Danae and Sonny are selling apples for a school fundraiser. Customers can buy small crates of apples and large crates of apples. Danae sold 3 small crates of apples and 14 large crates of apples for a total of $203. Sonny sold 11 small crates of apples and 11 large crates of apples for a total of $220. Find the cost each of one small crate of apples and one large box of apples.

Question

(2 points)

Small crates of apples cost $
Large crates of apples cost $

Answer 1

Let the cost of a small crate of apples be \( x \) and the cost of a large crate of apples be \( y \).

From the information given, we can set up the following equations based on Danae's and Sonny's sales:

1. For Danae: \[ 3x + 14y = 203 \]

2. For Sonny: \[ 11x + 11y = 220 \]

We can simplify Sonny's equation by dividing everything by 11: \[ x + y = 20 \]

Now we have a system of equations:

1. \( 3x + 14y = 203 \)
2. \( x + y = 20 \)

We can solve for \( y \) in the second equation: \[ y = 20 - x \]

Now, we substitute \( y \) into the first equation: \[ 3x + 14(20 - x) = 203 \]

Expanding this gives: \[ 3x + 280 - 14x = 203 \]

Combining like terms: \[ -11x + 280 = 203 \]

Subtracting 280 from both sides: \[ -11x = 203 - 280 \] \[ -11x = -77 \]

Dividing by -11: \[ x = 7 \]

Now, we substitute \( x \) back into the equation for \( y \): \[ y = 20 - 7 = 13 \]

Thus, the cost of one small crate of apples is \( \boxed{7} \) dollars and the cost of one large crate of apples is \( \boxed{13} \) dollars.