Final Answer:
Rachel can make 14/3 or 4 2/3 batches of the recipe with the amount of yogurt she has.
Step-by-step explanation:
To determine how many batches of the recipe Rachel can make with 3 1/2 cups of yogurt, first, we find out how many times the required amount (3/4 of a cup) fits into the total amount of yogurt she has.
3 1/2 cups of yogurt can be expressed as an improper fraction: 3 * 2 + 1 = 7/2 cups of yogurt.
Then, we divide the total amount of yogurt Rachel has by the amount needed for one batch of the recipe:
(7/2 cups) ÷ (3/4 cups/batch) = (7/2) ÷ (3/4)
To divide fractions, we invert and multiply:
(7/2) * (4/3) = 28/6 = 14/3
So, Rachel can make 14/3 batches of the recipe with the yogurt she has. This can also be expressed as a mixed number: 4 2/3 batches. Therefore, she can make approximately 4 2/3 batches of the recipe with the 3 1/2 cups of yogurt she possesses.