How to Solve Equations Involving Equality Distribution

Solving equations that involve the distribution of items equally among a group is a common problem in mathematics. One such example is a real-world scenario involving sweets distribution. Let's explore how to solve such problems using mathematical equations and provide multiple perspectives to enhance understanding.

Solving with a Specific Example: 720 Sweets

In a particular situation, 720 sweets were distributed equally among children in such a way that the number of sweets received by each child was 20% of the total number of children. We will use the equation method to solve for the number of sweets each child received and the total number of children.

Step 1: Let the number of children be denoted as n.

Step 2: According to the problem, the number of sweets received by each child is 20% of the total number of children. This can be expressed as:

Number of sweets per child 0.2n

Step 3: Since the total number of sweets distributed is 720, we can set up the equation:

n × 0.2n 720

Step 4: Simplify the equation to:

0.2n^2 720

Step 5: To eliminate the decimal, multiply both sides by 5:

n^2 3600

Step 6: Taking the square root of both sides:

n sqrt{3600} 60

Step 7: With n 60, we can find the number of sweets each child received:

Number of sweets per child 0.2n 0.2 × 60 12

Therefore, each child received 12 sweets.

Alternative Methods for Distribution Problems

Other scenarios involving the distribution of sweets may result in non-integer solutions, making the distribution unrealistic in a real-world context.

810 Sweets

In another example, 810 sweets were distributed equally among children such that the number of sweets each child received was 20% of the total number of children. Here, we will follow a similar process but account for the possibility of non-integer results.

Step 1: Let the number of children be C.

Step 2: According to the problem, the number of sweets each child receives is 20% of the total number of children, which can be expressed as:

Number of sweets per child 0.2C

Step 3: Since the total number of sweets distributed is 810, we can set up the equation:

C × 0.2C 810

Step 4: Simplify the equation to:

0.2C^2 810

Step 5: To eliminate the decimal, multiply both sides by 5:

C^2 4050

Step 6: Taking the square root of both sides:

C sqrt{4050} ≈ 63.64

Since the number of children cannot be a decimal, this example does not yield a practical solution.

Additional Examples and Insights

Explore another scenario where 405 sweets are distributed among a group of children with each receiving 20% of the total number of children. Here are the detailed steps to solve for the distribution:

Step 1: Let the number of children be a.

Step 2: According to the problem, each child receives 20% of a, which can be expressed as:

Number of sweets per child 0.2a

Step 3: Since the total number of sweets distributed is 405, we can set up the equation:

a × 0.2a 405

Step 4: Simplify the equation to:

0.2a^2 405

Step 5: To eliminate the decimal, multiply both sides by 5:

a^2 2025

Step 6: Taking the square root of both sides:

a sqrt{2025} 45

Step 7: With a 45, each child receives:

Number of sweets per child 0.2a 0.2 × 45 9

Each child received 9 sweets, which means 45 children distributed 405 sweets equally.

Conclusion

Solving distribution problems requires a well-defined setup of equations and careful handling of results. In most practical scenarios, the distribution must result in a whole number of items. These problems help develop logical reasoning and problem-solving skills, making them valuable in various academic and real-world contexts.