Understanding the Age Ratio Problem

The question asks us to find the sum of the present ages of two people, A and B. We are given information about the ratio of their ages at two different points in time: four years ago and eight years from now.

Setting Up the Equations

Let's denote the present age of A as $A$ and the present age of B as $B$. We can translate the information given into algebraic equations:

• Four years ago: The ages were $(A-4)$ and $(B-4)$. The ratio was 4:5.

  This gives us the equation: $\frac{A - 4}{B - 4} = \frac{4}{5}$

• Eight years from now: The ages will be $(A+8)$ and $(B+8)$. The ratio will be 11:13.

  This gives us the equation: $\frac{A + 8}{B + 8} = \frac{11}{13}$

Solving the First Ratio Equation

From the first condition:

$\frac{A - 4}{B - 4} = \frac{4}{5}$

Cross-multiplying gives:

$5(A - 4) = 4(B - 4)$

$5A - 20 = 4B - 16$

Rearranging the terms to form a standard linear equation:

$5A - 4B = 20 - 16$

$5A - 4B = 4$ (Equation 1)

Solving the Second Ratio Equation

From the second condition:

$\frac{A + 8}{B + 8} = \frac{11}{13}$

Cross-multiplying gives:

$13(A + 8) = 11(B + 8)$

$13A + 104 = 11B + 88$

Rearranging the terms:

$13A - 11B = 88 - 104$

$13A - 11B = -16$ (Equation 2)

Solving the System of Linear Equations

Now we need to solve the system of two linear equations:

1. $5A - 4B = 4$
2. $13A - 11B = -16$

We can use the method of elimination. Let's eliminate B. Multiply Equation 1 by 11 and Equation 2 by 4:

Subtract the second resulting equation from the first:

$(55A - 44B) - (52A - 44B) = 44 - (-64)$

$55A - 52A - 44B + 44B = 44 + 64$

$3A = 108$

Now, solve for A:

$A = \frac{108}{3}$

$A = 36$

Substitute the value of $A$ back into Equation 1 to find B:

$5(36) - 4B = 4$

$180 - 4B = 4$

$180 - 4 = 4B$

$176 = 4B$

$B = \frac{176}{4}$

$B = 44$

Calculating the Sum of Present Ages

The present age of A is 36 years, and the present age of B is 44 years.

The sum of their present ages is:

Sum = $A + B$

Sum = $36 + 44$

Sum = 80 years

Verification

Let's check if these ages satisfy the conditions:

• Four years ago: A was $36 - 4 = 32$, B was $44 - 4 = 40$. Ratio = $\frac{32}{40} = \frac{4}{5}$. (Correct)
• Eight years from now: A will be $36 + 8 = 44$, B will be $44 + 8 = 52$. Ratio = $\frac{44}{52} = \frac{11}{13}$. (Correct)

The calculated ages are correct.