1. **State the problem:** We surveyed 750 students about tattoos and body piercings. Given: - Total students: $750$ - Students with tattoos: $375$ - Students with body piercings: $187$ - Students with both tattoos and piercings: $102$ We want to find the number of students in each of the four Venn diagram regions: - Only tattoos - Only piercings - Both tattoos and piercings - Neither tattoos nor piercings 2. **Formula and rules:** - The number of students with only tattoos is the total with tattoos minus those with both: $$\text{Only tattoos} = 375 - 102$$ - The number of students with only piercings is the total with piercings minus those with both: $$\text{Only piercings} = 187 - 102$$ - The number of students with both is given as $102$. - The number of students with neither is the total minus those with tattoos or piercings: $$\text{Neither} = 750 - (\text{Only tattoos} + \text{Only piercings} + \text{Both})$$ 3. **Calculate each region:** - Only tattoos: $$375 - 102 = 273$$ - Only piercings: $$187 - 102 = 85$$ - Both tattoos and piercings: $$102$$ - Neither tattoos nor piercings: $$750 - (273 + 85 + 102) = 750 - 460 = 290$$ 4. **Summary:** - Only tattoos: $273$ - Only piercings: $85$ - Both: $102$ - Neither: $290$ These numbers fill the four regions of the Venn diagram accurately.