In this letter analogy, “DCEF is to QPRS as XWYZ is to ______”. Select the group of letters that completes the analogy by applying the same constant alphabet shift used in the first pair.

Introduction / Context:
This analogy uses a constant shift in alphabet positions, similar to a rotation or substitution cipher. The pair “DCEF : QPRS” shows how each letter in the first group is moved forward by the same number of positions to generate the second group. The task is to identify this fixed shift and then apply it to the group XWYZ, choosing the correct resulting sequence from the options. Mastery of alphabet positions and constant shifts is essential to solving such questions.

Given Data / Assumptions:

• First mapping: DCEF → QPRS. • Second mapping: XWYZ → ? • Options: KJLM, PQRS, STOV, IMGF. • Alphabet positions: A = 1, B = 2, ..., Z = 26, with wrap around when shifting beyond Z.

Concept / Approach:
We compare each corresponding letter in DCEF and QPRS: D with Q, C with P, E with R, and F with S. By converting these letters to numeric positions, we can check how many positions forward each letter moves. If the difference is the same for all four, this constitutes a constant shift (a rotation). Once we know the shift value, we apply it to each letter of XWYZ, wrapping around the alphabet when necessary, and then match the result to one of the options.

Step-by-Step Solution:
Step 1: Convert DCEF and QPRS to numeric positions. D = 4, C = 3, E = 5, F = 6. Q = 17, P = 16, R = 18, S = 19. Step 2: Compute the shift for each pair. 4 → 17: shift of +13. 3 → 16: shift of +13. 5 → 18: shift of +13. 6 → 19: shift of +13. Thus, each letter is shifted forward by 13 positions. Step 3: Apply the same shift to XWYZ. X = 24, W = 23, Y = 25, Z = 26. Add 13 to each and wrap around the alphabet by subtracting 26 whenever the sum exceeds 26. X: 24 + 13 = 37, 37 − 26 = 11 → K. W: 23 + 13 = 36, 36 − 26 = 10 → J. Y: 25 + 13 = 38, 38 − 26 = 12 → L. Z: 26 + 13 = 39, 39 − 26 = 13 → M. So XWYZ maps to KJLM.

Verification / Alternative check:
We can confirm the rule by reversing it. If we subtract 13 from Q, P, R, and S, we should return to D, C, E, and F respectively. Q (17) − 13 = 4 → D, P (16) − 13 = 3 → C, R (18) − 13 = 5 → E, S (19) − 13 = 6 → F. Similarly, subtracting 13 from KJLM should yield XWYZ: K (11) − 13 = −2, which after adding 26 gives 24 → X; J (10) − 13 = −3, plus 26 gives 23 → W; L (12) − 13 = −1, plus 26 gives 25 → Y; M (13) − 13 = 0, plus 26 gives 26 → Z. This confirms the mapping is consistent both forward and backward.

Why Other Options Are Wrong:
• PQRS: This is the mapping of DCEF itself and does not correspond to the second group XWYZ. • STOV and IMGF: These letter groups do not result from applying a +13 shift to each letter of XWYZ and do not match the consistent rotation pattern identified from the example pair.

Common Pitfalls:
A common mistake is to try to spot visual sequences like consecutive letters or simple reversals, ignoring the strong evidence from numeric differences. Another pitfall is to forget to wrap around the alphabet when the position goes beyond 26. Always remember that when using constant shifts, positions are effectively taken modulo 26, and subtracting or adding 26 keeps the letters within A to Z. This avoids incorrect mapping when working with letters near the end of the alphabet like X, Y, and Z.

Final Answer:
The group of letters that correctly completes the analogy is KJLM.