Solving Age Problems: Finding the Sum of Ages

This problem involves finding the ages of two individuals, A and B, based on two given conditions relating their ages. We need to determine the sum of their ages.

Setting Up the Algebraic Equations

Let $a$ represent the age of A (in years) and $b$ represent the age of B (in years).

From the first statement: "The sum of the age of A and $5\ times$ the age of B is $36\ years$." This translates to the equation:

$a + 5b = 36 \quad (1)$

From the second statement: "When $3\ times$ the age of A is added to $7\ times$ the age of B, the result is $62\ years$." This translates to the equation:

$3a + 7b = 62 \quad (2)$

Step-by-Step Solution Using Elimination Method

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

1. $a + 5b = 36$
2. $3a + 7b = 62$

To solve this system, we can use the elimination method. Let's eliminate $a$. Multiply equation (1) by 3:

$3 \times (a + 5b) = 3 \times 36$

$3a + 15b = 108 \quad (3)$

Now, subtract equation (2) from equation (3):

$\begin{array}{l} (3a + 15b) - (3a + 7b) = 108 - 62 \\ 3a + 15b - 3a - 7b = 46 \\ 8b = 46 \end{array}$

Solve for $b$:

$b = \frac{46}{8} = \frac{23}{4} = 5.75$

Now, substitute the value of $b$ back into equation (1) to find $a$:

$a + 5b = 36$

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

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

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

To subtract, find a common denominator:

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

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

$a = \frac{144 - 115}{4} = \frac{29}{4} = 7.25$

Calculating the Sum of Ages

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

$a + b = \frac{29}{4} + \frac{23}{4}$

$a + b = \frac{29 + 23}{4}$

$a + b = \frac{52}{4}$

$a + b = 13$

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