2021 AMC Fall 10B Problem 4

P1-10~2 minPrint

Linear Equations & Word Problems Absolute Value & Inequalities Split into exhaustive cases and handle each Set up the equation/formula and compute; no special trick needed

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 $N?$

Show solution without answering

Key insight

The gap shrinks by 5 + 3 = 8, so it becomes N - 8. That must be 2 or -2, giving N = 10 or N = 6.

Solution

Try a number first. Suppose St. Louis is at $50$ degrees at noon. Then Minneapolis is at $50 + N$ .

1. At 4:00. Minneapolis drops $5$ : it is at $50 + N - 5 = 45 + N$ . St. Louis rises $3$ : it is at $53$ .
2. The gap. Minneapolis minus St. Louis is
$(45 + N) - 53 = N - 8.$
The $50$ disappeared, so the starting temperature does not matter. (With a letter $S$ in place of $50$ , the same thing happens.) Makes sense: one city dropped $5$ and the other rose $3$ , so the gap closed by $8$ .
3. "Differ by 2" has two meanings. Either Minneapolis is still $2$ degrees warmer, or St. Louis is now $2$ degrees warmer.
- Minneapolis still warmer: $N - 8 = 2$ , so $N = 10$ .
- St. Louis now warmer: $N - 8 = -2$ , so $N = 6$ .
4. Both values are possible. Their product is $10 \times 6 = 60$ .

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

Why this works

Only the gap between the two temperatures matters, and each change moves the gap by a known amount. The words "differ by" do not say which one is bigger, so they always give two cases. When a problem asks for "all possible values," that is a hint that more than one case counts.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(C)}\ 60}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 20: Special Functions (§20.2, §20.5)
Introduction to Counting & Probability, Chapter 2: Basic Counting Techniques (§2.2)
Introduction to Algebra, Chapter 7: Proportion (§7.4)