Independant Contractors and Tax Law

This is the second request for information and we will try to be more concise in our wording. We are a partnership in NYS. Hypothetically we would like to solicit parents who need tutors for their children. We would then find a Tutor and match our client with the tutor. Our client would pay us 20 an hour for our INDEPENDANT CONTRACTOR to come in and tutor. At this point, we would pay our CONTRACTOR $10/hr. We understand that since the tutor is an independant contractor, we don't take taxes out of the $10/hr. When it comes time to file taxes for OUR company, do we pay taxes on the whole $20 dollars or do we deduct the $10 as a business expense and then pay taxes on the remainder? Isn't the IRS getting double tax? They are getting our tax from the $20 and they are collecting tax from the independant contractor. We are not lawyers so we hope that your response will be as simple as possible. Thanks in advance!