Solving Age Ratio Word Problems

This problem involves finding the present ages of two persons, P and Q, based on their current age ratio and their age ratio from a previous time. We will use algebraic equations to represent the given information and solve for the unknown ages.

Setting up the Equations for Ages

Let the present age of person P be \(P\) years and the present age of person Q be \(Q\) years.

According to the question, the present ages of P and Q are in the ratio 5 : 7. This can be written as:

\(\frac{P}{Q} = \frac{5}{7}\)

We can express P in terms of Q (or vice-versa) from this equation:

\(7P = 5Q\)

\(P = \frac{5}{7}Q \quad (Equation 1)\)

Alternatively, we can represent the present ages directly using a common multiplier. Let the common multiplier be \(x\). Then the present age of P is \(5x\) years and the present age of Q is \(7x\) years.

So, \(P = 5x\) and \(Q = 7x\).

Now consider the ages eight years ago.

• Eight years ago, P's age was \(P - 8\) or \(5x - 8\).
• Eight years ago, Q's age was \(Q - 8\) or \(7x - 8\).

The question states that eight years ago, the ratio of P and Q's ages was 7 : 13. This gives us a second equation:

\(\frac{P - 8}{Q - 8} = \frac{7}{13}\)

Substitute \(P = 5x\) and \(Q = 7x\) into this equation:

\(\frac{5x - 8}{7x - 8} = \frac{7}{13}\)

Solving the Equations to Find Ages

Now we need to solve the equation for \(x\):

\(13(5x - 8) = 7(7x - 8)\)

Distribute the numbers on both sides:

\(13 \times 5x - 13 \times 8 = 7 \times 7x - 7 \times 8\)

\(65x - 104 = 49x - 56\)

Now, gather the terms with \(x\) on one side and the constant terms on the other side:

\(65x - 49x = 104 - 56\)

\(16x = 48\)

Solve for \(x\):

\(x = \frac{48}{16}\)

\(x = 3\)

Now that we have the value of \(x\), we can find the present ages of P and Q using \(P = 5x\) and \(Q = 7x\).

• Present age of P = \(5x = 5 \times 3 = 15\) years.
• Present age of Q = \(7x = 7 \times 3 = 21\) years.

Verifying the Age Ratio

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

• Present age ratio: \(15 : 21\). Dividing both numbers by 3, we get \(5 : 7\). This matches the first condition.
• Ages eight years ago:
  • P's age: \(15 - 8 = 7\) years.
  • Q's age: \(21 - 8 = 13\) years.
• Ratio of ages eight years ago: \(7 : 13\). This matches the second condition.

Since both conditions are satisfied, the calculated present ages are correct.

Summary of Present Ages

The present ages of P and Q are 15 years and 21 years, respectively.

Revision Table: Age Ratio Problems

Additional Information: Solving Age Word Problems

Age word problems often involve setting up and solving linear equations. Here are some tips:

• Identify Variables: Assign variables (like \(x\), \(y\), or initials like P, Q) to the unknown ages you need to find, usually the present ages.
• Read Carefully: Pay close attention to keywords like "ago" (subtract from present age) and "from now" or "hence" (add to present age).
• Translate Ratios: A ratio like a:b can be written as \(\frac{\text{Quantity 1}}{\text{Quantity 2}} = \frac{a}{b}\) or by setting the quantities as \(ak\) and \(bk\) for some constant \(k\).
• Formulate Equations: Use the information given about ratios or differences/sums of ages at different points in time to create algebraic equations.
• Solve the System: If you have multiple variables, you'll need multiple equations (a system of equations). Substitute or eliminate variables to solve for their values. In this case, using a common multiplier \(x\) simplified it to a single variable equation.
• Check Your Answer: Plug the ages you found back into the original problem statement to make sure they satisfy all conditions. This helps catch errors.

Age problems are a common type in competitive exams and understanding how to translate the language into algebra is key.