Recover and solve (clarified condition): The difference between two numbers is 18. Also, “four times the second is less than three times the first by 20.” What is the sum of the two numbers?

Introduction / Context:
(Recovery-first clarification) The original wording was incomplete. We adopt a standard linear condition: “four times the second is less than three times the first by 20,” i.e., 3*first − 4*second = 20, along with a fixed difference of 18. This yields a unique solution for the pair and hence their sum.

Given Data / Assumptions:

• Let the numbers be x (first) and y (second).
• Difference: x − y = 18.
• Relation: 3x − 4y = 20 (“less by 20”).

Concept / Approach:
Solve the two linear equations in two unknowns. Substitution from the difference equation into the second relation simplifies the system quickly and gives integer values for x and y.

Step-by-Step Solution:
From x − y = 18 ⇒ x = y + 18.Substitute into 3x − 4y = 20: 3(y + 18) − 4y = 20.Simplify: 3y + 54 − 4y = 20 ⇒ −y + 54 = 20 ⇒ y = 34.Then x = y + 18 = 52.Sum: x + y = 52 + 34 = 86.

Verification / Alternative check:
Check 3x − 4y: 3*52 − 4*34 = 156 − 136 = 20; difference x − y = 18. Both hold, confirming the sum 86.

Why Other Options Are Wrong:
100, 80, and 92 do not satisfy both equations; 96 arises from common arithmetic slips but fails the verification.

Common Pitfalls:
Interpreting “less than” without the “by” amount; flipping x and y in the difference; arithmetic errors when substituting.

Final Answer:
86