Question

A recipe for 1 batch of muffins included 2/3 cup of raisins. Ina made 2 1/2 batches of muffins. How many cups of raisins did she use?

Answer 1

1 batch of muffins included 2/3 cups of raisins.  We can use this to set a proportion: 1 : 2/3.  That means for every 1 batch, she uses 2/3 cups of raisins.

Now, Ina made 2 1/2 batches.  To find out how, we need to look at our proportion again: 1 : 2/3.  Whatever operation you do to one side, you must do to the other.  So, to get 2 1/2 batches, we need to multiply the left side of the proportion, which is 1 batch, by 2 1/2 to get 2 1/2 batches.

BUT, we also need to multiply the other side, 2/3 cups, by 2 1/2.  We can start by converting this into an improper fraction, 5/2.  (Because 1=2/2, so 2=4/2, and 4/2+1/2=5/2.)  When multiplying fractions, you multiply the numerators together to get the new numerator, and the denominators to get the new denominator.  2*5=10, and 3*2=6.  Our new fraction is 10/6.  Since these are both divisible by 2, we simplify by dividing each side by 2 to get 5/3.  Finally, we can convert this into a mixed number: 1 2/3.  (Because 1=3/3 and 5/3 - 3/3 = 2/3, and 1 + 2/3 = 1 2/3.)  This is the cups of raisins she used.

Answer: She used 1 2/3 cups of raisins.