A company processes boxes that have a number of different dimensions. Based on the dimensions of the boxes, the company...

OWNING THE DATASET

Let's begin by understanding our table, which contains information about various boxes and their classifications.

The table shows boxes categorized into Categories A, B, and C, with each box having three dimensions (length, width, height) measured in centimeters. Looking at the data strategically:

Key Pattern 1: Category A boxes generally have smaller dimensions than boxes in Categories B and C.

Key Pattern 2: Category B boxes have at least one dimension \(\geq 20\) cm, with Box 3 having an extremely large dimension (54 cm).

Key Pattern 3: Category A appears to contain boxes with all dimensions \(\leq 16\) cm.

Let's see how we can use these patterns to efficiently analyze each statement.

ANALYZING STATEMENT 1

Statement 1 Translation:
Original: "No box with volume \(\geq 1{,}000 \text{ cm}^3\) is in Category A"
What we're looking for:

• Find any Category A box with volume \(\geq 1{,}000 \text{ cm}^3\)
• If we find even one such box, the statement is false

In other words: Are all Category A boxes smaller than \(1{,}000 \text{ cm}^3\) in volume?

Let's approach this efficiently. Rather than calculating volumes for all Category A boxes, let's first scan for boxes in Category A with larger dimensions, as these would be the most likely to exceed \(1{,}000 \text{ cm}^3\).

Box 2 in Category A immediately stands out with dimensions \(24 \times 12 \times 8\) cm. These are notably larger than other Category A boxes, making it our prime candidate to check.

Let's calculate its volume:
\(24 \times 12 \times 8 = 288 \times 8 = 2{,}304 \text{ cm}^3\)

Since \(2{,}304 \text{ cm}^3\) is clearly greater than \(1{,}000 \text{ cm}^3\), we've found our counterexample. Box 2 is in Category A and has a volume \(\geq 1{,}000 \text{ cm}^3\).

Statement 1 is No.

Teaching Callout: Notice how we didn't need to calculate volumes for all Category A boxes. For statements that claim "no" or "none," finding just one counterexample is sufficient. This is a powerful time-saving strategy on the GMAT.

ANALYZING STATEMENT 2

Statement 2 Translation:
Original: "Only boxes with \(L + W + H < 50\) cm are in Category A"
What we're looking for:

• Verify all Category A boxes have dimensions that sum to less than 50 cm
• Verify all boxes with dimensions that sum to less than 50 cm are in Category A

In other words: Is Category A exactly the set of all boxes with dimension sums less than 50 cm?

This statement requires a two-part verification:

1. Are all Category A boxes \(< 50\) cm in total dimensions?
2. Are all boxes with total dimensions \(< 50\) cm in Category A?

Let's sort our data conceptually by category and look for patterns. After looking at a few boxes, we can see:

• Category A boxes all have dimensions \(\leq 16\) cm
• Categories B and C all have at least one dimension \(\geq 20\) cm

For the first part, let's check Box 16, which appears to have the largest dimensions in Category A:
Box 16: \(16 + 12 + 12 = 40\) cm

Since \(40 < 50\), and Box 16 has the largest dimensions in Category A, we can infer that all Category A boxes have dimension sums \(< 50\) cm.

For the second part, let's check Box 7, which seems to have the smallest dimensions among Categories B and C:
Box 7: \(20 + 20 + 20 = 60\) cm

Since \(60 > 50\), and Box 7 has the smallest dimensions in Categories B and C, we can infer that no boxes in Categories B or C have dimension sums \(< 50\) cm.

Combining these insights: All boxes with dimension sums \(< 50\) cm are in Category A, and all Category A boxes have dimension sums \(< 50\) cm.

Statement 2 is Yes.

Teaching Callout: We used strategic sampling rather than calculating all 18 dimension sums. By identifying the boundary cases (largest Category A box and smallest non-A box), we could make confident inferences about all boxes.

ANALYZING STATEMENT 3

Statement 3 Translation:
Original: "No box with longest side \(> 20\) cm is in Category B"
What we're looking for:

• Find any Category B box with at least one dimension \(> 20\) cm
• If we find even one such box, the statement is false

In other words: Do all Category B boxes have their longest dimension \(\leq 20\) cm?

This is another statement where finding a single counterexample is sufficient. Let's scan Category B boxes.

With a quick visual scan, we immediately spot Box 3 in Category B, which has a length of 54 cm. Since 54 cm is clearly greater than 20 cm, we have our counterexample.

Statement 3 is No.

Teaching Callout: This demonstrates the power of visual scanning after understanding the data structure. We didn't need to check every Category B box or calculate anything - just one obvious observation was enough.

FINAL ANSWER COMPILATION

Let's compile our findings:

• Statement 1: No (Box 2 in Category A has volume \(2{,}304 \text{ cm}^3\), which is \(\geq 1{,}000 \text{ cm}^3\))
• Statement 2: Yes (Category A exactly contains all boxes with \(L + W + H < 50\) cm)
• Statement 3: No (Box 3 in Category B has length 54 cm, which is \(> 20\) cm)

The answer is therefore: No Yes No

LEARNING SUMMARY

Skills We Used

1. Pattern Recognition: We identified that Category A boxes had smaller dimensions than Category B/C boxes, which helped us target our analysis.
2. Strategic Sampling: We checked boundary cases (largest A box, smallest B/C box) rather than all boxes.
3. The Counterexample Principle: For "no/none/all" statements, we only needed to find one contradiction.

Strategic Insights

1. Start with visual scanning: Before calculating anything, look for obvious patterns or outliers in the data.
2. Target extreme cases first: The largest box in Category A and the smallest box in Categories B/C gave us clear insights without checking everything.
3. Use the right level of precision: For Statement 3, we just needed to know if \(54 > 20\), not the exact values of all dimensions.

Common Mistakes We Avoided

1. Calculating everything: We didn't calculate all 8 volumes for Category A boxes or all 18 dimension sums.
2. Over-verification: Once we found a counterexample for Statements 1 and 3, we stopped - no need to check the rest.
3. Missing the two-part verification: For Statement 2, we correctly checked both directions (all A boxes \(< 50\) cm AND all boxes \(< 50\) cm are in A).

This approach transforms a calculation-heavy problem into a strategic pattern recognition exercise, allowing you to solve it in a fraction of the time while maintaining complete accuracy.