Distributing Money According to Ratios: A Comprehensive Guide

Understanding how to distribute money based on given ratios is a key concept in mathematics and finance. This guide will walk you through the process of solving a specific problem where the ratios of As to Bs and Bs to Cs are given, and you need to find the total amount of money and C's share. By the end of this article, you will understand the steps to solve such problems and apply the same principles to various scenarios.

Introduction to Ratios

First, it's essential to understand what ratios are. A ratio is a comparison of two quantities, often expressed as a fraction or with a colon. When we say A:B is 5:4 and B:C is 3:2, it means for every 5 units of A, there are 4 units of B, and for every 3 units of B, there are 2 units of C.

Solving the Ratio Problem

Given: A:B is 5:4 and B:C is 3:2.

First, we need to find a common base for B in both ratios.

Step 1: Standardize the Ratios

We need to make the value of B in both ratios the same.

For A:B, the value of B is 4.

For B:C, the value of B is 3.

To make B the same in both ratios, we need to find a common multiple of 4 and 3, which is 12. Therefore, we will scale the ratios accordingly.

A:B 5:4 can be scaled to 15:12 (multiply both terms by 3).

B:C 3:2 can be scaled to 12:8 (multiply both terms by 4).

Thus, the combined ratio A:B:C is 15:12:8.

Step 2: Calculate the Total Distribution

Given that A has a total of 700, we can use the combined ratio to find the total amount and C's share.

Total A B C 15x 12x 8x 35x,

Where x is a common multiplier.

Since A's share is 15x 700, we can find x:

x 700 / 15.

Now, C's share is 8x:

C's share 8 * (700 / 15) 8 * 46.67 373.33.

Therefore, the total amount of money is 35 * 46.67 1633.43, and C's share is 373.33.

Alternative Methods

Method 1: Using Ratios Directly

We can also solve the problem by directly using the ratios and proportions:

A:B 5:4 implies A has 15 parts and B has 12 parts when the ratio is scaled to 15:12.

B:C 3:2 implies B has 12 parts and C has 8 parts when the ratio is scaled to 12:8.

The combined ratio A:B:C is 15:12:8.

Using the total of 700 for A:

MONEY WITH B 700 * (12/15) 560

MONEY WITH C 700 * (8/15) 1120 / 3 or 373.33

This method confirms our previous calculations.

Another Approach

Method 2: Algebraic Method

A 1.25B (since A:B 5:4 and A 5B/4), and B 3C/2 (since B:C 3:2), so:

A 1.25 * (3C/2) 15C/8.

So, A:B:C 15:12:8, and 15x 700, so x 700/15, and C's share 1.33 * 700 373.33.

Conclusion

Understanding how to tackle distribution problems in this manner can be very useful in various fields, such as economics, finance, and even daily budgeting. By applying the principles of ratios and proportional reasoning, you can accurately determine allocations and shares based on given conditions.

Remember, the key is to standardize the ratios to have a common base, and then use the given total to solve for individual shares.