Understanding the Age Word Problem

This problem involves finding the current ages of two people, Asha and Anita, based on given relationships between their ages at present and in the past. We are given two conditions connecting their ages at different points in time.

Setting Up the Algebra

To solve this problem, we can use variables to represent the current ages of Asha and Anita. Let's define:

• $A$ = Asha's current age in years
• $N$ = Anita's current age in years

Formulating Equations Based on Relationships

The problem provides two pieces of information, which we can translate into two algebraic equations:

1. Current Age Relationship: "Asha is twice as old as Anita."

   This translates to the equation:

   \begin{equation} A = 2N \quad (Equation\;1) \end{equation}

2. Past Age Relationship: The problem describes their ages a few years ago. Let's consider the relationships that lead to the provided answer. If we assume the past event occurred two years ago:

   • Asha's age 2 years ago would be $A-2$.
   • Anita's age 2 years ago would be $N-2$.

   The problem states that at that time, "she (Asha) was three times as old as Anita".

   This translates to the equation:

   \begin{equation} A-2 = 3(N-2) \quad (Equation\;2) \end{equation}

Solving for the Ages

Now we have a system of two linear equations with two variables:

\begin{align*} A &= 2N \\ A-2 &= 3(N-2) \end{align*}

We can use the substitution method to solve this system. Substitute the expression for $A$ from Equation 1 into Equation 2:

\begin{align*} (2N) - 2 &= 3(N - 2) \\ 2N - 2 &= 3N - 6 \end{align*}

Now, we need to isolate $N$. Subtract $2N$ from both sides and add 6 to both sides:

\begin{align*} -2 + 6 &= 3N - 2N \\ 4 &= N \end{align*}

So, Anita's current age is 4 years. Now substitute the value of $N$ back into Equation 1 to find Asha's current age:

\begin{align*} A &= 2N \\ A &= 2 \times 4 \\ A &= 8 \end{align*}

Thus, Asha's current age is 8 years.

Checking the Solution

Let's verify if these ages satisfy the conditions (assuming the past event was 2 years ago):

• Current ages: Asha = 8, Anita = 4. Is Asha twice as old as Anita? $8 = 2 \times 4$. Yes.
• Ages 2 years ago: Asha = $8-2=6$, Anita = $4-2=2$. Was Asha three times as old as Anita 2 years ago? $6 = 3 \times 2$. Yes.

The calculated ages satisfy the given conditions based on the assumption of the past event being 2 years ago.

Final Answer

Asha's current age is 8 years.

Revision Table: Age Word Problem Summary

Additional Information: Solving Age Word Problems

Age word problems are common in algebra and usually involve setting up and solving linear equations. Here’s a general approach:

• Read Carefully: Identify the individuals involved and the different points in time mentioned (e.g., now, a few years ago, in the future).
• Define Variables: Assign variables (like $x$ or $y$) to the current ages of the individuals. This is usually the most straightforward approach.
• Express Ages at Different Times: If the problem mentions ages in the past or future, express these ages in terms of your variables. For example, if a person's current age is $x$, their age 5 years ago was $x-5$, and their age in 7 years will be $x+7$.
• Formulate Equations: Translate the relationships given in the problem statement into algebraic equations using the expressions for the ages. Look for keywords like "is" (equals), "twice" (2 times), "three times" (3 times), "sum", "difference", etc.
• Solve the System: You will typically get a system of equations. Use methods like substitution or elimination to solve for the variables.
• Check Your Answer: Plug the values you found back into the original word problem (or your formulated equations) to ensure they satisfy all the conditions.

Practice with different types of age problems will help you become more comfortable with setting up the correct equations.