  # College Algebra - Union of Sets

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

### Comments for College Algebra - Union of Sets

 May 25, 2011 Word Problem - Union & Intersection of Sets by: Staff ----------------------------------------------------- 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 -------------------------------------------------------------- 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 IIStep 4: Write equations for: A, B, C, DWrite equations based on the information on the Venn Diagram185 offered camping:15 + B + C + D = 185140 offered camping and hiking trails:C + B = 140125 offered camping and picnicking:B + D = 125210 offered hiking trails20 + A + B + C = 210All the equations:15 + B + C + D = 185C + B = 140B + D = 12520 + A + B + C = 210Step 5: Solve the equations for all 4 unknown values: A, B, C, DSolve for C in terms of BC + B = 140C + B - B = 140 - BC + 0 = 140 - BC = 140 - BSolve for D in terms of BB + D = 125B - B + D = 125 - B0 + D = 125 - BD = 125 - BSimplify the following equation15 + B + C + D = 18515 - 15 + B + C + D = 185 - 150 + B + C + D = 185 ? 15B + C + D = 185 ? 15B + C + D = 170Simplify the following equation20 + A + B + C = 21020 - 20 + A + B + C = 210 - 200 + A + B + C = 210 - 20A + B + C = 210 - 20A + B + C = 190The final equations we will use to solve for A, B, C, and D:C = 140 - BD = 125 - BB + C + D = 170A + B + C = 190-------------------------------------------C = 140 - BD = 125 - BB + C + D = 170B + (140 - B) + (125 - B) = 170B + 140 - B + 125 - B = 170B - B - B + 140 + 125 = 1700 - B + 140 + 125 = 170- B + 140 + 125 = 170- B + 265 = 170- B + B + 265 = 170 + B0 + 265 = 170 + B265 = 170 + B265 - 170 = 170 - 170 + B95 = 170 - 170 + B95 = 0 + B95 = BB = 95C = 140 - BC = 140 - 95C = 45D = 125 - BD = 125 ? 95D = 30A + B + C = 190A + 95 + 45 = 190A + 95 - 95 + 45 - 45 = 190 - 95 ? 45A + 0 + 0 = 190 - 95 ? 45A = 190 - 95 ? 45A = 50The final values of A, B, C, and D:A = 50B = 95C = 45D = 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-------------------------------------------------------------------------------------------------------------------------