At a berry farm, Clayton picks $1 \frac{1}{4}$ cups of blackberries and twice as many raspberries. He takes the berries home and uses them to make mixed-berry muffins. If Clayton needs $\frac{3}{4}$ of a cup of mixed berries for each batch of muffins, how many batches can he make? Write your answer as a whole number, fraction, or mixed number. Simplify any fractions.

Step 1 ：First, we need to calculate the total amount of berries Clayton picked. He picked \(1 \frac{1}{4}\) cups of blackberries and twice as many raspberries, which means he picked \(2 \times 1 \frac{1}{4}\) cups of raspberries.

Step 2 ：Adding these two amounts together will give us the total amount of berries Clayton picked, which is \(1.25 + 2.5 = 3.75\) cups.

Step 3 ：Then, we need to divide this total amount by the amount of berries needed for each batch of muffins, which is \(\frac{3}{4}\) of a cup. So, \(\frac{3.75}{0.75} = 5.0\).

Step 4 ：Final Answer: Clayton can make \(\boxed{5}\) batches of muffins.