1. **State the problem:** We are given data about ownership of laptops, cell phones, and tablets from a survey. We need to find: - The percent of people who owned a cell phone. - The percent of people who owned only a cell phone. 2. **Identify given data:** - Number with laptops ($L$): 904 - Number with cell phones ($C$): 1261 - Number with tablets ($T$): 1053 - Number with all three ($L \cap C \cap T$): 186 - Number with none: 264 - Number with cell phones and tablets ($C \cap T$): 407 - Number with cell phones only (no laptop or tablet): 592 - Number with laptop and tablet but no cell phones ($L \cap T \setminus C$): 165 3. **Calculate total number surveyed:** Total surveyed = Number with none + Number with at least one device We find the total with at least one device by summing all disjoint groups: - Only cell phones: 592 - Laptop and tablet but no cell phones: 165 - All three: 186 - Cell phones and tablets (including those with laptops): 407 (includes the 186 with all three) We need to find the total number with at least one device. Since 264 had none, total surveyed = $N$. We use the formula for total surveyed: $$N = \text{none} + \text{only laptops} + \text{only cell phones} + \text{only tablets} + \text{laptop and cell phone only} + \text{cell phone and tablet only} + \text{laptop and tablet only} + \text{all three}$$ We know some values, but not all. Let's find missing values step by step. 4. **Find number with only laptops ($L$ only):** We know total laptops = 904, which includes: - Only laptops - Laptop and cell phone only - Laptop and tablet only (given 165) - All three (186) We don't have laptop and cell phone only directly, so let's find it. 5. **Find number with cell phones and tablets only:** Given $C \cap T = 407$ includes those with all three (186), so cell phones and tablets only is: $$407 - 186 = 221$$ 6. **Find number with cell phones and laptops only:** Total cell phones = 1261 includes: - Only cell phones (592) - Cell phones and tablets only (221) - Cell phones and laptops only (unknown) - All three (186) Calculate cell phones and laptops only: $$1261 = 592 + 221 + x + 186$$ $$x = 1261 - 592 - 221 - 186 = 262$$ 7. **Find number with only laptops:** Total laptops = 904 includes: - Only laptops (unknown) - Laptop and cell phone only (262) - Laptop and tablet only (165) - All three (186) Calculate only laptops: $$904 = y + 262 + 165 + 186$$ $$y = 904 - 262 - 165 - 186 = 291$$ 8. **Find number with only tablets:** Total tablets = 1053 includes: - Only tablets (unknown) - Laptop and tablet only (165) - Cell phones and tablets only (221) - All three (186) Calculate only tablets: $$1053 = z + 165 + 221 + 186$$ $$z = 1053 - 165 - 221 - 186 = 481$$ 9. **Calculate total surveyed:** Sum all disjoint groups plus none: $$N = 264 + 291 + 592 + 481 + 262 + 221 + 165 + 186 = 2462$$ 10. **Calculate percent who owned a cell phone:** Number owning cell phone = total surveyed - number with no cell phone Or sum all groups with cell phones: - Only cell phones: 592 - Cell phones and tablets only: 221 - Cell phones and laptops only: 262 - All three: 186 Sum: $$592 + 221 + 262 + 186 = 1261$$ Percent: $$\frac{1261}{2462} \times 100 = 51.22\%$$ 11. **Calculate percent who owned only a cell phone:** $$\frac{592}{2462} \times 100 = 24.05\%$$ **Final answers:** - Percent owning a cell phone: **51.22 percent** - Percent owning only a cell phone: **24.05 percent**