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A total of \(\mathrm{s}\) oranges are to be packaged in boxes that will hold \(\mathrm{r}\) oranges each, with no oranges left over. When \(\mathrm{n}\) of these boxes have been completely filled, what is the number of boxes that remain to be filled?
Let's break down what this problem is asking in everyday language:
Think of this like packing moving boxes. If you're moving your book collection and each box holds 20 books, you need to figure out how many more boxes you still need to pack after you've already filled some boxes.
Process Skill: TRANSLATE - Converting the problem language into concrete, understandable terms
To find how many boxes we need in total, we need to think: "How many groups of r oranges can we make from s oranges?"
For example, if we have 60 oranges and each box holds 12 oranges, we need \(\mathrm{60 ÷ 12 = 5}\) boxes total.
In general terms, when we have s oranges and each box holds r oranges, the total number of boxes needed is s divided by r.
Mathematically: Total boxes needed = \(\mathrm{s/r}\)
Now we know:
Using our moving example: if you need 5 boxes total and you've already packed 2 boxes, then you have 5 - 2 = 3 boxes left to pack.
In our orange problem: Boxes remaining = \(\mathrm{(s/r) - n}\)
Process Skill: INFER - Drawing the logical conclusion that remaining work equals total work minus completed work
Let's test our answer \(\mathrm{(s/r) - n}\) with simple numbers:
Suppose s = 24 oranges, r = 6 oranges per box, and n = 2 boxes already filled.
Using our formula: \(\mathrm{(s/r) - n = (24/6) - 2 = 4 - 2 = 2}\) boxes ✓
This makes perfect sense! We started with 24 oranges, filled 2 boxes with 12 oranges (2 × 6), so we have 12 oranges left, which need exactly 2 more boxes.
The number of boxes that remain to be filled is \(\mathrm{(s/r) - n}\).
Looking at our answer choices, this matches choice E: \(\mathrm{(s/r) - n}\)
Answer: E
1. Misunderstanding what the question is asking for
Students often confuse "boxes that remain to be filled" with "oranges that remain to be packaged." The question asks for the NUMBER OF BOXES, not the number of oranges. This leads to incorrect formulations like \(\mathrm{s - nr}\) (which gives remaining oranges, not remaining boxes).
2. Incorrectly interpreting "n boxes have been completely filled"
Some students misinterpret this as "n oranges have been used" rather than "n complete boxes have been filled." This confusion leads them to think that only n oranges (instead of n×r oranges) have been packaged, resulting in wrong approaches.
3. Not establishing the total framework first
Many students jump directly to subtracting without first determining how many total boxes are needed. They might think the answer is simply s - n, forgetting that s represents oranges while n represents boxes - you can't directly subtract boxes from oranges.
1. Arithmetic confusion with fractions
When working with \(\mathrm{s/r - n}\), students sometimes incorrectly manipulate the expression, writing it as \(\mathrm{(s-n)/r}\) or \(\mathrm{s/(r-n)}\), especially when trying to find a common denominator or simplify.
2. Misapplying the division
Students may correctly identify that division is needed but apply it incorrectly, calculating \(\mathrm{r/s}\) instead of \(\mathrm{s/r}\) when determining total boxes needed, leading them toward wrong answer choices like choice D.
1. Choosing an answer that gives oranges instead of boxes
Students who calculated \(\mathrm{s - nr}\) (remaining oranges) might select choice A, not realizing this gives the count of oranges left, not boxes left. They fail to do the final check of units.
2. Getting confused between similar-looking expressions
Students might correctly derive \(\mathrm{(s/r) - n}\) but then select choice B: \(\mathrm{s - (n/r)}\) because they misread the parentheses or think the expressions are equivalent. They don't verify their answer with concrete numbers to catch this error.
Step 1: Choose convenient concrete values
Let's select values that make calculations clean and logical:
• s = 20 oranges (total oranges)
• r = 4 oranges per box
• n = 3 boxes already filled
These numbers are chosen because:
• 20 is divisible by 4, ensuring no oranges are left over
• The values are small enough for easy calculation
• They represent a realistic scenario
Step 2: Calculate total boxes needed
Total boxes required = \(\mathrm{s ÷ r = 20 ÷ 4 = 5}\) boxes
Step 3: Calculate boxes remaining
Boxes already filled = n = 3
Boxes remaining to be filled = Total boxes - Boxes filled = \(\mathrm{5 - 3 = 2}\) boxes
Step 4: Test each answer choice with our concrete values
Step 5: Verify the logic
Choice E gives us \(\mathrm{(s/r) - n}\):
• \(\mathrm{s/r}\) represents total boxes needed
• n represents boxes already filled
• The difference gives boxes remaining to be filled
This matches our expected answer of 2 boxes remaining.