Actually there are a lot of factors involved in such a type of conversion. First of all you need to see what system of conversion you are using. Whether you want to convert a US standard cup size into a US standard teaspoon size or you are using a metric or UK system of conversion. Based on this you are going to do your conversions as the results for each one of them are different.
Assuming that you are using US system of conversion:
1 cup equals 47.99 teaspoons
1/6 cup will equal 7.99 teaspoons.