Pack 25 Pages With Minimum Packages

by Alex Johnson 36 views

Ever found yourself staring at a pile of items and wondering, "How do I pack all of this using the least amount of boxes or containers?" This is a classic problem that pops up in all sorts of situations, from moving house to organizing your shelves, and even in the fascinating world of mathematics. Today, we're going to tackle a specific scenario: how to pack 25 pages using the fewest packages. It sounds simple, but there's a clever way to think about it that can save you space and effort. We're given different packaging options, each holding a specific number of pages, and our goal is to reach exactly 25 pages with the fewest possible containers. Let's dive in and explore the options to find the most efficient solution. This isn't just about fitting things; it's about smart packing and understanding value, which are skills that come in handy everywhere!

Understanding the Packaging Options

Before we can figure out the best way to pack our 25 pages, it's crucial to understand the packaging options we have at our disposal. The problem provides us with a clear hierarchy of how items are packaged, and each level has a certain 'sticker' value, which in this context, we can think of as representing the capacity or the number of pages it holds. We have individual 'stickers' which represent 1 page. Then, we have 'pages' which contain 10 stickers (meaning 10 pages). Next, we have 'books,' where each book holds 10 pages (100 stickers in total). Finally, the largest unit is a 'bag,' which contains 10 books. Since each book has 10 pages, a bag holds 10 books * 10 pages/book = 100 pages (or 1,000 stickers). Our target is to pack exactly 25 pages. Let's break down what each option means in terms of pages:

  • 1 Sticker = 1 Page: This is our base unit. You can think of these as individual sheets of paper, each counted as one page.
  • 1 Page Container = 10 Pages: This container holds 10 individual pages.
  • 1 Book (10 pages) = 100 Pages: This is where things get interesting. The description says "1 book (10 pages) = 100". This is a potential point of confusion. Based on the structure, it's more likely that "1 book" is a unit that contains 100 pages (or 10 'page' containers). If we strictly follow "1 book (10 pages)", then it only holds 10 pages. However, the sticker count (100) strongly suggests it holds 100 pages. For the purpose of solving this problem and assuming the most logical hierarchy, we'll assume 1 book holds 100 pages. This is a common way to represent larger quantities efficiently.
  • 1 Bag (10 books) = 1,000 Pages: This is our largest unit. If a book holds 100 pages, then a bag holding 10 books would indeed hold 1,000 pages.

Given our target of 25 pages, we need to see which of these units can help us reach that number with the fewest packages. We're not dealing with stickers directly anymore; we're focused on the number of pages each unit can hold. The key is to use the largest possible units that fit within our 25-page target without exceeding it, and then fill the remainder with smaller units. This strategy is fundamental to making the most efficient use of packaging.

Evaluating the Packing Options for 25 Pages

Now that we've clarified what each package type represents in terms of pages, let's systematically evaluate the given options to see which one successfully packs exactly 25 pages using the fewest number of packages. The goal is to reach our target of 25 pages. We need to consider the total number of pages each combination provides and, critically, the total number of packages used.

Let's re-evaluate the definitions to be absolutely clear, as there was a slight ambiguity in the original prompt regarding "1 book (10 pages) = 100". Assuming the sticker count is the definitive measure of capacity:

  • Page Container: Holds 10 pages.
  • Book: Holds 100 pages (10 'page' containers).
  • Bag: Holds 1,000 pages (10 books).

We need to pack precisely 25 pages.

Let's examine the provided options:

Option A: 2 books and 5 pages

  • 2 books: If a book holds 100 pages, then 2 books hold 2 * 100 = 200 pages.
  • 5 pages: This likely refers to the base unit of 1 page, so 5 pages.

This option totals 200 + 5 = 205 pages. This is far more than our target of 25 pages. Furthermore, it uses 2 (books) + 5 (page containers) = 7 packages. This option is incorrect because it does not result in 25 pages.

Option B: 1 book and 15 pages

  • 1 book: If a book holds 100 pages, then 1 book holds 100 pages.
  • 15 pages: This likely refers to 15 individual page units.

This option totals 100 + 15 = 115 pages. This is also significantly more than our target of 25 pages. It uses 1 (book) + 15 (page containers) = 16 packages. This option is incorrect because it does not result in 25 pages.

Option C: 25 books

  • 25 books: If a book holds 100 pages, then 25 books hold 25 * 100 = 2,500 pages.

This option totals 2,500 pages, which is vastly over our target of 25 pages. It uses 25 packages. This option is incorrect because it does not result in 25 pages.

It seems there might be a misunderstanding of the options or the capacities based on the provided structure and the target of 25 pages. Let's re-interpret the problem assuming the options refer to combinations that add up to 25 pages, and we need to find the one that uses the fewest total number of packages. The capacities we've established are: Page Container (10 pages), Book (100 pages), Bag (1000 pages).

If the options are meant to represent actual page counts that sum to 25, then we need to break down 25 pages using the available units in the most efficient way.

To pack exactly 25 pages, we should use the largest units possible without exceeding the target.

  • Can we use a 'Book' (100 pages)? No, it's too large.
  • Can we use 'Page Containers' (10 pages each)? Yes.

To get 25 pages using 'Page Containers' and the base 'Page' unit:

  • We can use two 'Page Containers', which give us 2 * 10 = 20 pages.
  • We still need 25 - 20 = 5 more pages.
  • We can get these 5 pages using 5 individual 'Page' units.

So, the most efficient way to pack 25 pages using the defined units would be: 2 Page Containers + 5 Pages. This results in a total of 25 pages. The total number of packages used here is 2 (Page Containers) + 5 (individual Pages) = 7 packages.

Now let's look at the given options again and see if they can be interpreted differently to fit the target of 25 pages. It's possible the labels in the options (A, B, C) are meant to represent combinations that sum to 25 pages, not literal quantities of those package types. Let's assume this.

Re-evaluating Options as Combinations to Reach 25 Pages:

Let's assume the units are:

  • Page (1 page)
  • Page Container (10 pages)
  • Book (let's assume for a moment the "10 pages" in "1 book (10 pages)" was intended, making it similar to a Page Container but perhaps a different physical form).

If "1 book" means 10 pages (ignoring the "= 100" sticker count for a moment to see if the options make sense):

  • Page: 1 page
  • Page Container: 10 pages
  • Book: 10 pages

We need to reach 25 pages.

Option A: 2 books and 5 pages

  • 2 books * 10 pages/book = 20 pages
  • 5 pages * 1 page/page = 5 pages
  • Total pages = 20 + 5 = 25 pages
  • Total packages = 2 books + 5 pages = 7 packages

Option B: 1 book and 15 pages

  • 1 book * 10 pages/book = 10 pages
  • 15 pages * 1 page/page = 15 pages
  • Total pages = 10 + 15 = 25 pages
  • Total packages = 1 book + 15 pages = 16 packages

Option C: 25 books

  • 25 books * 10 pages/book = 250 pages
  • This option does not equal 25 pages.

Under this interpretation (where a book holds 10 pages), Option A gives us exactly 25 pages using 7 packages, and Option B gives us exactly 25 pages using 16 packages. Option C is invalid. Between A and B, Option A uses fewer packages (7 vs. 16).

This interpretation makes the options work as intended for a problem about reaching a specific number with minimal containers. The discrepancy with the "= 100" sticker count for a book is likely an error in the prompt's definition, or it implies a distinction between sticker count and actual page capacity that isn't relevant to solving the problem with the given options. Given the structure of the question asking us to choose from options A, B, and C, it's highly probable that the interpretation where a 'book' contributes 10 pages (like a 'page container') is the intended one, allowing us to reach 25 pages.

Therefore, Option A (2 books and 5 pages) is the correct answer because it precisely yields 25 pages and uses the fewest number of packages (7) compared to Option B (16 packages).

Why Option A is the Optimal Choice

Let's solidify why Option A is the best way to pack 25 pages using the fewest packages. We've analyzed the options by assuming that the goal is to reach exactly 25 pages, and that the terms 'book' and 'page' in the options refer to units that contribute to this total. The key insight here is that to minimize the number of packages, we should always try to use the largest available containers first, as they hold more items per package.

Our target is 25 pages. We have units that hold 10 pages (let's call them 'Page Containers' or 'Books' if they contain 10 pages) and units that hold 1 page (individual 'Pages').

To get 25 pages:

  • We want to use as many 10-page units as possible without going over 25.
  • We can use two 10-page units. This gives us 2 * 10 = 20 pages. This uses 2 packages.
  • We still need 25 - 20 = 5 more pages.
  • We get these remaining 5 pages by using 5 individual 1-page units. This uses 5 packages.
  • The total number of packages is 2 + 5 = 7 packages.

This combination (2 x 10-page units + 5 x 1-page units) perfectly reaches 25 pages and uses a total of 7 packages. This directly matches Option A: 2 books and 5 pages, assuming a 'book' in this context holds 10 pages.

Now, let's look at why other options are less efficient or incorrect:

  • Option B: 1 book and 15 pages:

    • This would mean 1 x 10-page unit + 15 x 1-page units.
    • Total pages = 10 + 15 = 25 pages. (It reaches the target).
    • Total packages = 1 + 15 = 16 packages.
    • Comparing Option A (7 packages) and Option B (16 packages), Option A clearly uses fewer packages.

  • Option C: 25 books:

    • If a 'book' holds 10 pages, this would be 25 * 10 = 250 pages. This doesn't equal 25 pages, so it's invalid.
    • If a 'book' holds 100 pages (based on sticker count), this would be 25 * 100 = 2,500 pages. Also invalid.

By using the largest possible units first, we ensure we minimize the total count of packages. This is a common strategy in problems involving optimization or making change – you always want to use as many of the highest denomination as possible.

Option A achieves the target of 25 pages by using two 10-page units (books) and five 1-page units (pages). This results in a total of 7 packages. This is the most efficient solution among the valid options presented, fulfilling both conditions: packing exactly 25 pages and using the fewest packages.

Conclusion

In our quest to pack exactly 25 pages using the fewest possible packages, we've systematically evaluated the provided options. The core principle we've applied is to prioritize larger containers to minimize the total number of packages used. After clarifying the potential ambiguity in the definition of a 'book,' we proceeded with the most logical interpretation that allows the options to function as intended for a problem of this nature – that a 'book' in the options represents a 10-page unit.

By doing so, Option A: 2 books and 5 pages emerged as the clear winner. This combination provides exactly 2 * 10 pages (from the books) + 5 * 1 page (from the pages) = 25 pages. The total number of packages used is 2 books + 5 pages = 7 packages.

Option B, while also totaling 25 pages (1 book of 10 pages + 15 individual pages), requires 1 + 15 = 16 packages, making it less efficient than Option A. Option C was not a valid way to pack exactly 25 pages under any reasonable interpretation.

This problem is a great example of how understanding value and quantity, and applying a strategy of using larger units first, can lead to the most efficient solution. It's a fundamental concept in mathematics, often seen in problems related to number bases or optimization.

For further exploration into optimization problems and efficient packing strategies, you can check out resources on Greedy Algorithms or Change-making problems. These mathematical concepts underpin many real-world solutions, from logistics to computer science.

For more on mathematical problem-solving strategies, you can visit Brilliant.org.