Math Problem Statement

A survey of 255 adults found that during the last year, 60 traveled by plane but not by train, 70 traveled by train but not by plane, 20 traveled by bus but not by plane or by train, 45 traveled by bus and plane, 25 traveled by all three, and 200 traveled by plane or train. How many did not travel by any of these modes of transportation?

Solution

We can solve this problem using the principle of inclusion and exclusion.

Let:

  • PP represent the set of adults who traveled by plane,
  • TT represent the set of adults who traveled by train,
  • BB represent the set of adults who traveled by bus.

We are given the following information:

  • 60 adults traveled by plane but not by train: PT=60|P \setminus T| = 60,
  • 70 adults traveled by train but not by plane: TP=70|T \setminus P| = 70,
  • 20 adults traveled by bus but not by plane or train: B(PT)=20|B \setminus (P \cup T)| = 20,
  • 45 adults traveled by both plane and bus: PB=45|P \cap B| = 45,
  • 25 adults traveled by all three modes (plane, train, bus): PTB=25|P \cap T \cap B| = 25,
  • 200 adults traveled by plane or train: PT=200|P \cup T| = 200.

Step 1: Determine the number of people who traveled by plane and train (but not necessarily by bus)

From the inclusion-exclusion formula for the union of two sets, we know:

PT=P+TPT|P \cup T| = |P| + |T| - |P \cap T|

From the problem, PT=200|P \cup T| = 200, so:

200=P+TPT200 = |P| + |T| - |P \cap T|

We don't yet know P|P|, T|T|, or PT|P \cap T|, but we can break these down further based on the provided information.

Step 2: Breakdown of the sets

  • PT=60|P \setminus T| = 60 (people who traveled by plane but not train),
  • TP=70|T \setminus P| = 70 (people who traveled by train but not plane),
  • PTB|P \cap T \setminus B| (people who traveled by both plane and train but not bus),
  • PB=45|P \cap B| = 45 (people who traveled by both plane and bus),
  • PTB=25|P \cap T \cap B| = 25 (people who traveled by all three).

Now we can calculate the remaining quantities.

Total number of people who traveled by plane:

P=PT+PTB+PB+PTB|P| = |P \setminus T| + |P \cap T \setminus B| + |P \cap B| + |P \cap T \cap B| Substitute the known values: P=60+(PTB)+45+25|P| = 60 + (|P \cap T \setminus B|) + 45 + 25 Since the people who traveled by both plane and train but not bus are part of the 200 total for plane or train, we can solve for PTB|P \cap T \setminus B|.

Total for Train:

Similar equations apply to the train set.

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Math Problem Analysis

Mathematical Concepts

Set Theory
Inclusion-Exclusion Principle
Counting

Formulas

Inclusion-Exclusion Formula: |A ∪ B| = |A| + |B| - |A ∩ B|
Extended Inclusion-Exclusion Formula for Three Sets: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |A ∩ C| - |B ∩ C| + |A ∩ B ∩ C|

Theorems

Principle of Inclusion-Exclusion

Suitable Grade Level

Grades 9-12