AMC 10 · 2018 · #8

Grade 7 arithmetic
Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search
systems-of-equationslinear-equations-two-varmulti-digit-arithmetic convert-to-algebra ↑ Prerequisites: systems-of-equationslinear-equations-one-var
📏 Medium solution 💡 2 insights
Problem
Joe owns 23 coins in three kinds: 5-cent, 10-cent, and 25-cent. He has 3 more 10-cent coins than 5-cent coins, and the whole collection is worth 320 cents. Find how many more 25-cent coins he has than 5-cent coins.

Pick an answer.

(A)
0
(B)
1
(C)
2
(D)
3
(E)
4

AMC 10 2018 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Three unknown counts are linked by two rules (the count rule and the value rule), so Tool #4 (Introduce a Variable) is the natural start: name the number of 5-cent coins, and the other two counts follow from the givens, leaving a single unknown. Tool #8 (Analyze the Units) keeps every quantity in cents so the value equation is honest — no mixing dollars and cents. Tool #13 (Convert to Algebra) then turns the word rules into one equation in that single unknown and solves it. With one variable the whole problem collapses to a single linear equation.

1STEP 1

Name one count, build the rest

Let n be the 5-cent count; then 10-cent is n+3, and the three summing to 23 forces the 25-cent count to be 20-2n.

5-cent: n, 10-cent: n+3, 25-cent: 23-n-(n+3)=20-2n
2STEP 2

Write the value rule in cents

Each coin kind adds count times worth in cents; matching the 320-cent total gives 5n + 10(n+3) + 25(20-2n) = 320.

5n + 10(n+3) + 25(20-2n) = 320
3STEP 3

Expand, combine, and solve

Expand and combine like terms: -35n + 530 = 320, so -35n = -210 and n = 6.

-35n + 530 = 320 → -35n = -210 → n = 6
4STEP 4

Find the counts and the gap

With n=6 the counts are 6, 9, 8 (sum 23); the 25-cent minus the 5-cent count is 8 - 6 = 2, choice (C).

25-cent: 8, 5-cent: 6, 8-6=2 → (C)
Answer
2
Check both rules with the counts 6, 9, 8. Coin count: 6+9+8=23, correct. Value: 5(6)+10(9)+25(8)=30+90+200=320 cents, correct. The 10-cent count is 9, which is indeed 3 more than the 5-cent count 6. Every condition holds, and the requested difference 8-6=2 matches choice (C). The answer is also a small whole number, exactly the size the choices (0 through 4) lead you to expect.
💡Key takeaway

Name the smallest unknown count with a letter, write every other count and the total value from it, then solve the one equation and plug back in.

  • Name one count, build the rest
  • Write the value rule in cents
  • Expand, combine, and solve
  • Find the counts and the gap

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