AMC 10 Daily Practice - Diophantine Equations

Complete problem set with solutions and individual problem pages

Problem 2 Medium

Jason's mother went shopping at the supermarket. It is known that buying 13 chicken eggs, 5 duck eggs, and 9 quail eggs costs 9.25 dollars. Buying 2 chicken eggs, 4 duck eggs, and 3 quail eggs costs 3.20 dollars. Jason's mother wants to know how much it would cost to buy one chicken egg, one duck egg, and one quail egg.

  • A.

    1.05 dollars

  • B.

    1 dollars

  • C.

    1.15 dollars

  • D.

    1.2 dollars

  • E.

    0.9 dollars

Answer:A

Let the prices of chicken eggs, duck eggs, and quail eggs be x, y, and z dollars, respectively. We have

\begin{cases}13x+5y+9=9.25 \\ 2x+4y+3z=3.2 \\ \end{cases}

\begin{cases}y=x+0.05 \\ z=1-2x \\ \end{cases}

Also,

\begin{cases}13x+5y+9=9.25 \\ 2x+4y+3z=3.2 \\ \end{cases}

\begin{cases}5\left( x+y+z \right)+4\left( 2x+z \right)=9.25 \\ 4\left( x+y+z \right)-\left( 2x+z \right)=3.2 \\ \end{cases}

Then, x+y+z=1.05.

Related Problems
Problem 1 AMC 10 Daily Practice - Diophantine Equations
Problem 3 AMC 10 Daily Practice - Diophantine Equations
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