Understanding the Age Problem

This problem asks us to find the sum of the ages of two people, A and B. We are given two pieces of information that relate their ages, which we can translate into mathematical equations. This is a classic example of solving a system of linear equations.

Setting Up the Equations

Let the age of person A be represented by the variable $A$, and the age of person B be represented by the variable $B$.

• The first statement says: "The sum of the age of A and 5 times the age of B is 36 years." This can be written as the equation:

  $A + 5B = 36 \quad (Equation \, 1)$

• The second statement says: "When 3 times the age of A is added to 7 times the age of B, the result is 62 years." This can be written as the equation:

  $3A + 7B = 62 \quad (Equation \, 2)$

Solving the System of Equations

We now have a system of two linear equations with two variables:

1. $A + 5B = 36$

2. $3A + 7B = 62$

We can solve this system using the elimination method. The goal is to eliminate one variable ($A$ or $B$) by manipulating the equations.

1. Multiply Equation 1 by 3 to make the coefficient of $A$ the same as in Equation 2:

   $3 \times (A + 5B) = 3 \times 36$

   $3A + 15B = 108 \quad (Equation \, 3)$

2. Subtract Equation 2 from Equation 3 to eliminate $A$:

   $(3A + 15B) - (3A + 7B) = 108 - 62$

   $3A + 15B - 3A - 7B = 46$

   $8B = 46$

3. Solve for $B$:

   $B = \frac{46}{8}$

   $B = \frac{23}{4}$

   $B = 5.75$

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

   $A + 5 \left( \frac{23}{4} \right) = 36$

   $A + \frac{115}{4} = 36$

5. Solve for $A$:

   $A = 36 - \frac{115}{4}$

   $A = \frac{36 \times 4}{4} - \frac{115}{4}$

   $A = \frac{144}{4} - \frac{115}{4}$

   $A = \frac{144 - 115}{4}$

   $A = \frac{29}{4}$

   $A = 7.25$

Calculating the Sum of Ages

The question asks for the sum of the ages of A and B, which is $A + B$.

• $A + B = \frac{29}{4} + \frac{23}{4}$
• $A + B = \frac{29 + 23}{4}$
• $A + B = \frac{52}{4}$
• $A + B = 13$

So, the sum of the ages of A and B is 13 years.