Question

a poll of 160 people determined that 139 had subscriptions to the hardcopy of a newspaper, 36 had hardcopy and online subscriptions, and 19 had neither. how many had only online subscriptions? (type a whole number.)

Explanation:

Step1: Find total with at least one subscription

Total people: 160, neither: 19. So with at least one: $160 - 19 = 141$.

Step2: Find only hardcopy subscriptions

Hardcopy total: 139, both: 36. So only hardcopy: $139 - 36 = 103$.

Step3: Find only online subscriptions

With at least one: 141, only hardcopy: 103, both: 36. So only online: $141 - 103 - 36 = 2$. Wait, no, correct formula: Let \( O \) be only online, \( H \) only hardcopy, \( B \) both. \( O + H + B = 141 \), \( H + B = 139 \), so \( O = 141 - 139 = 2 \)? Wait, no, wait: Wait, total with hardcopy is 139 (including both), total with online is \( O + B \). Wait, maybe better: Let \( x \) be only online. Then total with hardcopy: only hardcopy + both = 139, so only hardcopy = 139 - 36 = 103. Total with at least one: only hardcopy + only online + both = 103 + x + 36 = 139 + x. And total with at least one is 160 - 19 = 141. So 139 + x = 141 → x = 2? Wait, that can't be. Wait, no, maybe I messed up. Wait, total people: 160. Neither: 19. So total with subscription: 160 - 19 = 141. Hardcopy subscriptions: 139 (includes both). Online subscriptions: let \( O \) be only online, \( B = 36 \) both. So hardcopy is only hardcopy (\( H \)) + \( B = 139 \), so \( H = 139 - 36 = 103 \). Then total with subscription: \( H + O + B = 103 + O + 36 = 139 + O = 141 \). So \( O = 141 - 139 = 2 \). Wait, but that seems low. Wait, maybe the problem is: Wait, the poll is 160 people. 139 had hardcopy (including both), 36 had both, 19 had neither. So total with subscription: 160 - 19 = 141. So hardcopy (139) + online (only online + 36) - both (36) = 141 (by inclusion - exclusion). So \( 139 + (O + 36) - 36 = 141 \) → \( 139 + O = 141 \) → \( O = 2 \). Yes, that's correct. So only online is 2. Wait, but let's check again: Only hardcopy: 139 - 36 = 103. Only online: x. Both: 36. Neither: 19. So 103 + x + 36 + 19 = 160 → 103 + x + 55 = 160 → 158 + x = 160 → x = 2. Yes! That's the correct way. So 103 (only hardcopy) + x (only online) + 36 (both) + 19 (neither) = 160. So 103 + 36 + 19 = 158, so x = 160 - 158 = 2. Ah, there we go. So that's correct.

Answer:

2