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: (35x^2)/(7) + (84x)/(7) - (567)/(7) = (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 × c = 5 × -81 = -405 and add up to b = 12. The numbers are 27 and -15 (since 27 × -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 = -(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 × 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. (22)/(2) : (28)/(2) 11 : 14 The ratio of their present ages is 11 : 14. Detail Value Age Ratio (7 years ago) 5 : 7 Product of Present Ages 616 Calculated 'x' value 3 Ajith's Present Age 22 Ganesh's Present Age 28 Ratio of Present Ages 11 : 14 Revision Table: Key Concepts Concept Description Ratio A comparison of two quantities by division. Represented as a : b or a/b. Age Problems Word problems involving the ages of individuals at different points in time. Often involve setting up equations based on given conditions. Present Age An individual's age currently. Past Age An individual's age at a time in the past. Calculated by subtracting the number of past years from the present age. Quadratic Equation An equation of the form ax^2 + bx + c = 0, where a \neq 0. Can be solved by factoring, completing the square, or using the quadratic formula. 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.