Therefore,
.
To most people, it would not be immediately evident that the desired number:
, so
Therefore the answer is
.
.
, so we can multiply 6's until we get
16. Solution 1: Algebra
We multiply the first ratio by 8 on both sides, and the second ratio by 5 to get the same number for 8th graders, in order that we can put the two ratios together:
Therefore, the ratio of 8th graders to 7th graders to 6th graders is terms, the smallest number of students participating in the project is
. Since the ratio is in lowest
.
Solution 2: Fakesolving
The number of 8th graders has to be a multiple of 8 and 5, so assume it is 40 (the smallest possibility). Then there are
6th graders and
7th graders. The numbers of students is
17. Solution 1
The mean of these numbers is
, so the answer is
. Therefore the numbers are
Solution 2
Let the
number be . Then our desired number is
.
Our integers are , so we have that
.
Solution 3
Let the first term be . Our integers are
. We have,
18. Solution 1
There are
cubes on the base of the box. Then, for each of the 4 layers above the bottom (as
since each cube is 1 foot by 1 foot by 1 foot and the box is 5 feet tall, there are 4 feet left), there are
cubes. Hence, the answer is
.
Solution 2
We can just calculate the volume of the prism that was cut out of the original height will be feet. So the volume of the interior box is The volume of the original box is the fort is
.
.
box. Each
interior side of the fort will be feet shorter than each side of the outside. Since the floor is foot, the
. Therefore, the number of blocks contained in
19. If Hannah did better than Cassie, there would be no way she could know for sure that she didn't get
the lowest score in the class. Therefore, Hannah did worse than Cassie. Similarly, if Hannah did worse than Bridget, there is no way Bridget could have known that she didn't get the highest in the class. Therefore, Hannah did better than Bridget, so our order is
.
20.
A semicircle has symmetry, so the center is exactly at the midpoint of the 2 side on the rectangle, making the radius, by the Pythagorean Theorem,
. The area is
.
21.
The number of ways to get from Samantha's house to City Park is , and the number of ways to
get from City Park to school is . Since there's one way to go through City Park (just walking
straight through), the number of different ways to go from Samantha's house to City Park to school
.
22. There are
length of
vertical columns with a length of toothpicks, and there are horizontal rows with a
grid of toothpicks.
toothpicks. An effective way to verify this is to try a small case, i.e. a
.
Thus, our answer is
23. Solution 1
If the semicircle on AB were a full circle, the area would be 16pi. Therefore the diameter of the first circle is 8. The arc of the largest semicircle would normally have a complete diameter of 17. The Pythagorean theorem says that the other side has length 15, so the radius is
.
Solution 2
We go as in Solution 1, finding the diameter of the circle on AC and AB. Then, an extended version of the theorem says that the sum of the semicircles on the left is equal to the biggest one, so the area of the largest is
, and the middle one is
, so the radius is
.
24.
First let extension of is
(where is the side length of the squares) for simplicity. We can extend . Call this point
. The area of triangle
then is
until it hits the
The area of rectangle
. Thus, our desired area is . Now, the ratio of the shaded area to the combined area
of the three squares is .
Solution 2
Let the side length of each square be . Let the intersection of Since
congruent. We also have So we have
by and ,
be
. . Since
and
are vertical angles, they are
by definition. congruence. Therefore,
.
Since and are midpoints of sides,
.
. This combined with yields
The area of trapezoid is .
The area of triangle is .
So the area of the pentagon The area of the squares is
is .
.
Therefore, .