Solving the Age Word Problem: Finding Anamika's Present Age

This problem involves determining the present ages of three individuals: Anamika, Malini, and Srinidhi, based on given conditions relating their ages at different points in time. We need to find the present age of Anamika.

Setting Up the Variables for Present Ages

Let's represent the present ages of the three individuals with variables:

• Anamika's present age = \(A\) years
• Malini's present age = \(M\) years
• Srinidhi's present age = \(S\) years

Translating Statements into Equations

Now, let's convert the given information into mathematical equations:

1. "Seven years from now, Anamika will be as old as Malini was 4 years ago."
   • Anamika's age 7 years from now = \(A + 7\)
   • Malini's age 4 years ago = \(M - 4\)
   • Equation 1: \(A + 7 = M - 4\)
   • We can rearrange this equation to express \(M\) in terms of \(A\):
     \(M = A + 7 + 4\)
     \(M = A + 11\)
2. "Srinidhi was born 2 years ago."
   • This means Srinidhi's present age is 2 years.
   • Equation 2: \(S = 2\)
3. "The average age of Anamika, Malini and Srinidhi 10 years from now will be 33 years."
   • Anamika's age 10 years from now = \(A + 10\)
   • Malini's age 10 years from now = \(M + 10\)
   • Srinidhi's age 10 years from now = \(S + 10\)
   • Sum of their ages 10 years from now = \((A + 10) + (M + 10) + (S + 10) = A + M + S + 30\)
   • Average age 10 years from now = \(\frac{A + M + S + 30}{3}\)
   • Equation 3: \(\frac{A + M + S + 30}{3} = 33\)

Solving the System of Equations

We have a system of three equations with three variables. We can substitute the known values and relationships to solve for Anamika's present age (\(A\)).

1. Substitute the value of \(S\) from Equation 2 into Equation 3:
   \(\frac{A + M + 2 + 30}{3} = 33\)
   \(\frac{A + M + 32}{3} = 33\)
2. Multiply both sides of the equation by 3:
   \(A + M + 32 = 33 \times 3\)
   \(A + M + 32 = 99\)
3. Subtract 32 from both sides:
   \(A + M = 99 - 32\)
   \(A + M = 67\) (Let's call this Equation 4)
4. Now, substitute the expression for \(M\) from Equation 1 (\(M = A + 11\)) into Equation 4:
   \(A + (A + 11) = 67\)
5. Combine like terms:
   \(2A + 11 = 67\)
6. Subtract 11 from both sides:
   \(2A = 67 - 11\)
   \(2A = 56\)
7. Divide both sides by 2 to find the value of \(A\):
   \(A = \frac{56}{2}\)
   \(A = 28\)

So, the present age of Anamika is 28 years.

We can also find the present ages of Malini and Srinidhi for completeness:

• Malini's present age \(M = A + 11 = 28 + 11 = 39\) years.
• Srinidhi's present age \(S = 2\) years.

Let's quickly verify the average age 10 years from now:

• Anamika's age in 10 years = \(28 + 10 = 38\)
• Malini's age in 10 years = \(39 + 10 = 49\)
• Srinidhi's age in 10 years = \(2 + 10 = 12\)
• Sum of ages in 10 years = \(38 + 49 + 12 = 99\)
• Average age = \(\frac{99}{3} = 33\) years. This matches the condition given in the problem.

Therefore, the calculated present age of Anamika is correct.

Revision Table: Key Information

Additional Information: Solving Age Problems

Age problems are common in mathematics and can often be solved by setting up algebraic equations based on the given information. Here are some tips:

• Define variables: Always start by assigning variables (like \(x\) or \(A, M, S\)) to the present ages of the people involved.
• Read carefully: Pay close attention to phrases like "years from now," "years ago," "twice as old," "half as old," "sum of ages," and "average age."
• Formulate equations: Translate each sentence or condition in the problem into a mathematical equation using your defined variables. Remember that an age \(n\) years from now is the present age plus \(n\), and an age \(n\) years ago is the present age minus \(n\).
• Solve the system: Use algebraic methods like substitution or elimination to solve the system of equations you've created. The goal is to find the values of the variables, especially the one requested in the question (in this case, Anamika's present age).
• Verify your answer: Plug the values you found back into the original conditions or equations to ensure they hold true. This helps catch calculation errors.

These steps are crucial for accurately solving various types of age-related word problems.