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2021 AMC Fall 10B Problem 4

2021 AMC Fall 10B Problem 4AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 2 min · position P1-10

At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis. At $4{:}00$ the temperature in Minneapolis has fallen by $5$ degrees while the temperature in St. Louis has risen by $3$ degrees, at which time the temperatures in the two cities differ by $2$ degrees. What is the product of all possible values of $N?$

A)
$10$
B)
$30$
C)
$60$
D)
$100$
E)
$120$

Answer

Key insight

After the changes the gap is N - 8, and 'differ by 2' means N - 8 = ±2, so N is 10 or 6.

Solution

Let St. Louis be at temperature $S$ at noon, so Minneapolis is at $S + N$ .

At $4{:}00$ Minneapolis is at $S + N - 5$ and St. Louis is at $S + 3$ . The difference (Minneapolis minus St. Louis) is
$(S + N - 5) - (S + 3) = N - 8.$
The cities "differ by $2$ degrees," so $|N - 8| = 2$ . Either $N - 8 = 2$ , giving $N = 10$ (Minneapolis still warmer), or $N - 8 = -2$ , giving $N = 6$ (St. Louis now warmer).

Both are valid, and their product is $10 \cdot 6 = 60$ .

The answer is $\boxed{\textbf{(C)}\ 60}$ .

Why this works

Only the difference matters, so the unknown base temperature $S$ cancels immediately; the changes $-5$ and $+3$ shift the gap by $8$ . "Differ by" is an absolute value statement and always produces two cases. The phrase "all possible values" is the problem's hint that both cases count.

The trap

Only considering Minneapolis still being warmer (N = 10) and forgetting the case where St. Louis has become warmer (N = 6).

Common mistakes

Only considering Minneapolis still being warmer (N = 10) and forgetting the case where St. Louis has become warmer (N = 6).
Shifting the gap by $5 - 3 = 2$ instead of $5 + 3 = 8$ , because the two temperatures moved in opposite directions.

Techniques

Split into exhaustive cases and handle each · Set up the equation/formula and compute; no special trick needed