 4\). The ratio was 5 : 7.

  This gives us the equation: \(\frac{A_0 - 4}{B_0 - 4} = \frac{5}{7}\)

  Cross-multiplying, we get: \(7(A_0 - 4) = 5(B_0 - 4)\)

  \(7A_0 - 28 = 5B_0 - 20\)

  Rearranging gives our second linear equation: \(7A_0 - 5B_0 = 28 - 20 \implies 7A_0 - 5B_0 = 8\) (Equation 2)

Solving for Current Ages

Now we have a system of two linear equations with two variables, \(A_0\) and \(B_0\):

Equation 1: \(3A_0 - 2B_0 = 8\)

Equation 2: \(7A_0 - 5B_0 = 8\)

We can solve this system using methods like substitution or elimination. Let's use elimination:

1. Multiply Equation 1 by 5: \(5 \times (3A_0 - 2B_0) = 5 \times 8 \implies 15A_0 - 10B_0 = 40\) (Equation 3)
2. Multiply Equation 2 by 2: \(2 \times (7A_0 - 5B_0) = 2 \times 8 \implies 14A_0 - 10B_0 = 16\) (Equation 4)
3. Subtract Equation 4 from Equation 3:

   \((15A_0 - 10B_0) - (14A_0 - 10B_0) = 40 - 16\)

   \(15A_0 - 14A_0 - 10B_0 + 10B_0 = 24\)

   \(A_0 = 24\)

4. Substitute the value of \(A_0 = 24\) into Equation 1:

   \(3(24) - 2B_0 = 8\)

   \(72 - 2B_0 = 8\)

   \(72 - 8 = 2B_0\)

   \(64 = 2B_0\)

   \(B_0 = \frac{64}{2} = 32\)

So, the current age of A is 24 years, and the current age of B is 32 years.

Calculating Future Ages and Their Ratio

We need to find the ratio of their ages 8 years from now.

• Age of A in 8 years = \(A_0 + 8 = 24 + 8 = 32\) years.
• Age of B in 8 years = \(B_0 + 8 = 32 + 8 = 40\) years.

The ratio of their ages 8 years from now is:

\(\frac{\text{Age of A in 8 years}}{\text{Age of B in 8 years}} = \frac{32}{40}\)

Simplifying the ratio:

\(\frac{32}{40} = \frac{8 \times 4}{8 \times 5} = \frac{4}{5}\)

The ratio of their ages 8 years from now will be 4 : 5.

Summary of Ages

Revision Table: Key Concepts in Age Problems

Additional Information: Solving Systems of Linear Equations

Solving age problems often involves solving systems of linear equations. Here's a brief look at common methods:

• Substitution Method: Solve one equation for one variable (e.g., \(A_0\) in terms of \(B_0\)), then substitute that expression into the other equation. This reduces the system to a single equation with one variable, which can then be solved.
• Elimination Method: Multiply one or both equations by constants so that the coefficients of one variable become opposites or equal. Then add or subtract the equations to eliminate that variable, leaving a single equation with the other variable. This is the method used in the detailed solution above.
• Graphical Method: Graph both linear equations on the same coordinate plane. The point of intersection of the two lines represents the solution (\(A_0, B_0\)) that satisfies both equations. This method is less precise for non-integer solutions.
• Matrix Method: Systems of linear equations can also be solved using matrix operations, such as finding the inverse of the coefficient matrix or using Cramer's rule. This is typically used for larger systems.

Choosing the right method depends on the structure of the equations. For two equations with two variables, substitution or elimination are usually the most straightforward.