Question

A survey of 1200 people in a certain city indicates that 830 own microwave ovens, 740 own DVD players, and 570 own microwave ovens and DVD players. (A) How many people in the survey own either a microwave oven or a DVD player? (B) How many own neither a microwave oven nor a DVD player? (C) How many own a microwave oven and do not own a DVD player? (A) 0 people in the survey own either a microwave oven or a DVD player. (B) people in the survey own neither a microwave oven nor a DVD player. (C) O people in the survey own a microwave oven and do not own a DVD player.​

Answer 1

Answer: Let's use the principle of inclusion-exclusion to find the answers to the questions.

Given:

Total number of people surveyed (N) = 1200,

Number of people owning a microwave oven (M) = 830,

Number of people owning a DVD player (D) = 740,

Number of people owning both a microwave oven and a DVD player (M ∩ D) = 570.

(A) How many people in the survey own either a microwave oven or a DVD player?

To find the number of people who own either a microwave oven or a DVD player, we need to add the number of people who own a microwave oven (M) and the number of people who own a DVD player (D), and then subtract the number of people who own both (M ∩ D):

Number of people owning either a microwave oven or a DVD player = M + D - M ∩ D

Number of people owning either a microwave oven or a DVD player = 830 + 740 - 570

Number of people owning either a microwave oven or a DVD player = 970.

(B) How many own neither a microwave oven nor a DVD player?

To find the number of people who own neither a microwave oven nor a DVD player, we need to subtract the number of people owning either a microwave oven or a DVD player (found in part A) from the total number of people surveyed (N):

Number of people owning neither a microwave oven nor a DVD player = N - (M + D - M ∩ D)

Number of people owning neither a microwave oven nor a DVD player = 1200 - 970

Number of people owning neither a microwave oven nor a DVD player = 230.

(C) How many own a microwave oven and do not own a DVD player?

To find the number of people who own a microwave oven and do not own a DVD player, we need to subtract the number of people owning both a microwave oven and a DVD player (M ∩ D) from the number of people owning a microwave oven (M):

Number of people owning a microwave oven and not owning a DVD player = M - M ∩ D

Number of people owning a microwave oven and not owning a DVD player = 830 - 570

Number of people owning a microwave oven and not owning a DVD player = 260.

So, the correct answers are:

(A) The number of people owning either a microwave oven or a DVD player is 970.

(B) The number of people owning neither a microwave oven nor a DVD player is 230.

(C) The number of people owning a microwave oven and not owning a DVD player is 260.