AMC 10 · 2002 · #25

Grade 8 arithmetic
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
mean-median-mode-rangesystems-of-equationslinear-equations-two-var convert-to-algebra ↑ Prerequisites: mean-median-mode-rangesystems-of-equations
📏 Long solution 💡 3 insights
Problem
A list of integers has some mean. Adding 15 to the list raises the mean by 2. Adding 1 to that larger list lowers its mean by 1. Find how many integers the original list held.

Pick an answer.

(A)
4
(B)
5
(C)
6
(D)
7
(E)
8

AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The individual integers are unknown and never needed, but the count and the mean are exactly what the questions talk about. Tool #4 (Introduce a Variable) names the count n and the mean m, which pins down the sum as nm. Tool #13 (Convert to Algebra) then turns each English sentence about the mean into a linear equation, giving a small 2 × 2 system that solves cleanly. Tool #6 (Guess and Check) is held in reserve for the review: the answer choices are small whole numbers, so the winning count can be replayed on a concrete list to confirm both conditions.

1STEP 1

Name the count and the mean

Let n be the count of integers and m the mean, so the sum is nm — the individual values never have to be known.

sum = nm
2STEP 2

Turn the first change into an equation

Appending 15 gives count n+1, sum nm+15, mean m+2, so nm+15=(m+2)(n+1); the nm cancels and leaves m+2n=13.

(nm+15)/(n+1)=m+2 → m+2n=13
3STEP 3

Turn the second change into an equation

Appending 1 gives count n+2, sum nm+16, and mean m+1 — one below m+2 — so nm+16=(m+1)(n+2), leaving 2m+n=14.

(nm+16)/(n+2)=m+1 → 2m+n=14
4STEP 4

Solve the two equations together

Substituting m=13-2n into 2m+n=14 gives 26-3n=14, so n=4 and m=5 — the original list held four integers, choice (A).

m=13-2n → 26-3n=14 → n=4
Answer
4
Replay the answer on a concrete list of four 5s, whose mean is 5 and sum is 20. Appending 15 makes the sum 35 over 5 numbers, a mean of 7 — up by 2, as required. Appending 1 makes the sum 36 over 6 numbers, a mean of 6 — down by 1 from 7, as required. Both conditions hold, so n=4 is consistent.
💡Key takeaway

When only counts and means are given, name the count and the mean, write the total as count times mean, and each sentence about the mean becomes one clean equation.

  • Name the count and the mean
  • Turn the first change into an equation
  • Turn the second change into an equation
  • Solve the two equations together

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