Understanding the Age Ratio Problem

This question asks us to find the ratio of the present ages of two individuals, Ajith and Ganesh, given their age ratio seven years ago and the product of their present ages.

Let's break down the problem step-by-step:

• We are given the ratio of their ages seven years ago: Ajith : Ganesh was 5 : 7.
• We are given the product of their present ages: 616.
• We need to find the ratio of their present ages.

Setting up the Equations from Age Ratios

Let the common multiple of the ratio seven years ago be \(x\).

• Ajith's age seven years ago = \(5x\)
• Ganesh's age seven years ago = \(7x\)

Now, we can express their present ages:

• Ajith's present age = Age seven years ago + 7 years = \(5x + 7\)
• Ganesh's present age = Age seven years ago + 7 years = \(7x + 7\)

Using the Product of Present Ages

We are told that the product of their present ages is 616. So, we can write the equation:

\((5x + 7)(7x + 7) = 616\)

Now, let's expand and solve this equation for \(x\):

\(35x^2 + 35x + 49x + 49 = 616\)

Combine like terms:

\(35x^2 + 84x + 49 = 616\)

Subtract 616 from both sides to set the equation to zero:

\(35x^2 + 84x + 49 - 616 = 0\)

\(35x^2 + 84x - 567 = 0\)

We can divide the entire equation by 7 to simplify the coefficients:

\(\frac{35x^2}{7} + \frac{84x}{7} - \frac{567}{7} = \frac{0}{7}\)

\(5x^2 + 12x - 81 = 0\)

Solving the Quadratic Equation

We have a quadratic equation in the form \(ax^2 + bx + c = 0\). We can solve this by factoring.

We look for two numbers that multiply to \(a \times c = 5 \times -81 = -405\) and add up to \(b = 12\).

The numbers are 27 and -15 (since \(27 \times -15 = -405\) and \(27 + (-15) = 12\)).

Rewrite the middle term \(12x\) as \(27x - 15x\):

\(5x^2 + 27x - 15x - 81 = 0\)

Factor by grouping:

\(x(5x + 27) - 3(5x + 27) = 0\)

\((x - 3)(5x + 27) = 0\)

This gives us two possible values for \(x\):

• \(x - 3 = 0 \implies x = 3\)
• \(5x + 27 = 0 \implies 5x = -27 \implies x = -\frac{27}{5}\)

Since \(x\) represents a multiplier for age, it must be a positive value. Therefore, we discard the negative solution and take \(x = 3\).

Calculating Present Ages

Now that we have the value of \(x\), we can find their present ages:

• Ajith's present age = \(5x + 7 = 5(3) + 7 = 15 + 7 = 22\) years
• Ganesh's present age = \(7x + 7 = 7(3) + 7 = 21 + 7 = 28\) years

Let's quickly check if the product of their present ages is 616:

\(22 \times 28 = 616\). This matches the information given in the question.

Finding the Ratio of Present Ages

The question asks for the ratio of their present ages, which is Ajith's present age : Ganesh's present age.

Ratio = \(22 : 28\)

To simplify the ratio, divide both numbers by their greatest common divisor, which is 2.

\(\frac{22}{2} : \frac{28}{2}\)

\(11 : 14\)

The ratio of their present ages is 11 : 14.

Revision Table: Key Concepts

Additional Information: Solving Age-Based Ratio Questions

Age-based ratio questions are common in competitive exams. They often require setting up equations based on the information provided about ages at different times (past, present, future) and their ratios or products.

Steps usually involve:

• Assigning variables to represent the ages based on the given ratio (e.g., \(nx\) and \(my\), or if the ratio is at a specific time, \(nx\) and \(my\)).
• Expressing ages at other points in time relative to the assigned variables (e.g., adding/subtracting years).
• Forming an equation or a system of equations using the other given conditions (sum of ages, product of ages, difference in ages, another ratio at a different time).
• Solving the equation(s) to find the value of the variable(s).
• Calculating the required ages or ratios based on the variable value.
• Always check if the calculated ages are reasonable (positive values).

This problem involved a quadratic equation because the product of ages was given. Other problems might involve linear equations if sums or differences are given.