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At a beach resort, 90 guests each participate in at least one of three
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11 Aug 2026, 14:27
11 Aug 2026, 14:27
Expert Reply
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Question Stats:
33% (00:10) correct
66% (03:01) wrong based on 3 sessions
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At a beach resort, 90 guests each participate in at least one of three activities: surfing, kayaking, or snorkeling. 55 guests surf, 40 kayak, and 35 snorkel. Exactly 8 guests participate in all three activities, and the number of guests who participate in exactly two activities is three times the number who participate in all three. How many guests participate only in surfing?
(A) 17
(B) 21
(C) 23
(D) 25
(E) 29
(A) 17
(B) 21
(C) 23
(D) 25
(E) 29
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Re: At a beach resort, 90 guests each participate in at least one of three
[#permalink]
04 Sep 2026, 22:27
04 Sep 2026, 22:27
Hi Carcass, can you please provide a solution for this?
Struggling to understand how to segregate the "exactly two activities" among surfing, kayaking, and snorkelling since no info is given.
Struggling to understand how to segregate the "exactly two activities" among surfing, kayaking, and snorkelling since no info is given.
Re: At a beach resort, 90 guests each participate in at least one of three
[#permalink]
05 Sep 2026, 00:29
05 Sep 2026, 00:29
Expert Reply
To find the number of guests who participate only in surfing, we can use the 3-set Venn diagram formulas for overlap.
Step 1: Define the region variables
Let:
S=55 (Total surf)
K=40 (Total kayak)
N=35 (Total snorkel)
Total =90
Let the three overlapping regions for exactly two activities be x, y, z :
x= Surfing & Kayaking only
y= Surfing & Snorkeling only
z= Kayaking & Snorkeling only
Let E_2=x+y+z be the total number of guests participating in exactly two activities. Let E_3=8 be the guests participating in all three activities.
Step 2: Use the total set formula
The formula for the total number of unique elements across three sets is:
\(\text { Total }=\text { Exactly } 1+\text { Exactly } 2+\text { Exactly } 3\)
To find the number of guests who participate only in surfing, we can use the 3-set Venn diagram formulas for overlap.
Step 1: Define the region variables
Let:
S=55 (Total surf)
K=40 (Total kayak)
N=35 (Total snorkel)
Total =90
Let the three overlapping regions for exactly two activities be x, y, z :
x= Surfing & Kayaking only
y= Surfing & Snorkeling only
z= Kayaking & Snorkeling only
Let E_2=x+y+z be the total number of guests participating in exactly two activities. Let E_3=8 be the guests participating in all three activities.
Step 2: Use the total set formula
The formula for the total number of unique elements across three sets is:
\(\text { Total }=\text { Exactly } 1+\text { Exactly } 2+\text { Exactly } 3\)
\(58+2(24)+3(8)=58+48+24=130\)
Now, break down the individual sets:
S= Surf only \(+x+y+8=55 \Longrightarrow\) Surf only +x+y=47
K= Kayak only \(+x+z+8=40 \Longrightarrow\) Kayak only +x+z=32
N= Snorkel only \(+y+z+8=35 \Longrightarrow\) Snorkel only +y+z=27
Summing these three equations:
\(\begin{gathered}
(\text { Exactly } 1)+2(x+y+z)=47+32+27=106 \\
58+2(x+y+z)=106 \Longrightarrow 2(x+y+z)=48 \Longrightarrow x+y+z=24
\end{gathered}\)
Step 4: Isolate x+y to solve for "Surf Only"
Notice that:
\(\text { Surf Only }=55-(x+y+8)\)
We need to determine x+y . Let's express x+y in terms of set totals:
\quad( Kayak only + Snorkel only )= Exactly 1-$ Surf Only =58- Surf Only Using K+N :
\(58+2(24)+3(8)=58+48+24=130\)
Now, break down the individual sets:
S= Surf only \(+x+y+8=55 \Longrightarrow\) Surf only +x+y=47
K= Kayak only \(+x+z+8=40 \Longrightarrow\) Kayak only +x+z=32
N= Snorkel only \(+y+z+8=35 \Longrightarrow\) Snorkel only +y+z=27
Summing these three equations:
Exactly 1)+2(x+y+z)=47+32+27=106
58+2(x+y+z)=106\( \Longrightarrow\) 2(x+y+z)=48 \(\Longrightarrow\) x+y+z=24
Step 4: Isolate $x+y$ to solve for "Surf Only"
Notice that:
\(\text { Surf Only }=55-(x+y+8)
\)
We need to determine x+y . Let's express x+y in terms of set totals:
( Kayak only + Snorkel only )= Exactly 1- Surf Only =58- Surf Only Using $K+N :
\(\text { Surf Only }=23+0=23\)
Conclusion
The number of guests participating only in surfing is 23 .
Correct Answer: (C) 23
Step 1: Define the region variables
Let:
S=55 (Total surf)
K=40 (Total kayak)
N=35 (Total snorkel)
Total =90
Let the three overlapping regions for exactly two activities be x, y, z :
x= Surfing & Kayaking only
y= Surfing & Snorkeling only
z= Kayaking & Snorkeling only
Let E_2=x+y+z be the total number of guests participating in exactly two activities. Let E_3=8 be the guests participating in all three activities.
Step 2: Use the total set formula
The formula for the total number of unique elements across three sets is:
\(\text { Total }=\text { Exactly } 1+\text { Exactly } 2+\text { Exactly } 3\)
To find the number of guests who participate only in surfing, we can use the 3-set Venn diagram formulas for overlap.
Step 1: Define the region variables
Let:
S=55 (Total surf)
K=40 (Total kayak)
N=35 (Total snorkel)
Total =90
Let the three overlapping regions for exactly two activities be x, y, z :
x= Surfing & Kayaking only
y= Surfing & Snorkeling only
z= Kayaking & Snorkeling only
Let E_2=x+y+z be the total number of guests participating in exactly two activities. Let E_3=8 be the guests participating in all three activities.
Step 2: Use the total set formula
The formula for the total number of unique elements across three sets is:
\(\text { Total }=\text { Exactly } 1+\text { Exactly } 2+\text { Exactly } 3\)
\(58+2(24)+3(8)=58+48+24=130\)
Now, break down the individual sets:
S= Surf only \(+x+y+8=55 \Longrightarrow\) Surf only +x+y=47
K= Kayak only \(+x+z+8=40 \Longrightarrow\) Kayak only +x+z=32
N= Snorkel only \(+y+z+8=35 \Longrightarrow\) Snorkel only +y+z=27
Summing these three equations:
\(\begin{gathered}
(\text { Exactly } 1)+2(x+y+z)=47+32+27=106 \\
58+2(x+y+z)=106 \Longrightarrow 2(x+y+z)=48 \Longrightarrow x+y+z=24
\end{gathered}\)
Step 4: Isolate x+y to solve for "Surf Only"
Notice that:
\(\text { Surf Only }=55-(x+y+8)\)
We need to determine x+y . Let's express x+y in terms of set totals:
\quad( Kayak only + Snorkel only )= Exactly 1-$ Surf Only =58- Surf Only Using K+N :
\(58+2(24)+3(8)=58+48+24=130\)
Now, break down the individual sets:
S= Surf only \(+x+y+8=55 \Longrightarrow\) Surf only +x+y=47
K= Kayak only \(+x+z+8=40 \Longrightarrow\) Kayak only +x+z=32
N= Snorkel only \(+y+z+8=35 \Longrightarrow\) Snorkel only +y+z=27
Summing these three equations:
Exactly 1)+2(x+y+z)=47+32+27=106
58+2(x+y+z)=106\( \Longrightarrow\) 2(x+y+z)=48 \(\Longrightarrow\) x+y+z=24
Step 4: Isolate $x+y$ to solve for "Surf Only"
Notice that:
\(\text { Surf Only }=55-(x+y+8)
\)
We need to determine x+y . Let's express x+y in terms of set totals:
( Kayak only + Snorkel only )= Exactly 1- Surf Only =58- Surf Only Using $K+N :
\(\text { Surf Only }=23+0=23\)
Conclusion
The number of guests participating only in surfing is 23 .
Correct Answer: (C) 23
