Question

1. -/1 points of 200 clock radios with digital tuners and/or cd players sold recently in a department store, 140 had digital tuners and 100 had cd players. how many radios had both digital tuners and cd players?

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1. -/4 points

in a survey of 350 members of a local sports club, 175 members indicated that they plan to attend the next summer olympic games, 105 indicated that they plan to attend the next winter olympic games, and 70 indicated that they plan to attend both games. how many members of the club plan to attend
(a) at least one of the two games?
members
(b) exactly one of the games?
members
(c) the summer olympic games only?
members
(d) none of the games?
members

Explanation:

Step1: Define set formula for Q9

Use principle of inclusion-exclusion: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$
Rearrange to find $n(A \cap B)$: $n(A \cap B) = n(A) + n(B) - n(A \cup B)$
Given $n(A \cup B)=200$, $n(A)=140$, $n(B)=160$

Step2: Calculate Q9 intersection

$n(A \cap B) = 140 + 160 - 200$
$n(A \cap B) = 100$

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Step3: Calculate Q10(a) at least one game

Use inclusion-exclusion: $n(S \cup W) = n(S) + n(W) - n(S \cap W)$
Given $n(S)=175$, $n(W)=105$, $n(S \cap W)=70$
$n(S \cup W) = 175 + 105 - 70$
$n(S \cup W) = 210$

Step4: Calculate Q10(b) exactly one game

Subtract 2×both from total attendees: $n(\text{exactly one}) = n(S \cup W) - 2n(S \cap W)$
$n(\text{exactly one}) = 210 - 2\times70$
$n(\text{exactly one}) = 70$

Step5: Calculate Q10(c) Summer only

Subtract both from Summer attendees: $n(S \text{ only}) = n(S) - n(S \cap W)$
$n(S \text{ only}) = 175 - 70$
$n(S \text{ only}) = 105$

Step6: Calculate Q10(d) none of the games

Subtract total attendees from total members: $n(\text{none}) = 350 - n(S \cup W)$
$n(\text{none}) = 350 - 210$
$n(\text{none}) = 140$

Answer:

Question 9: 100 radios
Question 10:
(a) 210 members
(b) 70 members
(c) 105 members
(d) 140 members