Understanding the Age Problem

This question involves finding the present ages of two individuals, A and B, based on a given ratio of their ages and the sum of their ages after a certain number of years.

Setting Up the Equations

Let the present age of A be \(A_p\) and the present age of B be \(B_p\).

We are given the ratio of their present ages:

\(\frac{A_p}{B_p} = \frac{3}{2}\)

We can express their present ages in terms of a common variable, say \(x\). Let:

• Present age of A \( = 3x\) years
• Present age of B \( = 2x\) years

Next, we need to consider their ages after 10 years. Their ages will increase by 10 years.

• Age of A after 10 years \( = 3x + 10\) years
• Age of B after 10 years \( = 2x + 10\) years

We are also given that the sum of their ages after 10 years will be 80 years.

So, we can write the equation:

\((3x + 10) + (2x + 10) = 80\)

Solving the Equation

Now, let's solve this linear equation for \(x\):

\(3x + 10 + 2x + 10 = 80\)

Combine the like terms:

\((3x + 2x) + (10 + 10) = 80\)

\(5x + 20 = 80\)

Subtract 20 from both sides of the equation:

\(5x = 80 - 20\)

\(5x = 60\)

Divide both sides by 5 to find the value of \(x\):

\(x = \frac{60}{5}\)

\(x = 12\)

Calculating Present Ages

Now that we have the value of \(x\), we can find their present ages:

• Present age of A \( = 3x = 3 \times 12 = 36\) years
• Present age of B \( = 2x = 2 \times 12 = 24\) years

Verification

Let's check if these present ages satisfy the condition about the sum of their ages after 10 years.

• Age of A after 10 years \( = 36 + 10 = 46\) years
• Age of B after 10 years \( = 24 + 10 = 34\) years

Sum of their ages after 10 years \( = 46 + 34 = 80\) years.

This matches the information given in the problem, confirming our calculated present ages are correct.

Summary of Ages

The present ages of A and B are 36 years and 24 years, respectively.

Revision Table: Key Concepts

Additional Information: Solving Age Problems

Age problems are common in mathematics and quantitative aptitude tests. They often involve relationships between ages at different points in time (past, present, future).

Key strategies for solving age problems:

• Read the problem carefully and identify the unknown quantities, usually the present ages.
• Assign variables to the unknown ages, often based on a given ratio or relationship.
• Write expressions for the ages at different points in time (e.g., age 'n' years ago, age 'm' years from now). Remember that the difference in age between two people remains constant.
• Formulate equations based on the given conditions (sum, difference, ratio, product of ages).
• Solve the system of equations (usually linear equations) to find the values of the variables.
• Calculate the required ages (present or future) using the solved variable values.
• Verify your answer by plugging the calculated ages back into the original problem statement to ensure all conditions are met.

This specific problem used a ratio and a sum of future ages, leading to a single linear equation in one variable.