Distributed Sweets: A Comprehensive Guide

When dealing with problems involving the distribution of items based on a given ratio, it is crucial to understand the basic principles of proportion and how to manipulate these principles to find the total amount. We'll explore a detailed solution to a classic problem: if a box of sweets is distributed between A and B in the ratio 3:4, and A received 36 sweets, what was the total number of sweets?

Understanding the Ratio

To solve the problem, let's first define what a ratio represents. A ratio of 3:4 means that for every 3 parts that A gets, B gets 4 parts. This problem can be approached by breaking it down into several steps:

Step-by-Step Solution

Step 1: Define the Ratio in Terms of Parts

Let's denote the number of parts A gets as 3 parts, and the number of parts B gets as 4 parts. The total number of parts is then 3 4 7 parts.

Step 2: Determine the Value of One Part

Since A received 36 sweets, and this corresponds to 3 parts, we can find the value of one part by dividing the number of sweets A received by the number of parts A has:

One part 36 / 3 12

Step 3: Calculate the Total Number of Sweets

Now, knowing that one part is equal to 12 sweets, we can find the total number of sweets by multiplying the total number of parts by the value of one part:

Total sweets 7 * 12 84

This method works for any similar problem where items are distributed in a given ratio.

Alternative Methods

There are several ways to approach this problem, and we will explore a few more methods to solidify your understanding.

Method 1: Using the Common Multiplication Factor

Since A got 36 sweets, and this corresponds to 3 parts, we can determine the common multiplication factor:

Common multiplication factor 36 / 3 12

Now, since B gets 4 parts, we can determine the number of sweets B received:

B's sweets 4 * 12 48

Finally, the total number of sweets is the sum of A and B's sweets:

Total sweets 36 48 84

Method 2: Using Variables to Represent Sweets Allocated to Each Part

Let's denote the number of sweets received by A as 3x and the number of sweets received by B as 4x. Given that A received 36 sweets, we can set up the following equation:

3x 36

Solving for x, we get:

x 36 / 3 12

Now, we can find the number of sweets for B:

4x 4 * 12 48

The total number of sweets is the sum of A's and B's sweets:

Total sweets 36 48 84

Advanced Problem-Solving Techniques

This problem showcases a practical application of proportional reasoning. Here are some key points to remember:

To solve problems involving ratios, always define the total parts first. Use the given information to find the value of one part. Scale the parts to find the total quantity. Verify your answer by checking if the total parts add up correctly.

Conclusion

The total number of sweets distributed between A and B, given that A received 36 sweets in a 3:4 ratio, is 84 sweets. By understanding and applying the principles of ratios and proportions, you can solve a wide variety of similar problems.

Further Reading

For more advanced problem-solving techniques and detailed explanations, consider exploring additional resources on algebraic expressions and proportional reasoning.