Seven years ago, the ratio of the ages of Ajith and Ganesh was 5 : 7. If the product of their present ages is
616, find the ratio of their present age.

Let's denote the present ages of Ajith and Ganesh as \( A \) and \( G \) respectively.

### Step 1: Seven years ago
According to the problem, the ratio of Ajith and Ganesh's ages seven years ago was 5:7. This means:
\[
\frac{A - 7}{G - 7} = \frac{5}{7}
\]

Cross-multiplying gives:
\[
7(A - 7) = 5(G - 7)
\]
Expanding both sides:
\[
7A - 49 = 5G - 35
\]
Rearranging to isolate variables:
\[
7A - 5G = 14 \quad \text{(Equation 1)}
\]

### Step 2: Product of their present ages
We are also given that the product of their present ages is 616:
\[
A \times G = 616 \quad \text{(Equation 2)}
\]

### Step 3: Solving the system of equations
We now have the system of equations:
1. \( 7A - 5G = 14 \)
2. \( A \times G = 616 \)

We'll solve this system step by step.

First, solve Equation 1 for \( G \) in terms of \( A \):
\[
7A - 5G = 14
\]
\[
5G = 7A - 14
\]
\[
G = \frac{7A - 14}{5}
\]

Now, substitute this expression for \( G \) into Equation 2:
\[
A \times \frac{7A - 14}{5} = 616
\]
Multiply both sides by 5 to eliminate the denominator:
\[
A(7A - 14) = 3080
\]
Expanding the left-hand side:
\[
7A^2 - 14A = 3080
\]
Dividing the entire equation by 7:
\[
A^2 - 2A = 440
\]
Rearranging the equation:
\[
A^2 - 2A - 440 = 0
\]

### Step 4: Solving the quadratic equation
We now solve the quadratic equation \( A^2 - 2A - 440 = 0 \) using the quadratic formula:
\[
A = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-440)}}{2(1)}
\]
\[
A = \frac{2 \pm \sqrt{4 + 1760}}{2}
\]
\[
A = \frac{2 \pm \sqrt{1764}}{2}
\]
\[
A = \frac{2 \pm 42}{2}
\]

Thus, we have two possible solutions for \( A \):
\[
A = \frac{2 + 42}{2} = 22 \quad \text{or} \quad A = \frac{2 - 42}{2} = -20
\]
Since age cannot be negative, we have \( A = 22 \).

### Step 5: Finding \( G \)
Now, substitute \( A = 22 \) into the expression for \( G \):
\[
G = \frac{7(22) - 14}{5} = \frac{154 - 14}{5} = \frac{140}{5} = 28
\]

### Step 6: Finding the ratio of their present ages
The ratio of their present ages is:
\[
\frac{A}{G} = \frac{22}{28} = \frac{11}{14}
\]

Thus, the ratio of Ajith and Ganesh's present ages is \( 11:14 \).

Would you like any additional details or clarification?

Here are 5 related questions to further explore the concept:
1. How would the problem change if the product of their present ages was 700?
2. What is the sum of Ajith and Ganesh's present ages?
3. Can you express Ganesh's age in terms of Ajith's age based on their present age ratio?
4. How would the solution change if the age ratio seven years ago was 3:4 instead of 5:7?
5. Can we use another method to solve the system of equations?

**Tip:** When solving quadratic equations, always check if the discriminant is a perfect square for easier calculations.

Seven years ago, the ratio of the ages of Ajith and Ganesh was 5:7. If the product of their present ages is 616, find the ratio of their present ages.

Learn how to solve a real-world problem involving the ages of Ajith and Ganesh. Seven years ago, their age ratio was 5:7, and their present ages multiply to 616. Follow a detailed, step-by-step guide using algebra and quadratic equations to find the ratio of their current ages.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation