Marvin is making lemon bread and pumpkin bread for a school bake sale. He needs \( 4 \frac{1}{4} \) cups of flour for the lemon bread and \( 1 \frac{1}{2} \) times as much flour for the pumpkin bread. He only has \( 2 \frac{1}{8} \) cups of flour left in his pantry, so he buys a new bag from the grocery store. How much flour will Marvin use from the new bag? Write your answer as a whole number, fraction, or mixed number. Simplify any fractions. cups

First, let's find out how much flour Marvin needs for the pumpkin bread. He needs \( 4 \frac{1}{4} \) cups for the lemon bread. To convert this mixed number to an improper fraction: \[ 4 \frac{1}{4} = \frac{17}{4} \] Now, since he needs \( 1 \frac{1}{2} \) times as much for the pumpkin bread, we can convert \( 1 \frac{1}{2} \) to an improper fraction as well: \[ 1 \frac{1}{2} = \frac{3}{2} \] Next, we multiply the flour needed for the lemon bread by the factor for the pumpkin bread: \[ \frac{17}{4} \times \frac{3}{2} = \frac{17 \times 3}{4 \times 2} = \frac{51}{8} \] Now, let's convert \( \frac{51}{8} \) into a mixed number: \[ \frac{51}{8} = 6 \frac{3}{8} \] So Marvin needs \( 4 \frac{1}{4} \) cups for lemon bread and \( 6 \frac{3}{8} \) cups for pumpkin bread. To find out the total flour he needs, we add these two amounts: Convert \( 4 \frac{1}{4} \) to an improper fraction again: \[ 4 \frac{1}{4} = \frac{17}{4} \] Next, we convert \( 6 \frac{3}{8} \): \[ 6 \frac{3}{8} = \frac{51}{8} \] Now, we need a common denominator to add these fractions. The least common multiple of 4 and 8 is 8. Thus: \[ \frac{17}{4} = \frac{34}{8} \] Now we can add: \[ \frac{34}{8} + \frac{51}{8} = \frac{85}{8} \] Convert \( \frac{85}{8} \) to a mixed number: \[ \frac{85}{8} = 10 \frac{5}{8} \] Now, let's check how much flour Marvin has left. He has \( 2 \frac{1}{8} \) cups or: \[ 2 \frac{1}{8} = \frac{17}{8} \] Now, we subtract the amount Marvin already has from the total amount he needs: \[ \frac{85}{8} - \frac{17}{8} = \frac{68}{8} = 8 \frac{4}{8} = 8 \frac{1}{2} \] Thus, the amount of flour Marvin will need from the new bag is \( 8 \frac{1}{2} \) cups.