Question

scientists have found a way to distinguish chronic - fatigue syndrome (cfs) from post - treatment lyme disease by different proteins in a patients spinal fluid. in a study of 4064 patients, the results to the right were found. (a) how many had the protein for lyme only? (b) how many had the normal protein only? (c) how many had none of the proteins? 1655 had the protein for normal patients, cfs, and post - treatment lyme disease, 1855 had the protein for cfs and lyme, 647 had the protein for cfs only, 2642 had the protein for cfs, 1844 had the normal protein as well as the one for lyme, 2693 had the protein for lyme, 2568 had the normal protein. (a) there were patients who only had the protein for lyme. (type a whole number.)

Explanation:

Step1: Find number of patients with Lyme - CFS overlap

We know that 1855 had the protein for CFS and Lyme, and 647 had the protein for CFS only. Since the number of patients with CFS is the sum of those with CFS only and those with CFS and Lyme, and we know the number of patients with CFS is 2642. But we can also use the Lyme - related data.
The number of patients with Lyme is 2693 and 1855 had the protein for CFS and Lyme.
Let \(x\) be the number of patients with Lyme only.
We know that the number of patients with Lyme is the sum of those with Lyme only and those with CFS and Lyme. So \(x+1855 = 2693\).

Step2: Solve for \(x\) (Lyme only)

Subtract 1855 from both sides of the equation \(x+1855 = 2693\).
\(x=2693 - 1855\)
\(x = 838\)

Answer:

838