Nineteen different mouthwash products make the following claims: 12 claim to freshen breath, 10 claim to prevent gingivitis, 11 claim to reduce plaque, 6 claim to both freshen breath and reduce plaque, 5 claim to both prevent gingivitis and freshen breath, and 5 claim to both prevent gingivitis and reduce plaque. a. how many products make all three claims? b. how many products claim to freshen breath but do not claim to reduce plaque?

a.) Number of products that makes all three claims x = 2

b.) Number of products that claims to freshen breath but not reduce plaque = 6

Explanation:

a.)

Let total number of products be S  = 19

let FB be for fresh breath = 12

let Gin be for prevention of gingivitis = 10

let Pl be for reduced plaque = 11

let FBnGIN be for freshen breath and prevention of gingivitis = 5

let FBnPL be for freshen breath and reduced of plaque = 6

let GINnPL be for fprevention of gingivitis and reduced of plaque = 5

let x be for All three claims

for Products that make all three claims

S = FB + GIN + PL + X - (FBnGIN + FBnPL + GINnPL)

19 = 12 + 10 + 11 + X - (5 + 6 + 5)

19 = 17 + X

X = 2

b.) For number of products that claim to freshen breath but do not claim to reduce plaque = FB - FBnPL = 12 - 6 = 6