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 $

Answer 1

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

From the information provided, we can create the following system of equations:

1. From Danae's sales: \[ 3x + 14y = 203 \]

2. From Sonny's sales: \[ 11x + 11y = 220 \]

Now, we can simplify the second equation by dividing everything by 11: \[ x + y = 20 \quad \text{(1)} \]

Next, we will solve for \( y \) in terms of \( x \) using equation (1): \[ y = 20 - x \quad \text{(2)} \]

Now, we can substitute equation (2) into the first equation: \[ 3x + 14(20 - x) = 203 \]

Expanding the equation: \[ 3x + 280 - 14x = 203 \]

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

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

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

Now, substituting \( x = 7 \) back into equation (2) to find \( y \): \[ y = 20 - 7 = 13 \]

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