Solving Age Ratio Problems in Mathematics
Mathematics often involves solving problems based on given ratios and proportions. This article delves into a specific type of problem involving the ages of individuals and how to solve it using algebraic equations. The problem at hand is to find the difference in the ages of two people, Anu and Binu, when given their current age ratio and the ratio of their ages after several years.
Introduction to the Problem
Suppose the current ages of Anu and Binu are in the ratio 7:10. After six years, their ages will be in the ratio 17:23. The task is to find the difference in their ages.
Setting Up the Equations
To solve this problem, let's denote the current ages of Anu and Binu by 7x and 1 respectively, where x is a positive integer.
After six years, Anu's age will be 7x 6 and Binu's age will be 1 6. The problem states that the new ratio of their ages is 17:23. Therefore, we can set up the following proportion:
(frac{7x 6}{1 6} frac{17}{23})
Cross-multiplying the terms gives:
23(7x 6) 17(1 6)
Expanding the terms:
161x 138 17 102
Rearranging the equation to solve for x:
161x - 17 102 - 138
-9x -36
x (frac{-36}{-9}) 4
Now that we have the value of x, we can find the current ages of Anu and Binu:
Current age of Anu: 7x 7 × 4 28
Current age of Binu: 1 10 × 4 40
The difference in their ages is:
40 - 28 12
Summary of the Solution
By setting up and solving the algebraic equations, we have determined that the difference in the current ages of Anu and Binu is 12 years. The problem was approached systematically using proportional reasoning and algebraic manipulation.
Conclusion
The solution to this type of age ratio problem involves setting up and solving equations based on the given ratios. By breaking down the problem into smaller steps and using algebraic techniques, we can find the desired solution effectively. This method is not only useful for solving similar age problems but also serves as a fundamental skill in algebraic problem-solving.