In Virginia, 120 lame dogs were tested for arthritis and Lyme disease. The results showed that 42 of the dogs had Lyme disease, and 96 had arthritis. How many dogs had both Lyme disease and arthritis? Explain.

The number of dogs that had both Lyme disease and arthritis is 18

Step-by-step explanation:

Let x represent dogs that have both Lyme disease and arthritis

Dog with only Lyme disease = 42 - x

Dog with only arthritis = 96 - x

The sum of the 3 category should give 120

x + (42 - x) + (96 - x) = 120

Collect like terms and solve

x - x - x + 42 + 96 = 120

-x + 138 = 120

138 - 120 = x

x = 18

Dog with only Lyme disease = 42 - x = 42 - 18 = 24

Dog with only arthritis = 96 - x = 96 - 18 = 78

Therefore, the number of dogs with both Lyme disease and anthritis is 18.