Distributing an Amount in Given Ratios

Understanding how to distribute an amount in given ratios is a fundamental concept in mathematics with practical applications in finance, accounting, and everyday life. In this article, we will explore how to distribute a sum of 555 among four friends, P, Q, R, and S, based on specific ratios and solve for each individual’s share.

Given Ratios and Simplification

The first ratio given is P:Q 2:3. From P:Q 2:3, we can express this ratio in terms of Q in terms of P: Q 3P/2. The next ratio given is Q:R 1:2, which can be written as R 2Q. The final ratio, R:S 3:2, can be simplified to S (2/3)R or S (2/3)(2Q) (4/3)Q.

Multiplying the Ratios

To combine these ratios, we need to make the same variable (Q) in each ratio have the same coefficient. By multiplying each part of the ratios together:

P:Q 2:3 becomes 2:3. Q:R 1:2 becomes 3:6 (since 3 * 2 6). R:S 3:2 becomes 6:4 (since 6 * 3/2 9 and 4 * 3/2 6).

Final Combined Ratios

By combining all these ratios, we get the combined ratio P:Q:R:S 2:3:6:4.

Calculating Individual Shares

The total sum to be shared is 555. To find the share of each individual, we need to divide the total sum by the sum of the parts of the combined ratio (2 3 6 4 15). The share of P is calculated as 555 * (2/15) 74. The share of Q is calculated as 555 * (3/15) 111. The share of R is calculated as 555 * (6/15) 222. The share of S is calculated as 555 * (4/15) 148.

Verification and Solved Example

To verify, we can substitute the values back into the given ratios:

P:Q 2:3 74:111, which is correct since 74 * 3 111 * 2. Q:R 1:2 111:222, which is correct since 111 * 2 222 * 1. R:S 3:2 222:148, which is correct since 222 * 2/3 148 * 3/2.

Simplified Ratios and Solutions

Alternatively, we can simplify the ratios to make the calculations easier. Using substitution:

P:Q 2:3 → Q (3/2)P. Q:R 1:2 → R 2Q 2(3/2)P 3P. R:S 3:2 → S (2/3)R (2/3)(3P) 2P.

Substituting these into the total sum (P Q R S 555):

P   (3/2)P   3P   2P  555
(2   3   6   4)P/15  555
15P/15  555
P  74

From P, we can now find the shares of Q, R, and S:

Q (3/2)P (3/2) * 74 111. R 3P 3 * 74 222. S 2P 2 * 74 148.

The shares are P:Q:R:S 74:111:222:148.

Conclusion

By understanding and solving these mathematical problems, we can distribute an amount in given ratios accurately. This method is not only useful in academic settings but also in real-world scenarios such as dividing profits, assets, or even in financial planning.