Solving Age Ratio Problems: Finding the Sum of Ages

This problem involves finding the sum of the ages of two individuals, A and B, after a certain number of years, based on their age ratios at different points in time. We are given the ratio of their ages eight years ago and the ratio of their present ages.

Understanding the Age Ratios

We are given two key pieces of information about the ages of A and B:

• Ratio 8 years ago: The ratio of A's age to B's age eight years ago was 5 : 4.
• Present Ratio: The ratio of A's present age to B's present age is 6 : 5.

We need to use these ratios to find their current ages and then calculate the sum of their ages seven years from now.

Step-by-Step Solution

Let's break down the problem into manageable steps.

Step 1: Represent Ages Using Variables

Let the ages of A and B eight years ago be $5x$ and $4x$ years respectively, based on the given ratio 5:4.

If their ages eight years ago were $5x$ and $4x$, then their present ages must be:

• Present age of A $= 5x + 8$ years
• Present age of B $= 4x + 8$ years

Step 2: Use the Present Age Ratio to Form an Equation

We are given that the ratio of their present ages is 6 : 5. So, we can write the equation:

\begin{equation*} \frac{\text{Present age of A}}{\text{Present age of B}} = \frac{6}{5} \end{equation*}

Substituting the expressions for their present ages:

\begin{equation*} \frac{5x + 8}{4x + 8} = \frac{6}{5} \end{equation*}

Step 3: Solve the Equation for x

Now, we solve this equation for the variable $x$ by cross-multiplication:

\begin{align*} 5 \times (5x + 8) &= 6 \times (4x + 8) \\ 25x + 40 &= 24x + 48 \end{align*}

Subtract $24x$ from both sides:

\begin{align*} 25x - 24x + 40 &= 48 \\ x + 40 &= 48 \end{align*}

Subtract 40 from both sides:

\begin{equation*} x = 48 - 40 \\ x = 8 \end{equation*}

Step 4: Calculate the Present Ages

Now that we have the value of $x$, we can find their present ages:

• Present age of A $= 5x + 8 = 5(8) + 8 = 40 + 8 = 48$ years.
• Present age of B $= 4x + 8 = 4(8) + 8 = 32 + 8 = 40$ years.

Let's quickly check the present age ratio: $48 : 40$. Dividing both by 8, we get $6 : 5$, which matches the given information.

Step 5: Calculate Ages After 7 Years

We need to find the sum of their ages after 7 years from now. First, let's find their ages after 7 years:

• Age of A after 7 years $= \text{Present age of A} + 7 = 48 + 7 = 55$ years.
• Age of B after 7 years $= \text{Present age of B} + 7 = 40 + 7 = 47$ years.

Step 6: Calculate the Sum of Ages After 7 Years

Finally, we find the sum of their ages after 7 years:

Sum of ages after 7 years = Age of A after 7 years + Age of B after 7 years

Sum $= 55 + 47 = 102$ years.

Therefore, the sum of the ages of A and B after 7 years from now will be 102 years.

Revision Table: Key Information

Additional Information: Solving Age Problems

Age-related problems in mathematics often involve ratios and setting up linear equations. Here are some tips:

• Always define variables clearly, usually representing an unknown multiplier for the ratio or the current age.
• Translate the information given for different time periods (past, present, future) into algebraic expressions involving the variables.
• Use the given ratios or sums/differences of ages to form equations.
• Solve the equations to find the value of the variable.
• Once the variable is found, calculate the required ages at the specified time periods.
• Pay attention to whether the problem asks for individual ages or the sum/difference of ages.
• Always check your final answer against the conditions given in the original problem.

These types of problems are common in competitive exams and help test your ability to translate word problems into algebraic models and solve them systematically.