Understanding the Age Problem

This problem involves finding the sum of the ages of two individuals, A and B, based on two given conditions relating their ages. We can represent their ages using variables and set up a system of linear equations to solve for their individual ages and then their sum.

Setting Up the Equations

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

From the first statement, "The sum of the age of A and 5 times the age of B is 55 years", we can write the first equation:

$ A + 5B = 55 \quad (Equation \, 1) $

From the second statement, "When 3 times the age of A is added to 7 times the age of B, the result is 99 years", we can write the second equation:

$ 3A + 7B = 99 \quad (Equation \, 2) $

Solving the System of Equations

We need to find the value of $A + B$. We can solve the system of equations using either the substitution method or the elimination method. Let's use the elimination method.

1. Multiply Equation 1 by 3 so that the coefficient of $A$ matches in both equations:

   $ 3 \times (A + 5B) = 3 \times 55 $

   $ 3A + 15B = 165 \quad (Equation \, 3) $

2. Now, subtract Equation 2 from Equation 3 to eliminate $A$:

   $ (3A + 15B) - (3A + 7B) = 165 - 99 $

   $ 3A + 15B - 3A - 7B = 66 $

   $ 8B = 66 $

3. Solve for $B$:

   $ B = \frac{66}{8} $

   $ B = \frac{33}{4} $

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

   $ A + 5 \left( \frac{33}{4} \right) = 55 $

   $ A + \frac{165}{4} = 55 $

   $ A = 55 - \frac{165}{4} $

   To subtract, find a common denominator:

   $ A = \frac{55 \times 4}{4} - \frac{165}{4} $

   $ A = \frac{220}{4} - \frac{165}{4} $

   $ A = \frac{220 - 165}{4} $

   $ A = \frac{55}{4} $

Calculating the Sum of Ages

Now that we have the values for $A$ and $B$, we can find their sum:

$ \text{Sum of Ages} = A + B $

$ A + B = \frac{55}{4} + \frac{33}{4} $

$ A + B = \frac{55 + 33}{4} $

$ A + B = \frac{88}{4} $

$ A + B = 22 $

The sum of the ages of A and B is 22 years.