Solving Age Ratio Problems: A Step-by-Step Guide
Introduction:
Age ratio problems are a common type of algebraic problem that can be encountered in various educational and real-life scenarios. This article provides a detailed guide on how to solve age ratio problems using a step-by-step approach, making it accessible for students, teachers, and anyone else interested in improving their problem-solving skills.
Understanding the Problem
A typical age ratio problem involves establishing a relationship between the ages of two individuals based on given ratios and future scenarios. One well-known example is the problem where the ratio of Pratibha’s age to Aradhana’s age is 5:6, and after twelve years, the ratio becomes 9:10.
Defining Variables and Setting Up Equations
To solve such problems, we start by defining the variables. Let’s denote:
Pratibha’s current age as x Aradhana’s current age as yGiven information:
The ratio of Pratibhas age to Aradhanas age is 5:6, so we have the equation: x/y 5/6 After 12 years, the ratio of their ages will be 9:10, so we have the equation: ((x 12)/(y 12)) 9/10Solving the Equations
Set up the first equation using the given ratio:
x/y 5/6 which can be rewritten as: nx 5/6y
Set up the second equation using the ratio after 12 years:
(x 12)/(y 12) 9/10
Multiplying through by (y 12) we get: 10(x 12) 9(y 12)
Expanding and rearranging gives:
1 120 9y 108
Simplifying further:
1 - 9y 120 - 108
Solving for y in the first equation: x (5/6)y or x (5y/6)
Substitute x in the second equation:
10(5y/6) - 9y 12
Multiplying through by 6 to clear the fraction:
50y - 54y 72
Simplifying:
-4y 72
Therefore:
y -72 / -4 18
Substituting y 18 into the first equation to find x:
x (5/6) × 18 15
Therefore, Pratibha’s current age is 15 years and Aradhana’s current age is 18 years.
Alternative Solutions
There are alternative methods to solve this problem, making use of simplifying techniques and assumptions.
Multiplying the present ages by the current ratio: The present ages of Pratibha is 5x and that of Aradhana 6x. After twelve years, the ages become 5x 12 and 6x 12 respectively. The equation can be written as: 5x 12 : 6x 12 9 : 10. Multiplying both sides by 10 gives: 5 120 54x 108, and simplifying this equation results in: 4x 12, hence x 3. So, the present age of Pratibha and Aradhana are 15 years and 18 years respectively.
Another method involves recognizing that the difference in the ratio increases by 4 over 12 years. Hence, 12/4 3, which is the multiplier for the ages. Therefore, the present ages are 5(3) 15 and 6(3) 18 years.
Conclusion
Age ratio problems require systematic and careful approach to ensure accurate solutions. By understanding how to set up and solve algebraic equations, you can tackle similar problems with confidence. Whether you're a student, educator, or just keen on improving your math skills, these techniques can be extremely helpful. Remember, practice makes perfect!
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Sharma