A rectangular garden is surrounded on all four sides by a concrete walking path of uniform… — State PSC Exam

A rectangular garden is surrounded on all four sides by a concrete walking path of uniform width $w$ meters. What is the area of the concrete walking path in square meters?

Statement I: The perimeter of the outer boundary of the walking path is $80\text{ meters}$ .
Statement II: The area of the rectangular garden is $300\text{ square meters}$ , and its length is $5\text{ meters}$ greater than its width.

Which of the following options correctly describes the sufficiency of the statements to answer the question?

A
Statement I alone is sufficient, but Statement II alone is not sufficient.
B
Statement II alone is sufficient, but Statement I alone is not sufficient.
C
Either Statement I alone or Statement II alone is sufficient.
D
Neither Statement I nor Statement II is sufficient, even when combined.
Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.Answer

Answer

Both Statement I and Statement II together are sufficient, but neither statement alone is sufficient.

The correct answer states that both statements together are sufficient, but neither alone is sufficient. Statement I leaves the path width and garden dimensions undetermined, while Statement II gives the garden dimensions without the path width. Combining them allows the path width and consequently the exact path area to be calculated uniquely.

Step-by-Step Solution

Analyze Statement I independently

Let the length and width of the garden be

L

and

W

meters, and the uniform path width be

w

meters. The outer dimensions are

(L + 2w)

and

(W + 2w)

. The outer perimeter is

2[(L + 2w) + (W + 2w)] = 2(L + W + 4w) = 80

, which simplifies to

L + W + 4w = 40

. The area of the path is

(L + 2w)(W + 2w) - LW = 2w(L + W + 2w)

. Since

L + W

and

w

can take multiple values satisfying

L + W + 4w = 40

, the area of the path cannot be uniquely determined. Hence, Statement I alone is NOT sufficient.

Evaluating whether Statement I alone fixes a single numerical value for the path area.

Analyze Statement II independently

Statement II states that

L \cdot W = 300

and

L = W + 5

. Substituting gives

(W + 5)W = 300

, so

W^2 + 5W - 300 = 0

. Factoring gives

(W + 20)(W - 15) = 0

, yielding

W = 15\text{ m}

and

L = 20\text{ m}

. However, no information is given about the path width

w

. Thus, the path area

2w(35 + 2w)

cannot be computed without

w

. Hence, Statement II alone is NOT sufficient.

Evaluating whether Statement II alone fixes a single numerical value for the path area.

Analyze Statements I and II combined

From Statement II,

L = 20\text{ m}

and

W = 15\text{ m}

, so

L + W = 35\text{ m}

. Substitute this into the relation from Statement I:

35 + 4w = 40 \Rightarrow 4w = 5 \Rightarrow w = 1.25\text{ meters}

. The area of the path is

2(1.25)(35 + 2.5) = 2.5 \times 37.5 = 93.75\text{ square meters}

, which is a unique value. Therefore, both statements together are sufficient.

Determining if combining both statements resolves all unknown variables uniquely.

Key Concept

Evaluating individual statement sufficiency for multi-variable geometric systems prior to combination.

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