5. On a busy Saturday afternoon, an ice cream shop earns \( \$ 231.25 \) from selling small cups of ice cream and large cups of ice cream. The shop charges \( \$ 4.25 \) per large cup of ice cream and \( \$ 2.50 \) per small cup of ice cream. The ice cream shop sells twice as many small cups of ice cream as large cups of ice cream. a. Which system of equations represents \( x \), the number of large cups of ice cream sold and \( y \), the number of small cups of ice cream sol ( \( \begin{array}{l}y=x+2 \\ 2.50 x+4.25 y=231.25 \\ y=2 x \\ 2.50 x+4.25 y=231.25 \\ y=2 x \\ 4.25 x+2.50 y=231.25\end{array} \) \[ \begin{array}{l}x=2 y \\ 4.25 x+2.50 y=231.25\end{array} \] b. How many small cups of ice cream did the shop sell?

To represent the situation with equations, let's define \( x \) as the number of large cups sold and \( y \) as the number of small cups sold. Since the shop sells twice as many small cups as large cups, we can express this relationship as \( y = 2x \). The total sales amount gives us a second equation: \( 4.25x + 2.50y = 231.25 \). So the system of equations is: \[ \begin{cases} y = 2x \\ 4.25x + 2.50y = 231.25 \end{cases} \] To find the number of small cups sold, substitute \( y = 2x \) into the second equation. You get: \[ 4.25x + 2.50(2x) = 231.25 \] This simplifies to: \[ 4.25x + 5.00x = 231.25 \] Combining like terms gives: \[ 9.25x = 231.25 \] Divide both sides by 9.25: \[ x = 25 \] Now, substituting back to find \( y \): \[ y = 2(25) = 50 \] So, the shop sold **50 small cups of ice cream!** Ice cream has been a beloved treat for centuries, with evidence of its consumption dating back to ancient China, where a mixture of milk and rice was frozen in the snow. In Europe, it became popular among the aristocracy in the 17th century, and by the 18th century, the first ice cream recipes were published, bringing this cool delight to the masses! If you ever run an ice cream shop or a similar business, managing inventory and pricing can make a big difference in profits! Keep track of your bestsellers and adjust the number of ingredients accordingly. Also, be cautious about pricing—small changes can lead to substantial impacts on overall sales, so consider running a few tests to find what works best!