1/6 X 60 can be done several ways. Which way makes the most sense to you? I think of it as "one-sixth OF sixty," which is 60 divided into 6 parts. 60/6 = 10.
Alternatively, 1/6 X 60 is 1/6 X 60/1. To multiply 2 fractions, take the top numbers times each other (1 X 60) to get the top number of your answer. Take the bottom numbers times each other (6 X 1) to get the bottom number of your answer. 60/6 is your answer. Reduce it to 10.
