An idealized square section is exactly 1 mile on each side and is divided into four equal squares; one of those is again divided into four equal squares. What is the side length of a final quarter-quarter square?
Correct Answer
C) 0.25 miles
Idealized side length =1 mile×1/2×1/2=0.25 mile. Common errors: stopping at the quarter section gives 0.50 miles; reporting the area fraction (1/16) as a side length gives 0.06 miles; halving the side three times gives 0.13 miles.
Why This Is the Correct Answer
Option C is correct because dividing a square into four equal squares halves its side, so 1 mile × 1/2 × 1/2 = 0.25 mile. The first division gives a quarter section with a 0.50-mile side, and the second gives a quarter-quarter with a 0.25-mile side. The question asks for side length, not area.
Why the Other Options Are Wrong
Option A: 0.06 miles
Option A, 0.06 miles, comes from reporting the area fraction 1/16 (about 0.06) as a side length. The quarter-quarter covers 1/16 of the area, but its side is 1 × 1/2 × 1/2 = 0.25 mile.
Option B: 0.13 miles
Option B, 0.13 miles, comes from halving the side three times (1/8, about 0.13). Only two divisions occur, so the side is 1 × 1/2 × 1/2 = 0.25 mile.
Option D: 0.50 miles
Option D, 0.50 miles, stops at the quarter section after one division. The second division halves the side again: 0.50 × 1/2 = 0.25 mile.
Quarter the area, halve the side
Each split into four equal squares cuts area to one-fourth but cuts side length only in half.
How to use: Use this when converting survey subdivisions between area and side length.
Exam Tip
Check whether the question asks for area or side length; the two change at different rates.
Common Mistakes to Avoid
- -Using the area fraction as the side length
- -Stopping after one division
- -Halving the side one too many times
Concept Deep Dive
Analysis
This question tests rectangular survey arithmetic for side lengths. Each division of a square into four equal squares halves the side length. Halve the 1-mile side twice to reach a quarter-quarter.
Background Knowledge
In the rectangular survey system, each township contains 36 sections of approximately one square mile (640 acres). Quarter sections and quarter-quarter sections are common subdivisions, and in an idealized section a quarter-quarter is 40 acres with sides of 0.25 mile.
Real-World Application
An appraiser reviews a rural legal description for a quarter-quarter section. Knowing that each side is about a quarter mile helps the appraiser check that the frontage on the plat makes sense.
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