On checking with 100 families, it was found that 25 families subscribe to Time, 30 to Newsweek, and 65 to neither magazine. How many subscribe to both?
To find the number of families that subscribe to both Time and Newsweek, we can use the principle of inclusion-exclusion.
Let's denote:
A = Number of families that subscribe to Time.
B = Number of families that subscribe to Newsweek.
N(A) = Number of families that do not subscribe to Time.
N(B) = Number of families that do not subscribe to Newsweek.
N(A∪B) = Number of families that do not subscribe to either Time or Newsweek.
Given:
A = 25
B = 30
N(A∪B) = 65
We can use the formula:
N(A∪B) = N(A) + N(B) - N(A∩B)
Substituting the given values:
65 = N(A) + N(B) - N(A∩B)
Since N(A∩B) is the number of families that subscribe to both magazines, let's solve for it:
65 = 25 + 30 - N(A∩B)
Rearranging the equation:
N(A∩B) = 25 + 30 - 65
N(A∩B) = 55 - 65
N(A∩B) = -10
Since negative values are not meaningful in this context, it is not possible for -10 families to subscribe to both magazines. Therefore, the number of families that subscribe to both Time and Newsweek is 0.
To find out how many families subscribe to both Time and Newsweek magazines, we need to use the principle of inclusion-exclusion.
Given:
- Total number of families = 100
- Number of families subscribing to Time magazine = 25
- Number of families subscribing to Newsweek magazine = 30
- Number of families subscribing to neither magazine = 65
Step 1: Identify the total number of families subscribing to either Time or Newsweek:
- Total number of families subscribing to either Time or Newsweek = Number of families subscribing to Time + Number of families subscribing to Newsweek
- Total number of families subscribing to either Time or Newsweek = 25 + 30 = 55
Step 2: Subtract the number of families subscribing to neither magazine from the total number of families subscribing to either Time or Newsweek:
- Number of families subscribing to both magazines = Total number of families subscribing to either Time or Newsweek - Number of families subscribing to neither magazine
- Number of families subscribing to both magazines = 55 - 65 = -10
This means that there is no overlap between the subscribers of both magazines. Therefore, there are no families that subscribe to both Time and Newsweek magazines.