Two-number system — The difference of two numbers is 18. Also, four times the second is 18 less than three times the first. Find the sum of the two numbers.

Introduction / Context:
Solving two linear equations in two unknowns is a staple of aptitude exams. Here, one equation gives the difference of the two numbers; the other relates multiples of the numbers with an offset. The objective is the sum of the numbers once they are determined.

Given Data / Assumptions:

• Let the numbers be x (first) and y (second).
• x - y = 18.
• Four times the second is 18 less than three times the first: 4y = 3x - 18.

Concept / Approach:
Use substitution: from x - y = 18 → x = y + 18. Substitute into 4y = 3x - 18 and solve for y, then back-substitute for x. Finally, compute x + y.

Step-by-Step Solution:
From x - y = 18 → x = y + 18.Substitute: 4y = 3(y + 18) - 18 → 4y = 3y + 54 - 18 → 4y = 3y + 36 → y = 36.Then x = y + 18 = 36 + 18 = 54.Sum = x + y = 54 + 36 = 90.

Verification / Alternative check:
Check the second condition: three times the first is 162; four times the second is 144; indeed 4y is 18 less than 3x.

Why Other Options Are Wrong:

• 100 / 80 / 86 / 96: These come from algebra slips or misreading “less than” direction; none matches the computed sum 90.

Common Pitfalls:
Reversing “18 less than” into “+18”; forgetting to add the difference when expressing x in terms of y; arithmetic errors in substitution.

Final Answer:
90