Understanding the Age Word Problem

This problem involves the ages of two people, A and B, at different points in time: the present and 4 years ago. We are given the age difference between them now and the ratio of their ages 4 years ago. Our goal is to find the present age of B.

Setting Up the Equations for the Age Problem

Let's use variables to represent their present ages:

• Let the present age of A be \(A\) years.
• Let the present age of B be \(B\) years.

We are given two conditions:

1. At present, A is younger than B by 8 years.
2. 4 years ago, their ages were in the ratio 1 ∶ 2.

Let's translate these conditions into mathematical equations.

Condition 1: Present Age Difference

A is 8 years younger than B. This means the difference between B's age and A's age is 8 years.

\(B - A = 8\)

We can express A's age in terms of B's age from this equation:

\(A = B - 8\) (Equation 1)

Condition 2: Age Ratio 4 Years Ago

First, let's determine their ages 4 years ago:

• A's age 4 years ago was \(A - 4\) years.
• B's age 4 years ago was \(B - 4\) years.

The ratio of their ages 4 years ago was 1 ∶ 2. This can be written as a fraction:

\(\frac{A - 4}{B - 4} = \frac{1}{2}\) (Equation 2)

Solving the System of Equations to Find Present Age

Now we have a system of two equations with two variables (\(A\) and \(B\)):

1. \(A = B - 8\)
2. \(\frac{A - 4}{B - 4} = \frac{1}{2}\)

We can substitute the expression for \(A\) from Equation 1 into Equation 2.

Substitute \(A = B - 8\) into \(\frac{A - 4}{B - 4} = \frac{1}{2}\):

\(\frac{(B - 8) - 4}{B - 4} = \frac{1}{2}\)

Simplify the numerator:

\(\frac{B - 12}{B - 4} = \frac{1}{2}\)

Now, we can cross-multiply to solve for \(B\):

\(2 \times (B - 12) = 1 \times (B - 4)\)

\(2B - 24 = B - 4\)

Collect the \(B\) terms on one side and the constant terms on the other side:

\(2B - B = 24 - 4\)

\(B = 20\)

So, the present age of B is 20 years.

Verifying the Solution

Let's check if this value of B satisfies the original conditions.

• If B's present age is 20, then A's present age (using \(A = B - 8\)) is \(20 - 8 = 12\) years.
• 4 years ago, A's age was \(12 - 4 = 8\) years.
• 4 years ago, B's age was \(20 - 4 = 16\) years.
• The ratio of their ages 4 years ago was A:B = \(8:16\).

Simplify the ratio \(8:16\) by dividing both numbers by their greatest common divisor, which is 8:

\(\frac{8}{8} : \frac{16}{8} = 1 : 2\)

The ratio 4 years ago is 1:2, which matches the second condition given in the problem. The age difference now is \(20 - 12 = 8\), which matches the first condition.

Both conditions are satisfied, confirming that the present age of B is 20 years.

The present age of B is 20 years.

Revision Table: Key Steps in Solving Age Problems

Additional Information: Solving Age Problem Concepts

Age problems are common in quantitative aptitude. They typically involve setting up linear equations based on given conditions about ages at different times (past, present, future).

• Representing Ages: Always start by defining variables, usually for the present ages.
• Handling Time Shifts: If a condition refers to an age \(x\) years ago, subtract \(x\) from the present age. If it refers to an age \(y\) years from now, add \(y\) to the present age. This applies to everyone mentioned in the problem.
• Forming Equations: Use the relationships given (differences, sums, ratios, products) to form equations involving the ages at the specified times.
• Solving Equations: Use substitution or elimination methods to solve the system of equations and find the unknown ages.
• Checking the Answer: Always plug the calculated ages back into the original problem statement to ensure all conditions are met.

Ratio problems, like the one discussed, often lead to fractional equations that can be solved by cross-multiplication.