An aviation safety committee audited 240 international flights for operational delays caus… — GRE General Test

An aviation safety committee audited $240$ international flights for operational delays caused by Weather ( $W$ ), Mechanical issues ( $M$ ), and Air Traffic Control ( $T$ ). The audit revealed the following data:

- $112$ flights experienced Weather delays.
- $98$ flights experienced Mechanical delays.
- $104$ flights experienced Air Traffic Control delays.
- $28$ flights experienced both Weather and Mechanical delays.
- $34$ flights experienced both Mechanical and Air Traffic Control delays.
- $42$ flights experienced both Weather and Air Traffic Control delays.
- The number of flights that experienced none of these three types of delays was equal to twice the number of flights that experienced all three types of delays.

How many flights experienced exactly one of these three types of delays?

Answer: 136 flights

Answer

The number of flights that experienced exactly one of the three types of delays is 136.

The total number of audited flights is partitioned into those experiencing at least one delay and those experiencing none. By applying the Inclusion-Exclusion formula

|W \cup M \cup T| = 112 + 98 + 104 - (28 + 34 + 42) + x = 210 + x

, and setting the total audited flights to

240 = (210 + x) + 2x

, we solve for the triple intersection

x = 10

. Isolating the exclusive single-delay regions yields

52

flights for Weather only,

46

for Mechanical only, and

38

for ATC only, giving a total of

136

flights.

Step-by-Step Solution

Define the unknown intersection and external set quantities in terms of a single variable.

Let

x = |W \cap M \cap T|

represent the number of flights experiencing all three types of delays. Consequently, the number of flights experiencing none of the delays is

2x

Relating the unknown triple intersection to the 'neither' category allows all regions to be expressed in terms of

x

Apply the Principle of Inclusion-Exclusion for three overlapping sets to find the total number of flights with at least one delay.

|W \cup M \cup T| = |W| + |M| + |T| - (|W \cap M| + |M \cap T| + |W \cap T|) + |W \cap M \cap T| = 112 + 98 + 104 - (28 + 34 + 42) + x = 210 + x

Subtracting pairwise intersections corrects for double-counting, and adding back the triple intersection corrects for over-subtraction.

Formulate and solve the linear equation for the total sample population of 240 flights.

\text{Total Flights} = |W \cup M \cup T| + |\text{Neither}| \implies 240 = (210 + x) + 2x \implies 240 = 210 + 3x \implies 3x = 30 \implies x = 10

The sum of flights in the union of the sets and flights outside all three sets must equal the total number of flights audited.

Determine the number of flights in the two-set intersection regions excluding the triple intersection.

Weather and Mechanical only:

28 - 10 = 18

; Mechanical and ATC only:

34 - 10 = 24

; Weather and ATC only:

42 - 10 = 32

Each pairwise intersection given in the stem includes the triple intersection of 10 flights.

Compute the number of flights belonging exclusively to single categories and sum them.

Weather only:

112 - (18 + 32 + 10) = 52

; Mechanical only:

98 - (18 + 24 + 10) = 46

; ATC only:

104 - (32 + 24 + 10) = 38

. Sum of single categories:

52 + 46 + 38 = 136

Subtracting all overlapping components from each set total isolates the exclusive single-category elements.

Key Concept

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Partitioning

Estimated Time:2m 30s

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