Directions : Study the following information carefully and answer the given…

2024

Directions : Study the following information carefully and answer the given questions.
In a certain coded language:
M@6N means M is 4m north of N.
M%9N means M is 6m south of N.
M#4N means M is 2m east of N.
M$5N means M is 2m west of N.

Conditions: A%8B, C$12B, Z#17D, E$19F, C%11E, Z@9F, G%4D, A$7R, R%14W

What is the shortest distance between Point W and Point C and Point W is in which direction with respect to Point E?

  1. A.

    2√59m, South-east

  2. B.

    2√58m, South-east

  3. C.

    3√59m, South-west

  4. D.

    58m, South

  5. E.

    None of these

Show answer & explanation

Correct answer: B

Concept

In a coded-direction puzzle the symbol fixes the direction and the number fixes the distance through a parity rule. From the legend, @ = north, % = south, # = east and $ = west. The legend also encodes distance: an EVEN number means (number - 2) metres and an ODD number means (number - 3) metres — verify with the four examples: @6 -> 6-2 = 4, %9 -> 9-3 = 6, #4 -> 4-2 = 2, $5 -> 5-3 = 2. Put one point at the origin on a north-up grid (north = +y, east = +x), turn every condition into a coordinate shift, then use the distance formula for the straight-line gap and the signs of the coordinate difference for the compass direction.

Applying the code (decode each condition, with B at the origin):

  1. Decode the distances first (even -> n-2, odd -> n-3): 8->6, 12->10, 17->14, 19->16, 11->8, 9->6, 4->2, 7->4, 14->12.

  2. A%8B: A is 6 m south of B, so with B(0, 0), A = (0, -6).

  3. C$12B: C is 10 m west of B, so C = (-10, 0).

  4. C%11E: C is 8 m south of E, so E = (-10, 8).

  5. E$19F: E is 16 m west of F, so F = (6, 8).

  6. Z@9F: Z is 6 m north of F, so Z = (6, 14).

  7. Z#17D: Z is 14 m east of D, so D = (-8, 14).

  8. A$7R: A is 4 m west of R, so R = (4, -6).

  9. R%14W: R is 12 m south of W, so W = (4, 6).

  10. G%4D: G is 2 m south of D, giving G = (-8, 12); point G is decoded for completeness but is not used in the W-C-E comparison.

Distance W to C with W(4, 6) and C(-10, 0): horizontal gap = 4 - (-10) = 14, vertical gap = 6 - 0 = 6, so the straight-line distance = √(142 + 62) = √(196 + 36) = √232 = 2√58 m.

Direction of W relative to E with W(4, 6) and E(-10, 8): x-difference = 4 - (-10) = +14 (W is east of E) and y-difference = 6 - 8 = -2 (W is south of E). East with south gives South-east.

Cross-check

232 = 4 x 58, so √232 = 2√58 exactly. It is distinct from √236 (= 2√59) because the squared gaps total 196 + 36 = 232, not 236. Reading the parity code on another line confirms the setup: $7 is odd, giving 7-3 = 4 m for A to R, which keeps W's x-coordinate at 4 and preserves the +14 eastward offset from E.

Result: the shortest distance between W and C is 2√58 m, and W is South-east of E.

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