Solving the Age Ratio Problem for Rohit and Sahil

Introduction

Today, we're exploring a fascinating problem in algebra and ratio analysis. The problem involves two individuals, Rohit and Sahil, with an initial age ratio and a future age ratio. By solving this problem, you'll gain insights into how to manipulate ratios and algebraic equations to find unknown values. Let's dive in!

Solving the Problem: Current Ages of Rohit and Sahil

The problem provides two key pieces of information about the ages of Rohit and Sahil. Currently, their ages are in the ratio 6:7. Four years later, their ages will be in the ratio 7:8. We need to determine the current ages of Sahil and Rohit.

Step 1: Define the Variables

Let's denote the current ages of Rohit and Sahil as R and S, respectively. Thus, we have the following equations based on the given ratios:

One-year ago: (R-1) / (S-1) 6 / 7

Four years from now: (R 4) / (S 4) 7 / 8

Step 2: Simplify and Solve the Equations

Let's start by simplifying the first equation:

$$frac{R-1}{S-1} frac{6}{7} 7(R-1) 6(S-1) 7R - 7 6S - 6 7R - 6S 1 R frac{6S}{7} frac{1}{7} $$

Next, let's simplify the second equation:

$$frac{R 4}{S 4} frac{7}{8} 8(R 4) 7(S 4) 8R 32 7S 28 8R - 7S -4 $$

Now, we have two simultaneous equations:

$$begin{align*} 7R - 6S 1 8R - 7S -4 end{align*}$$

To solve these equations, we can use the elimination method. First, let's eliminate R. Multiply the first equation by 8 and the second by 7:

$$begin{align*} 56R - 48S 8 56R - 49S -28 end{align*}$$

Subtract the second equation from the first:

$$-S 36 S 36 $$

Now, substitute the value of S back into one of the original equations to find R. Using the first equation:

$$7R - 6(36) 1 7R - 216 1 7R 217 R 31 $$

Thus, Sahil's current age is 36 years, and Rohit's current age is 31 years.

Step 3: Verification

To ensure our solution is correct, let's check the given conditions:

One year ago: (R-1): (S-1) 30 : 35 6 : 7

Four years from now: (R 4): (S 4) 35 : 40 7 : 8

Both conditions are satisfied, confirming our solution.

Conclusion

In this problem, we successfully solved for the current ages of Rohit and Sahil using algebraic equations and ratios. By defining the ages and setting up the equations, we were able to determine their current ages accurately.

Additional Insights

The solution demonstrates the power of algebra in solving real-world problems. Understanding and manipulating ratios and algebraic expressions are crucial skills in various fields, from mathematics to finance and engineering. This problem also highlights the importance of verifying your solution to ensure accuracy.

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