Solving the Dhahiri and Eunice Age Word Problem

This problem involves finding the present ages of two people, Dhahiri and Eunice, based on two conditions related to their present and future ages. We can solve this by setting up a system of linear equations.

Defining Variables

Let's represent their present ages using variables:

• Let \(D\) be Dhahiri's present age in years.
• Let \(E\) be Eunice's present age in years.

Setting Up Equations from the Conditions

The problem gives us two main pieces of information:

1. "The sum of their present ages is 61 years."
   • This translates directly to the equation: \(D + E = 61\) (Equation 1)
2. "Three years from now Dhahiri’s age will be eight years less than twice Eunice’s age."
   • Dhahiri's age in three years will be \(D + 3\).
   • Eunice's age in three years will be \(E + 3\).
   • Twice Eunice's age in three years will be \(2 \times (E + 3)\).
   • Eight years less than twice Eunice's age in three years is \(2(E + 3) - 8\).
   • So, the second condition translates to the equation: \(D + 3 = 2(E + 3) - 8\) (Equation 2)

Solving the System of Equations

Now we have a system of two linear equations:

1. \(D + E = 61\)
2. \(D + 3 = 2(E + 3) - 8\)

Let's simplify Equation 2:

\(D + 3 = 2E + 6 - 8\)

\(D + 3 = 2E - 2\)

\(D = 2E - 2 - 3\)

\(D = 2E - 5\) (Simplified Equation 2)

Now we can use the substitution method. Substitute the expression for \(D\) from the simplified Equation 2 into Equation 1:

\((2E - 5) + E = 61\)

\(3E - 5 = 61\)

Add 5 to both sides:

\(3E = 61 + 5\)

\(3E = 66\)

Divide by 3:

\(E = \frac{66}{3}\)

\(E = 22\)

So, Eunice's present age is 22 years.

Now substitute the value of \(E\) (22) back into Equation 1 (\(D + E = 61\)) to find Dhahiri's present age (\(D\)):

\(D + 22 = 61\)

Subtract 22 from both sides:

\(D = 61 - 22\)

\(D = 39\)

So, Dhahiri's present age is 39 years.

Verification of the Solution

Let's check if our present ages satisfy both original conditions:

• Condition 1: Sum of present ages is 61.
  • \(D + E = 39 + 22 = 61\). This is correct.
• Condition 2: Three years from now, Dhahiri’s age will be eight years less than twice Eunice’s age.
  • Dhahiri's age in 3 years: \(39 + 3 = 42\).
  • Eunice's age in 3 years: \(22 + 3 = 25\).
  • Twice Eunice's age in 3 years: \(2 \times 25 = 50\).
  • Eight years less than twice Eunice's age in 3 years: \(50 - 8 = 42\).
  • Is Dhahiri's age in 3 years equal to this value? \(42 = 42\). This is also correct.

Both conditions are satisfied, confirming our solution.

Final Answer

Dhahiri's present age is 39 years.

Revision Table: Key Concepts in Age Problems

Additional Information: Solving Age Word Problems

Age word problems are common in algebra. They usually involve relationships between people's ages at different points in time (present, past, or future).

Here's a general approach to solving them:

• Read Carefully: Understand the relationships between the ages and the timelines (how many years from now or how many years ago).
• Assign Variables: Use variables (like \(x\), \(y\), \(A\), \(B\)) to represent the unknown present ages.
• Express Ages at Different Times: Write expressions for their ages in the past or future based on your variables (e.g., if present age is \(x\), age in 5 years is \(x+5\), age 5 years ago was \(x-5\)).
• Formulate Equations: Translate the sentences describing the relationships between the ages into algebraic equations. Each distinct piece of information often gives you one equation.
• Solve the System: Use methods like substitution or elimination to solve the system of equations you've created.
• Answer the Question: Make sure you answer what the question asks for (e.g., present age, future age, age difference).
• Check Your Answer: Plug the values you found back into the original word problem's conditions to ensure they make sense and satisfy all statements.

These problems often test your ability to translate written language into mathematical expressions and solve simultaneous equations.