Phase 1: Owning the Dataset
Visual Representation
Let's draw Andy's garden transformation options:
Original Garden:
+--------+
| | W
| LW |
+--------+
L
Option 1 (Length +5m):
+-------------+
| | W Area = 2LW
| |
+-------------+
L + 5
Option 2 (Width +2m):
+--------+
| |
| | W+2 Area = 1.5LW
| |
+--------+
L
Phase 2: Understanding the Question
Setting Up Equations
Let \(\mathrm{L}\) = original length and \(\mathrm{W}\) = original width
From Option 1:
- New area = \(\mathrm{(L + 5) \times W = 2LW}\)
- Expanding: \(\mathrm{LW + 5W = 2LW}\)
- Simplifying: \(\mathrm{5W = LW}\)
- Therefore: \(\mathrm{L = 5}\)
From Option 2:
- New area = \(\mathrm{L \times (W + 2) = 1.5LW}\)
- Expanding: \(\mathrm{LW + 2L = 1.5LW}\)
- Simplifying: \(\mathrm{2L = 0.5LW}\)
- Therefore: \(\mathrm{W = 4}\)
Key Insight
We've determined that the existing garden must have:
- Length = 5 meters
- Width = 4 meters
Phase 3: Finding the Answer
Verification
Let's verify our answer works for both scenarios:
Original area: \(\mathrm{5 \times 4 = 20 \, m^2}\)
Option 1 check: \(\mathrm{(5 + 5) \times 4 = 10 \times 4 = 40 \, m^2 = 2 \times 20}\) ✓
Option 2 check: \(\mathrm{5 \times (4 + 2) = 5 \times 6 = 30 \, m^2 = 1.5 \times 20}\) ✓
Both conditions are satisfied!
Phase 4: Solution
Final Answer
- Statement 1 (Length): 5
- Statement 2 (Width): 4
Our answer perfectly satisfies both area expansion requirements. The existing garden has dimensions of \(\mathrm{5m \times 4m}\).
