Understanding the Age Difference Problem

This problem involves finding the present age of the younger of two people, A and B, based on two pieces of information: the difference in their present ages and a relationship between their ages six years ago.

We need to set up mathematical equations to represent the given conditions and then solve them to find the required age.

Setting Up Equations for Ages

Let's denote the present age of the younger person as \(Y\) years and the present age of the elder person as \(E\) years.

According to the first condition:

• The ages of A and B differ by 16 years.
• This means the difference between the elder's age and the younger's age is 16.

So, we can write the equation:

\(E - Y = 16\)

From this, we can express the elder's age in terms of the younger's age:

\(E = Y + 16\)

Now, let's consider the ages 6 years ago:

• The younger person's age 6 years ago was \(Y - 6\).
• The elder person's age 6 years ago was \(E - 6\).

According to the second condition:

• 6 years ago, the elder one was 3 times as old as the younger one.

So, we can write the equation:

\(E - 6 = 3 \times (Y - 6)\)

Solving the Age Equations

We now have a system of two equations:

1. \(E = Y + 16\)
2. \(E - 6 = 3(Y - 6)\)

We can substitute the expression for \(E\) from equation (1) into equation (2):

\((Y + 16) - 6 = 3(Y - 6)\)

Now, simplify and solve for \(Y\):

\(Y + 10 = 3Y - 18\)

To isolate \(Y\) terms on one side and constants on the other, add 18 to both sides and subtract \(Y\) from both sides:

\(10 + 18 = 3Y - Y\)

\(28 = 2Y\)

Now, divide by 2 to find the value of \(Y\):

\(Y = \frac{28}{2}\)

\(Y = 14\)

The value of \(Y\) represents the present age of the younger person.

Verifying the Solution

Let's check if our answer satisfies both conditions:

• If the younger person's present age (\(Y\)) is 14 years, the elder person's present age (\(E\)) is \(Y + 16 = 14 + 16 = 30\) years.
• The difference in ages is \(30 - 14 = 16\) years. This matches the first condition.
• 6 years ago, the younger person's age was \(14 - 6 = 8\) years.
• 6 years ago, the elder person's age was \(30 - 6 = 24\) years.
• Is the elder one's age (24) three times the younger one's age (8) 6 years ago? \(3 \times 8 = 24\). Yes, it is. This matches the second condition.

Both conditions are satisfied, so the present age of the younger person is indeed 14 years.

Present Age of the Younger Person

Based on our calculations, the present age of the younger person is 14 years.

Checking the conditions:

• Present difference: \(30 - 14 = 16\). Correct.
• 6 years ago relationship: \(24 = 3 \times 8\). Correct.

Revision Table: Age Problem Concepts

Additional Information: Solving Word Problems

Solving word problems, especially age problems or problems involving linear equations, often follows a structured approach:

1. Read Carefully: Understand the problem, identify the knowns and the unknowns. What is the question asking for?
2. Assign Variables: Use letters to represent the unknown quantities. Be specific (e.g., "let \(x\) be the present age of A").
3. Formulate Equations: Translate the sentences and conditions given in the problem into mathematical equations using the variables.
4. Solve the Equations: Use appropriate algebraic methods (like substitution, elimination, simplification) to find the values of the variables.
5. Answer the Question: Make sure you have found the specific value that the question asked for. Sometimes you solve for a variable, but the question asks for something else (e.g., the sum of ages, or the age in the future).
6. Verify: Check if your solution makes sense in the context of the original problem and satisfies all the given conditions.

Age problems are common examples of linear equations in one or two variables and are fundamental in algebra.