A rectangular garden has a perimeter of 28 meters and an area of 45 square meters. What is the ratio of the length of the shorter side to the length of the longer side?

This ratio would imply side lengths of \( 3k \) and \( 5k \) for some \( k \). However, using the perimeter equation \( 3k + 5k = 14 \) leads to \( 8k = 14 \), giving \( k = 1.75 \). Calculating the area \( (3 \times 1.75)(5 \times 1.75) = 15.75 \), which does not satisfy the area condition of 45 square meters.

If the sides were in a ratio of 4 to 5, we would have lengths of \( 4k \) and \( 5k \). The perimeter equation \( 4k + 5k = 14 \) results in \( 9k = 14 \), so \( k \) would equal approximately \( 1.56 \). The area calculated from these dimensions would be \( (4 \times 1.56)(5 \times 1.56) \approx 31.2 \), which is again inconsistent with the area of 45 square meters.

This choice suggests side lengths of \( 5k \) and \( 9k \). From the perimeter condition, \( 5k + 9k = 14 \) leads to \( 14k = 14 \), so \( k = 1 \). The area becomes \( 5 \times 1 \times 9 \times 1 = 45 \), which is correct, but the ratio of the shorter side (5) to the longer side (9) is \( \frac{5}{9} \), not \( \frac{2}{