# Solutions

**Lower Elementary:***Answer:* 27 inches tall*Solution:* The plant grew 2 inches the first week, 5 – 2 = 3 inches the second week, 9 – 5 = 4 inches the third week, 14 – 9 = 5 inches the fourth week, and 20 – 14 = 6 inches the fifth week. The pattern is that it grows 1 more inch each week than it did the last. So, it will grow 7 inches the sixth week, making it 20 + 7 = 27 inches tall.

**Upper Elementary:***Answer:* 11^{1}⁄_{3 }mosquito larvae*Solution:* To find the average, we add up all the numbers of larvae and then divide by the number of samples. Natalie collected a total of 12 + 22 + 0 + 13 + 8 + 13 = 68 mosquito larvae in her samples. So, the average number of mosquito larvae in a sample is 68 ÷ 6 = 11^{1}⁄_{3}.

**Middle School:***Answer:* 23^{5}⁄_{8 }square centimeters*Solution:* Over three days, the human bacteria colony triples in area 3 times. So, the human bacteria colony will be 1 × 3 × 3 × 3 = 27 square centimeters. The dog bacteria does the same thing, but its factor is 1^{1}⁄_{2}, so it will be 1 × 1 ^{1}⁄_{2}× 1^{1}⁄_{2} × 1^{1}⁄_{2} = 1 × ^{3}⁄_{2} × ^{3}⁄_{2} × ^{3}⁄_{2} = ^{27}⁄_{8} = 3 ^{3}⁄_{8}. So, the human bacteria colony is 27 – 3^{3}⁄_{8 }= 23^{5}⁄_{8} square centimeters bigger.

**Algebra and Up:***Answer:* 573^{1}⁄_{3 }π cubic inches*Solution: *Here’s one way to find the difference in volume between the cut-off cone and the volume of the whole cone: The whole cone is ^{1}⁄_{3 }× 144π × 12 = 576π cubic inches before the top is removed. The dimensions of the cut-off cone are 2 ÷ 12 = ^{1}⁄_{6 }those of the whole. Since the base is in two dimensions, its area is ^{1}⁄_{6 }× ^{1}⁄_{6 }× 144π = 4π square inches. The volume of the cut-off cone is ^{1}⁄_{3 }× 4π × 2 = 2^{2}⁄_{3 }π cubic inches, and the difference is 573^{1}⁄_{3 }π cubic inches.