Dividing 680 Among A, B, and C Based on Given Ratios

In many real-world scenarios, it's often necessary to divide a certain sum based on given ratios. Let's explore the problem of dividing Rs. 680 among A, B, and C such that A gets 2/3 of what B gets, and B gets 1/4 of what C gets. This involves understanding ratios, solving equations, and applying these concepts to find the solution.

Understanding the Problem

Let's denote the share of C as x. According to the problem:

B's share 1/4 x A's share 2/3 of B's share 2/3 (1/4 x) x/6 The total sum is 680.

To solve this, we combine the shares of A, B, and C:

A B C x/6 x/4 x 680.

In order to combine the fractions, we need a common denominator, which would be 12 in this case:

A B C 2x/12 3x/12 12x/12 17x/12.

Given that the total sum is 680, we can set up the equation:

17x/12 680.

Solving the Equation

By solving for x, we get:

x 680 * 12/17 480.

Thus, the share of C is 480.

Confirmation and Generalization

Let's verify our solution and apply the concept to a more general scenario:

Verification

If C's share is 480, then B's share 1/4 * 480 120, and A's share 2/3 * 120 80.

The total is 480 120 80 680, which matches the given sum.

General Scenario

Now, consider a different sum, say Rs. 1900. If we scale the same ratios (A:B:C 6:4:3) to Rs. 1900:

The ratio of A:B:C 6:4:3, which simplifies to the same setup as before:

A 6x, B 4x, C 3x.

Using the ratio, we can find the value of x:

Total amount 6x 4x 3x 1900.

So, 13x 1900.

Solving for x, we get:

x 1900/13 150.

Hence, C's share 3 * 150 450.

Formulas and Key Concepts

The key formulas and concepts used here are the ratios in algebra, the combined ratios theorem, and solving equations. Knowing these concepts enables us to solve a variety of related problems effectively.

Conclusion

Understanding and correctly applying the principles of ratio and algebraic division can help solve complex problems in various fields, from finance to engineering. By breaking down the problem into smaller, manageable parts and systematically solving for the unknowns, we can ensure accurate and reliable solutions.