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College Algebra - Union of Sets











































A survey of 300 parks showed the following.
• 15 had only camping.
• 20 had only hiking trails.
• 35 had only picnicking.
• 185 had camping.
• 140 had camping and hiking trails.
• 125 had camping and picnicking.
• 210 had hiking trails.
Find the number of parks that
• a)had at least one of these features.
• b)had all three features.
• c)did not have any of these features.
• d)had exactly two of these features.

Comments for College Algebra - Union of Sets

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May 25, 2011
Word Problem - Union & Intersection of Sets
by: Staff

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Part III


The answers to your original questions:

Find the number of parks that

• a) had at least one of these features = 290 parks.

Total of parks with at least 1 feature = sum of all portions of the Venn Diagram

= 15 + 20 + 35 + A + B + C + D

= 15 + 20 + 35 + 50 + 95 + 45 + 30

= 290

• b) had all three features = 95 parks.

Total of parks with all 3 features = B

= 95

• c) did not have any of these features = 10 parks.

300 parks took part in the survey

300 - (15 + 20 + 35 + A + B + C + D) = parks without any of the three features

300 - (15 + 20 + 35 + 50 + 95 + 45 + 30) = 10

There are 10 other parks in the survey which do not offer any of the three types of recreation: no camping, no hiking, and no picnicking.


• d) had exactly two of these features = 125 parks.

Total of parks with 2 features = A + C + D

= A + C + D

= 50 + 45 + 30

= 125



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Check the values on the Venn Diagram by substituting the values of A, B, C, and D in the original equations:

185 offered camping:

15 + B + C + D = 185

15 + 95 + 45 + 30 = 185

185 = 185, OK


140 offered camping and hiking trails:

C + B = 140

45 + 95 = 140

140 = 140, OK


125 offered camping and picnicking:

B + D = 125

95 + 30 = 125

125 = 125, OK




210 offered hiking trails

20 + A + B + C = 210

20 + 50 + 95 + 45 = 210

220 = 210, OK



Thanks for writing.

Staff
www.solving-math-problems.com


May 25, 2011
Word Problem - Union & Intersection of Sets
by: Staff

-----------------------------------------------------

Part II



Step 4: Write equations for: A, B, C, D


Write equations based on the information on the Venn Diagram


185 offered camping:

15 + B + C + D = 185


140 offered camping and hiking trails:

C + B = 140


125 offered camping and picnicking:

B + D = 125


210 offered hiking trails

20 + A + B + C = 210



All the equations:

15 + B + C + D = 185

C + B = 140

B + D = 125

20 + A + B + C = 210


Step 5: Solve the equations for all 4 unknown values: A, B, C, D


Solve for C in terms of B

C + B = 140

C + B - B = 140 - B

C + 0 = 140 - B

C = 140 - B


Solve for D in terms of B

B + D = 125

B - B + D = 125 - B

0 + D = 125 - B

D = 125 - B

Simplify the following equation

15 + B + C + D = 185

15 - 15 + B + C + D = 185 - 15

0 + B + C + D = 185 ? 15

B + C + D = 185 ? 15

B + C + D = 170


Simplify the following equation

20 + A + B + C = 210

20 - 20 + A + B + C = 210 - 20

0 + A + B + C = 210 - 20

A + B + C = 210 - 20

A + B + C = 190



The final equations we will use to solve for A, B, C, and D:



C = 140 - B


D = 125 - B


B + C + D = 170


A + B + C = 190


-------------------------------------------

C = 140 - B

D = 125 - B

B + C + D = 170

B + (140 - B) + (125 - B) = 170

B + 140 - B + 125 - B = 170

B - B - B + 140 + 125 = 170

0 - B + 140 + 125 = 170

- B + 140 + 125 = 170

- B + 265 = 170

- B + B + 265 = 170 + B

0 + 265 = 170 + B

265 = 170 + B

265 - 170 = 170 - 170 + B

95 = 170 - 170 + B

95 = 0 + B

95 = B

B = 95

C = 140 - B

C = 140 - 95

C = 45


D = 125 - B

D = 125 ? 95

D = 30


A + B + C = 190

A + 95 + 45 = 190

A + 95 - 95 + 45 - 45 = 190 - 95 ? 45

A + 0 + 0 = 190 - 95 ? 45

A = 190 - 95 ? 45


A = 50



The final values of A, B, C, and D:

A = 50

B = 95

C = 45

D = 30



(1) Click the following link to VIEW the VENN DIAGRAM with the FINAL VALUES shown; or (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page:

http://www.solving-math-problems.com/images/venn-diagram-3cir-final-values.png

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May 25, 2011
Word Problem - Union & Intersection of Sets
by: Staff


Part I

The question:

A survey of 300 parks showed the following.
• 15 had only camping.
• 20 had only hiking trails.
• 35 had only picnicking.
• 185 had camping.
• 140 had camping and hiking trails.
• 125 had camping and picnicking.
• 210 had hiking trails.

Find the number of parks that
• a)had at least one of these features.
• b)had all three features.
• c)did not have any of these features.
• d)had exactly two of these features.


The answer:


A Venn Diagram is very helpful in visualizing the solution to this problem.


Step 1: Draw a 3-circle Venn Diagram with all three circles overlapping.

Label one circle “camping”, one circle “hiking”, and one circle “picnicking”


(1) Click the following link to VIEW the VENN DIAGRAM; or (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page:

http://www.solving-math-problems.com/images/venn-diagram-3cir-camping-hiking-picnicking.png



Step 2: Label the overlapping portions of the three circles separately.

I used the following labels:

A = parks which offer both hiking and picnicking, but do not offer camping

B = parks which offer all three activities: camping, hiking and picnicking

C = parks which offer both camping and hiking, but do not offer picnicking

D = parks which offer both camping and picnicking, but do not offer hiking


(1) Click the following link to VIEW the OVERLAP LABELS on the VENN DIAGRAM; or (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page:

http://www.solving-math-problems.com/images/venn-diagram-3cir-overlap-label.png




Step 3: Add the known values to the Venn Diagram.

I used the following labels:

15 parks offer only camping

20 parks offer only hiking

35 parks offer only picnicking


(1) Click the following link to VIEW the OVERLAP LABELS and the KNOWN VALUES on the VENN DIAGRAM; or (2A) highlight and copy the link, then (2B) paste the link into your browser Address bar & press enter:

Use the Backspace key to return to this page:

http://www.solving-math-problems.com/images/venn-diagram-3cir-overlap-label-known-values.png

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