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2018 AMC 10A Problem 8

2018 AMC 10A Problem 8AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 2 of 5 · about 3 min · position P1-10

Joe has a collection of $23$ coins, consisting of $5$ -cent coins, $10$ -cent coins, and $25$ -cent coins. He has $3$ more $10$ -cent coins than $5$ -cent coins, and the total value of his collection is $320$ cents. How many more $25$ -cent coins does Joe have than $5$ -cent coins?

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

Answer

Key insight

Express every count in terms of the number of nickels x: dimes x + 3, quarters 20 - 2x; the value equation then has one unknown.

Solution

Let $x$ be the number of $5$ -cent coins. Then there are $x + 3$ ten-cent coins, and the quarters make up the rest:
$23 - x - (x + 3) = 20 - 2x.$

Now write the value in cents:
$5x + 10(x+3) + 25(20 - 2x) = 320.$
Expanding: $5x + 10x + 30 + 500 - 50x = 320$ , so $530 - 35x = 320$ , giving $35x = 210$ and $x = 6$ .

Joe has $6$ nickels, $9$ dimes, and $20 - 12 = 8$ quarters. Check: $30 + 90 + 200 = 320$ cents and $6 + 9 + 8 = 23$ coins.

The question asks for quarters minus nickels: $8 - 6 = 2$ .

The answer is $\boxed{\textbf{(C)}\ 2}$ .

Why this works

Three unknowns with three relationships (count, "3 more than", value) is a linear system, but two of the relationships are simple enough to substitute immediately, collapsing everything into one equation in one variable. Always name the quantity that the other quantities are described relative to (here the nickels) as the variable.

The trap

Reporting the number of quarters (8) or nickels (6) instead of their difference, or writing the quarter count as 23 - x instead of 23 - x - (x + 3).

Common mistakes

Reporting the number of quarters (8) or nickels (6) instead of their difference, or writing the quarter count as 23 - x instead of 23 - x - (x + 3).
Forgetting to distribute the $25$ across $20 - 2x$ , which produces a non-integer coin count and signals an error.

Techniques

Set up the equation/formula and compute; no special trick needed