Question

Scientists have found a way to distinguish​ chronic-fatigue syndrome​ (CFS) from​ post-treatment Lyme disease by different proteins in a​ patient's spinal fluid. In a study of 3991 ​patients, the results to the right were found.

1538 had the protein for normal​ patients, CFS, and​ post-treatment Lyme​ disease;
1884 had the protein for CFS and​ Lyme;
408 had the protein for CFS​ only;
2602 had the protein for​ CFS;
1731 had the normal protein as well as the one for​ Lyme;
2798 had the protein for​ Lyme;
2516 had the normal protein.

Part 1
​(a) There were ---- patients who only had the protein for Lyme.
​(Type a whole​ number.)
Part 2
​(b) There were ---- patients who only had the normal protein.
​(Type a whole​ number.)
Part 3
​(c) There were ---- patients who had none of the proteins.

Answer 1

Answer: There are no patients who had none of the proteins.

Step-by-step explanation: Part 1

(a) There were 408 patients who only had the protein for Lyme.

This information can be inferred from the data given: 1884 had the protein for CFS and Lyme, and 2798 had the protein for Lyme. If we subtract 1884 from 2798, we get 914 which is the number of patients who only had the protein for Lyme.

Part 2

(b) There were 2516 patients who only had the normal protein.

This information can be inferred from the data given: 1538 had the protein for normal patients, CFS, and post-treatment Lyme disease, 1884 had the protein for CFS and Lyme, and 2602 had the protein for CFS. If we add 1538, 1884 and 2602, we get 6124, then if we subtract 6124 from the total number of patients (3991) we get 2516 patients who only had the normal protein.

Part 3

(c) There were 0 patients who had none of the proteins.

This information can be inferred from the data given: 1538 had the protein for normal patients, CFS, and post-treatment Lyme disease, 1884 had the protein for CFS and Lyme, 408 had the protein for CFS only, 2602 had the protein for CFS, 1731 had the normal protein as well as the one for Lyme, 2798 had the protein for Lyme, and 2516 had the normal protein. All patients have at least one protein, therefore there are no patients who had none of the proteins.

Answer 2

Part 1

(a) There were 2798 patients who only had the protein for Lyme.

Part 2

(b) There were 2516 patients who only had the normal protein.

Part 3

(c) It is not specified in the given information how many patients had none of the proteins. It is only specified that 1538 had the protein for normal patients, CFS, and post-treatment Lyme disease; 1884 had the protein for CFS and Lyme; 408 had the protein for CFS only; 2602 had the protein for CFS; 1731 had the normal protein as well as the one for Lyme; 2798 had the protein for Lyme; 2516 had the normal protein. Based on this information, it is not possible to know how many patients had none of the proteins.