Three numbers satisfy: the sum of the first and second is 45, the sum of the second and third is 55, and the sum of the third and thrice the first is 90. Find the third number.

Introduction / Context:
Word problems with multiple sums over three unknowns can be untangled by expressing each variable in terms of another and substituting step by step. This yields a simple linear equation whose solution leads directly to the requested value.

Given Data / Assumptions:

• Let the numbers be a (first), b (second), c (third).
• a + b = 45.
• b + c = 55.
• c + 3a = 90.

Concept / Approach:
Express b from the first equation and c from the second in terms of a, then substitute into the third equation. Solve for a, then compute c. This approach minimizes simultaneous juggling of all three variables at once.

Step-by-Step Solution:

Verification / Alternative check:
Check sums: a + b = 45 → b = 25; then b + c = 25 + 30 = 55; and c + 3a = 30 + 60 = 90. All conditions are satisfied.

Why Other Options Are Wrong:
20 and 25 are values of a and b respectively, not the third number. 35 and 40 do not satisfy the third equation when combined with the derived values.

Common Pitfalls:
Swapping which variable is expressed or sign mistakes when substituting b into the expression for c, leading to off-by-10 errors.

Final Answer:
30