Solving Age Ratio Problems: Finding Present Age

This problem involves using ratios to determine the present ages of two individuals, A and B, based on their current age ratio and their age ratio after a certain number of years.

Understanding the Age Ratios

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

• The ratio of their present ages is 3 : 5.
• The ratio of their ages five years from now will be 13 : 20.

A ratio like 3:5 means that for some common factor, say \(x\), the present age of A is \(3x\) and the present age of B is \(5x\).

Setting Up the Equations

Let the common factor for their present ages be \(x\).

• Present age of A = \(3x\) years
• Present age of B = \(5x\) years

Now, let's consider their ages after five years:

• Age of A after 5 years = \((3x + 5)\) years
• Age of B after 5 years = \((5x + 5)\) years

We are told that the ratio of their ages five years after becomes 13 : 20. We can write this as an equation:

\(\frac{3x + 5}{5x + 5} = \frac{13}{20}\)

Solving for the Unknown Variable \(x\)

To solve for \(x\), we can cross-multiply the equation:

\(20 \times (3x + 5) = 13 \times (5x + 5)\)

Now, distribute the numbers on both sides:

\(20 \times 3x + 20 \times 5 = 13 \times 5x + 13 \times 5\)

\(60x + 100 = 65x + 65\)

Next, we need to isolate \(x\) on one side of the equation. Subtract \(60x\) from both sides:

\(100 = 65x - 60x + 65\)

\(100 = 5x + 65\)

Now, subtract 65 from both sides:

\(100 - 65 = 5x\)

\(35 = 5x\)

Finally, divide by 5 to find the value of \(x\):

\(x = \frac{35}{5}\)

\(x = 7\)

Calculating the Present Age of B

We defined the present age of B as \(5x\). Now that we have the value of \(x\), we can calculate B's present age:

Present age of B = \(5 \times x = 5 \times 7\)

Present age of B = \(35\) years.

Verification

Let's verify if these ages fit the condition for five years later:

• Present age of A = \(3 \times 7 = 21\) years
• Present age of B = \(5 \times 7 = 35\) years
• Age of A after 5 years = \(21 + 5 = 26\) years
• Age of B after 5 years = \(35 + 5 = 40\) years

The ratio of their ages after 5 years is 26 : 40. Dividing both parts by their greatest common divisor, 2, we get 13 : 20, which matches the ratio given in the problem. This confirms our calculation is correct.

Conclusion on Present Age

Based on the calculations, the present age of B is 35 years.

Revision Table: Key Concepts for Age Problems

Additional Information: Solving Age Ratio Problems

Age-based problems are common in quantitative aptitude tests. They typically involve ratios or differences between ages at different points in time (past, present, or future).

Here are some tips for tackling age ratio problems:

• Always use a variable (like \(x\)) to represent the common part of the ratio when dealing with unknown ages.
• Clearly define the ages for each person at each specified time point (present, future, past).
• Formulate an equation based on the given information, usually involving the ages at a specific time or the difference/sum of ages.
• Solve the algebraic equation carefully to find the value of the variable.
• Substitute the value of the variable back into the expressions for the ages to find the required age(s).
• It's a good practice to verify your answer by plugging the calculated ages back into the original problem statement to ensure all conditions are met.

These problems often require careful reading to distinguish between present, past, and future ages and their corresponding ratios or differences. Setting up the initial variables correctly based on the present ratio is a crucial first step.