Solving Age Word Problems: Finding Present Ages

This problem involves finding the present age of Krish using information about his age relative to Parthiv's age at two different points in time: the present and eight years ago. We can solve this by setting up algebraic equations based on the given relationships.

Defining Variables for Present Ages

Let's represent the unknown present ages using variables:

• Let Krish's present age be \(K\) years.
• Let Parthiv's present age be \(P\) years.

Setting Up Equations from the Problem Statement

The problem gives us two pieces of information that can be translated into equations:

1. Relationship between present ages: "Krish is 5 years younger than Parthiv."
   This means Krish's age is Parthiv's age minus 5 years.
   Equation 1: \(K = P - 5\)
2. Relationship between ages eight years ago: "Eight years ago, three times the age of Krish was 10 more than twice the age of Parthiv."
   First, let's find their ages eight years ago:
   • Krish's age eight years ago = \(K - 8\)
   • Parthiv's age eight years ago = \(P - 8\)
   Now, we translate the relationship: three times Krish's age eight years ago is equal to twice Parthiv's age eight years ago plus 10.
   Equation 2: \(3(K - 8) = 2(P - 8) + 10\)

Solving the System of Linear Equations

We now have a system of two linear equations with two variables:

1. \(K = P - 5\)

2. \(3(K - 8) = 2(P - 8) + 10\)

We can use the substitution method to solve this system. Substitute the expression for \(K\) from Equation 1 into Equation 2:

Substitute \(K = P - 5\) into \(3(K - 8) = 2(P - 8) + 10\):

\(3((P - 5) - 8) = 2(P - 8) + 10\)

Simplify the expression inside the parentheses on the left side:

\(3(P - 13) = 2(P - 8) + 10\)

Now, distribute on both sides:

\(3P - 3 \times 13 = 2 \times P - 2 \times 8 + 10\)

\(3P - 39 = 2P - 16 + 10\)

Combine the constant terms on the right side:

\(3P - 39 = 2P - 6\)

Now, isolate the variable \(P\) by moving terms. Subtract \(2P\) from both sides:

\(3P - 2P - 39 = -6\)

\(P - 39 = -6\)

Add 39 to both sides to solve for \(P\):

\(P = -6 + 39\)

\(P = 33\)

So, Parthiv's present age is 33 years.

Now that we have Parthiv's present age, we can find Krish's present age using Equation 1: \(K = P - 5\)

\(K = 33 - 5\)

\(K = 28\)

Therefore, Krish's present age is 28 years.

Verification of the Solution

Let's check if these ages satisfy the conditions given in the problem:

• Krish's present age = 28, Parthiv's present age = 33. Is Krish 5 years younger than Parthiv? \(28 = 33 - 5\), Yes.
• Eight years ago: Krish was \(28 - 8 = 20\), Parthiv was \(33 - 8 = 25\).
• Is three times Krish's age 8 years ago (3 * 20 = 60) equal to 10 more than twice Parthiv's age 8 years ago (2 * 25 + 10 = 50 + 10 = 60)? Yes, \(60 = 60\).

The ages satisfy both conditions, so the solution is correct.

Final Answer for Krish's Present Age

Based on the calculations, Krish's present age is 28 years.

Revision Table: Key Concepts for Age Problems

Additional Information: Tips for Solving Age Word Problems

Age word problems are common in mathematics and tests. Here are some tips to solve them effectively:

• Read Carefully: Understand the relationship between the ages and the specific time periods mentioned (present, past, future).
• Assign Variables: Choose variables for the unknown ages, usually the present ages.
• Write Down Ages at Different Times: Clearly state how the ages change in the past or future (e.g., age N years ago is Present Age - N).
• Formulate Equations: Translate each statement in the problem into an algebraic equation. Make sure the equations accurately reflect the relationships.
• Solve the Equations: Use standard algebraic techniques (substitution, elimination) to solve the system of equations.
• Check Your Answer: Substitute the values you found back into the original statements or equations in the problem to ensure they hold true. This helps catch errors.
• Answer the Specific Question: Make sure you provide the age the question asks for (e.g., present age of Krish, not Parthiv, unless asked).

Practice with various age problems will help you become more comfortable setting up and solving the equations quickly and accurately.