2004 AMC 10B Problem 15

Below is the professionally curated solution for Problem 15 of the 2004 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2004 AMC 10B solutions, or check the answer key.

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Concepts:system of equations money

Difficulty rating: 1240

15.

Patty has $20$ coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $70$ cents more. How much are her coins worth?

$\$1.15$

$\$1.20$

$\$1.25$

$\$1.30$

$\$1.35$

Solution:

Swapping increases the value, so Patty has more nickels than dimes. Each swapped coin changes the total by $5$ cents, so she has $70 / 5 = 14$ more nickels than dimes.

With $n + d = 20$ and $n - d = 14,$ she has $17$ nickels and $3$ dimes.

Her coins are worth $17 \cdot 5 + 3 \cdot 10 = 115$ cents, or $\$1.15.$

Thus, the correct answer is A.

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