A garden bed consists of two adjacent square plots that share one complete side. The side length of the larger...
GMAT Problem-Solving and Data Analysis : (PS_DA) Questions
A garden bed consists of two adjacent square plots that share one complete side. The side length of the larger square is 3 times the side length of the smaller square. Fertilizer is spread uniformly across the entire garden bed at a rate of 25 grams per square meter. If a total of 2,500 grams of fertilizer is used on the garden bed, what mass of fertilizer, in grams, is applied to the larger square? (Enter your answer as an integer.)
Fill-in-the-blank (enter an integer). No units needed.
1. TRANSLATE the problem information
- Given information:
- Two adjacent squares share one side
- Larger square side = 3 × smaller square side
- Fertilizer rate = 25 grams per square meter
- Total fertilizer used = 2,500 grams
- Need: fertilizer mass on larger square only
2. TRANSLATE into mathematical variables
- Let \(\mathrm{s}\) = side length of smaller square
- Then \(\mathrm{3s}\) = side length of larger square
- This gives us areas: \(\mathrm{s^2}\) (small) and \(\mathrm{(3s)^2 = 9s^2}\) (large)
3. INFER the solution strategy
- We have fertilizer information but need area information
- Key insight: Use the total fertilizer to find total garden area first
- Then we can find individual square areas
4. Find total garden area using fertilizer data
- Total area = Total fertilizer ÷ Rate = \(\mathrm{2,500 ÷ 25 = 100}\) square meters
- Combined areas: \(\mathrm{s^2 + 9s^2 = 10s^2 = 100}\)
- Therefore: \(\mathrm{s^2 = 10}\)
5. SIMPLIFY to find the larger square's area
- Larger square area = \(\mathrm{9s^2 = 9(10) = 90}\) square meters
6. Calculate fertilizer on larger square
- Fertilizer mass = Area × Rate = \(\mathrm{90 × 25 = 2,250}\) grams
Answer: 2250
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak INFER skill: Students try to work directly with the \(\mathrm{3:1}\) side ratio without realizing they need the \(\mathrm{9:1}\) area ratio.
They might think: "If the side ratio is \(\mathrm{3:1}\), then the larger square gets 3/4 of the fertilizer." This would give them \(\mathrm{(3/4) × 2,500 = 1,875}\) grams. However, this ignores that area scales by the square of the linear dimension.
This leads to confusion and an incorrect answer.
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret "share one complete side" and try to subtract the shared area, not realizing this is already accounted for in the problem setup.
They might calculate total area as something other than \(\mathrm{s^2 + 9s^2}\), leading to wrong values for \(\mathrm{s^2}\) and ultimately the wrong fertilizer amount.
This causes them to get stuck with inconsistent equations.
The Bottom Line:
This problem requires recognizing that linear scaling (\(\mathrm{3:1}\) ratio) becomes quadratic scaling (\(\mathrm{9:1}\) ratio) for areas. Students who miss this crucial relationship between side length ratios and area ratios will struggle throughout the solution.