For an art project, Amy cut a 30 ft piece of yarn into different sized pieces. 10 pieces were \( \frac{7}{12} \mathrm{ft} \) long and 18 pieces were \( \frac{3}{4} \mathrm{ft} \) long. How long is the leftover piece of yarn in feet?

To find out how long the leftover piece of yarn is, we first need to calculate the total length of the yarn used for the pieces. Amy has 10 pieces that are each \( \frac{7}{12} \) ft long. The total length for these pieces is: \[ 10 \times \frac{7}{12} = \frac{70}{12} = \frac{35}{6} \text{ ft} \] Next, she has 18 pieces that are \( \frac{3}{4} \) ft long. The total length for these pieces is: \[ 18 \times \frac{3}{4} = \frac{54}{4} = \frac{27}{2} \text{ ft} \] Now we need a common denominator to add \( \frac{35}{6} \) and \( \frac{27}{2} \). The common denominator for 6 and 2 is 6. Convert \( \frac{27}{2} \) to sixths: \[ \frac{27}{2} = \frac{27 \times 3}{2 \times 3} = \frac{81}{6} \] Now, add the two lengths of yarn used: \[ \frac{35}{6} + \frac{81}{6} = \frac{116}{6} = \frac{58}{3} \text{ ft} \] The total length of yarn used is \( \frac{58}{3} \) ft. Now, subtract this from the original 30 ft of yarn to find the leftover yarn: Convert 30 ft to sixths: \[ 30 = \frac{30 \times 6}{1 \times 6} = \frac{180}{6} \] Now, subtract the total used from the initial length: \[ \frac{180}{6} - \frac{116}{6} = \frac{64}{6} = \frac{32}{3} \text{ ft} \] So, the leftover piece of yarn is \( \frac{32}{3} \) ft long, which is approximately 10.67 ft.