Question

Parks 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.

Answer 1

Draw Venn diagram with large outer circle (for park, A) enclosing 3 interlocking circles for hiking (H), camping (C) and picnicking (P). There are eight zones (Z1 to Z8) created by the circles and overlaps:

1. Only camping (15)
2. Only hiking (20)
3. Only picnicking (35)
4. Overlap C and H less Z7 (140)
5. Overlap C and P less Z7 (125)
6. Overlap H and P less Z7
7. Overlap C, H and P
8. A outside all 3 circles.

Write the numbers in their relevant zones.

H=Z2+Z4+Z6+Z7=210; Z1+...+Z8=300; C=Z1+Z4+Z5+Z7=185.

We can start plugging in actual numbers: 

C=Z1+Z4+Z5+Z7 and Z4+Z7=140, so 185=C=15+140+125-Z7=280-Z7, so Z7=280-185=95.

Z5+Z7=125 and P=Z3+Z5+Z7+Z6=35+125+Z6=160+Z6; H=210=Z2+Z4+Z7+Z6, so Z6=210-20-140 and Z6=50 and therefore P=210. From Z7=95, Z5=125-95=30; Z4=140-95=45.

300=15+20+35+140+125+50+Z7+Z8=385-Z7+Z8 so Z8=Z7-85=95-85=10.

Since Z8=10 (no features), 300-10=290 parks have at least one feature.

Z7 is the zone where all three features overlap, so 95 parks have all three features.