Distributing Pencils and Sweets: A Mathematical Exploration

Mathematics is a powerful tool for solving real-world problems. In this article, we explore the intricacies of distributing items in a way that satisfies specific conditions. We'll delve into two scenarios involving pencils and sweets, applying mathematical reasoning to find the most suitable solutions.

Distributing Pencils

Consider the following scenario: around 300 pencils were distributed equally among a group of children such that the number of pencils received by each child is 20% of the total number of children. Let's denote the number of children as ( n ). According to the problem, each child receives 20% of the total number of children, which can be mathematically expressed as:

[text{Pencils per child} 0.2n]

The total number of pencils distributed can be expressed as:

[n times 0.2n 0.2n^2]

Given that the total number of pencils is approximately 300, we can set up the equation:

[0.2n^2 300]

Solving for ( n^2 ), we multiply both sides by 5:

[n^2 1500]

Next, we take the square root of both sides:

[n sqrt{1500} approx 38.73]

Since ( n ) must be a whole number, we round it to the nearest whole number, which is 39. Now, we can find out how many pencils each child receives:

[text{Pencils per child} 0.2n 0.2 times 39 7.8]

Since each child must receive a whole number of pencils, we need to check if we can distribute 300 pencils among 39 children evenly. Calculating the total pencils distributed if each child receives 7 pencils:

[39 times 7 273]

Calculating for 8 pencils:

[39 times 8 312]

Since 312 exceeds 300, we consider that each child receives 7 pencils. This means:

[text{Total pencils distributed} 273]

Thus, while each child receives 7 pencils, the distribution does not utilize all 300 pencils, as indicated by 27 pencils remaining. Therefore, the final answer is:

Each child received 7 pencils.

Distributing Sweets

Now, let's turn our attention to a similar problem involving sweets. In another distribution, 810 sweets were given to children such that the number of sweets each child received is 20% of the total number of children. Let's denote the number of children as ( C ).

Mathematically, this can be expressed as:

[C times 0.2C 810]

or simplified to:

[C^2 4050]

Solving for ( C ), we get:

[C sqrt{4050} approx 63.64]

Since ( C ) must be a whole number, we cannot have a fractional number of children. Thus, the distribution of sweets in such a way does not yield a perfect whole number result.

Balancing Distribution

To illustrate with a practical example, consider the following scenario: try with 100 children. In this case, you would need 100/5 100 2000 pencils. But you have only 500 pencils. If you multiply both terms by 1/2, the ratio still holds, and you will end up with 500 pencils: 1/2 100/5 1/2 100 500. So we get 50 children with each 10 pens.

This illustrates how practical constraints can influence the distribution, ensuring that the final distribution is both fair and feasible.