Answer: D

Small Hint:

Round each price to the nearest dollar before adding.

Big Hint:

The exact cents are not needed because the question asks for the nearest dollar.

Solution:

Since all the values are already close to a whole number, we can just add the sums of the rounded numbers.

These prices round to $\$ 2,$ $\$ 5,$ and $\$ 10,$ which add to $\$ 17.$

Thus, D is the correct answer.

Answer: C

Small Hint:

On the AMC $8,$ only correct answers add to the score.

Big Hint:

Incorrect and blank answers do not change the number of correct answers.

Solution:

Since the AMC $8$ only awards $1$ point for each correct question, Billy will get $13$ points.

Thus, C is the correct answer.

Answer: A

Small Hint:

Find Elisa’s minutes per lap before and after the improvement.

Big Hint:

Compare $\frac{25}{10}$ minutes per lap with $\frac{24}{12}$ minutes per lap.

Solution:

Initially, Elisa swam one lap in $\frac{25}{10}=2.5$ minutes. Now she swims one lap in $\frac{24}{12}=2$ minutes.

Therefore, she improved her lap time by $2.5-2=0.5=\dfrac{1}{2}$ minute.

Thus, A is the correct answer.

Answer: B

Small Hint:

Whole revolutions do not change the final direction.

Big Hint:

After removing whole turns, compare a clockwise quarter-turn with a counterclockwise three-quarter-turn.

Solution:

If the spinner goes $2 \frac{1}{4}$ revolutions clockwise and $3 \frac{3}{4}$ revolutions counterclockwise, it goes $1 \frac{1}{2}$ revolutions counterclockwise.

This means that the spinner will be pointing east.

Thus, B is the correct answer.

Answer: D

Small Hint:

The four corner triangles around the smaller square are congruent.

Big Hint:

Those four outside triangles can be rearranged to cover the smaller square.

Solution:

Note that we can fold all the triangles in to perfectly cover the smaller square. This means that area of the smaller square is half the area of the larger square.

This makes the area of the smaller square $60 \div 2 = 30.$

Thus, D is the correct answer.

Answer: C

Small Hint:

Start with the perimeters of the two rectangles separately.

Big Hint:

The shared interior segments are counted in the separate perimeters but not in the outside perimeter.

Solution:

If we found the total perimeter of the two rectangles separately, we would have gotten $$2(2(2 + 4)) = 2 \cdot 2 \cdot 6 = 24.$$

In the letter T, we can see that their intersection removes a piece of length $2$ from each of the rectangles. Therefore, the perimeter of the T is $$24 - 2 \cdot 2 = 20.$$

Thus, C is the correct answer.

Answer: B

Small Hint:

Convert all three pieces of information into radii.

Big Hint:

Use $C=2\pi r$ for circle $Y$ and $A=\pi r^2$ for circle $Z$.

Solution:

Recall that $C = 2 \pi r$ and $A = \pi r^2.$ Using these formulas we get that the radius of $Y$ is $$8 \pi \div (2 \pi) = 4.$$ We also get that the radius of $Z$ is $$\sqrt{9 \pi \div \pi} = 3.$$ As $\pi$ is greater than $3$ and less than $4,$ the correct order is $Z,$ $X,$ $Y.$

Thus, the answer is B .

Answer: E

Small Hint:

Use the totals to fill in the number of males surveyed.

Big Hint:

Find the male listeners by subtracting the male non-listeners from the male total.

Solution:

The total number of males surveyed is the total number surveyed minus the number of females surveyed: $200-96=104$.

Of those males, $26$ do not listen, so $104-26=78$ males listen. The desired percentage is $\dfrac{78}{104}\cdot 100\%=75\%$.

Thus, E is the correct answer.

Answer: C

Small Hint:

Most adjacent numerators and denominators cancel.

Big Hint:

After cancellation, only the first denominator and the last numerator remain.

Solution:

Note that the numerator of every fraction cancels with the denominator of the following fraction. This leaves two numbers: $$\dfrac{2006}{2} = 1003.$$

Thus, C is the correct answer.

Answer: A

Small Hint:

List the positive integer factor pairs of $12$.

Big Hint:

Remember that $(w,l)$ and $(l,w)$ are different plotted points when the factors differ.

Solution:

The positive integer factor pairs $(w,l)$ with $wl=12$ are $(1,12),$ $(2,6),$ $(3,4),$ $(4,3),$ $(6,2),$ and $(12,1)$.

These points form a decreasing set of six points, and only graph A matches them.

Thus, A is the correct answer.

Answer: C

Small Hint:

The largest possible digit sum is $18$.

Big Hint:

Count the two-digit numbers whose digit sums are $1,$ $4,$ $9,$ and $16$.

Solution:

There is $1$ number whose digit sum is $1$: $10.$

There are $4$ numbers whose digit sum is $4$: $13,$ $22,$ $31,$ and $40.$

There are $9$ numbers whose digit sum is $9$: $18,$ $27,$ $36,$ $45,$ $54,$ $63,$ $72,$ $81,$ and $90.$

There are $3$ numbers whose digit sum is $16$: $79,$ $88,$ and $97.$

Therefore, there are $17$ numbers that satisfy the problem statement.

Thus, C is the correct answer.

Answer: D

Small Hint:

Convert each test percentage into a number of correct answers.

Big Hint:

This is a weighted average by number of problems, not a plain average of the three percentages.

Solution:

Antonette got $0.7\cdot 10=7$ questions right on the first test. Similarly, she got $0.8\cdot 20=16$ and $0.9\cdot 30=27$ problems right on her second and third tests.

Adding these up yields $50$ correct questions. Her score on the combined test would have been $100\cdot \dfrac{50}{60}=83\frac{1}{3}\%$, closest to $83\%$.

Thus, D is the correct answer.

Answer: D

Small Hint:

First account for Cassie’s half-hour head start.

Big Hint:

After Brian starts, the distance between them closes at the sum of their speeds.

Solution:

By the time Brian starts biking, Cassie has already traveled $\dfrac{1}{2}\cdot 12=6$ miles. This means that Cassie and Brian are then $62-6=56$ miles apart.

Together, they close the distance at $12+16=28$ miles per hour. This means that they meet $56\div 28=2$ hours after $9{:}00$ AM.

This means that they meet at $11:00$ AM.

Thus, D is the correct answer.

Answer: B

Small Hint:

Compare Bob and Chandra by seconds per page.

Big Hint:

Bob takes the same extra number of seconds on each of the $760$ pages.

Solution:

Bob will take $760 \cdot 45$ seconds to read the book, and Chandra will take $760 \cdot 30$ seconds.

The difference between the time they spent reading is $$760 \cdot 45 - 760 \cdot 30$$$$= 760(45 - 30)$$$$= 760 \cdot 15$$$$= 11,400$$ seconds.

Thus, B is the correct answer.

Answer: C

Small Hint:

Let $x$ be the number of pages Chandra reads.

Big Hint:

Set Chandra’s time for $x$ pages equal to Bob’s time for the remaining pages.

Solution:

Let $x$ be the number of pages that Chandra will read. Then Bob will read $760 - x$ pages.

For them to read for the same amount of time, $$\begin{align*} 30x &= 45(760 - x) \\ 30x &= 45 \cdot 760 - 45x \\ 75x &= 45 \cdot 760 \\ 5x &= 3 \cdot 760 \\ x &= 3 \cdot 152 \\ x &= 456. \end{align*}$$

Thus, C is the correct answer.

Answer: E

Small Hint:

Compare how many pages each person reads in the same amount of time.

Big Hint:

A common time such as $180$ seconds makes all three page counts whole numbers.

Solution:

If all $3$ individuals read the same amount of time, then the number of pages Bob, Chandra, and Alice will read will be in the ratio $4:6:9$ respectively.

This means that they will read $160,$ $240,$ and $360$ pages respectively. Since they all read for the same amount of time, we can just calculate how long it takes for Bob to read his portion.

Bob will take $45 \cdot 160 = 7200$ seconds to read his portion.

Thus, E is the correct answer.

Answer: B

Small Hint:

Track only odd and even values, not the exact sums.

Big Hint:

Spinner $Q$ always gives an even number, and spinner $R$ always gives an odd number.

Solution:

Spinner $Q$ always lands on an even number, so it does not change the parity of the total. Spinner $R$ always lands on an odd number.

For the full sum to be odd, the number from spinner $P$ must be even. Only one of the three equal sectors of spinner $P$ contains an even number.

Therefore the probability is $\dfrac{1}{3}$.

Thus, B is the correct answer.

Answer: D

Small Hint:

Look at one face of the large cube.

Big Hint:

Each face has four corner unit squares from black corner cubes and five other unit squares.

Solution:

Since each face has the same black and white surface area, we can analyze what fraction of one side is white.

On one side, there are $9$ unit squares. $4$ of them are black. This means that $\dfrac{5}{9}$ of each face is white.

Thus, D is the correct answer.

Answer: D

Small Hint:

Use the congruent triangles to relate $EC$ and $AB$.

Big Hint:

Then use $AB=BC$ and the fact that $D$ is the midpoint of $BC$.

Solution:

By the congruency condition, we know that $$AB = EC = 11.$$

Also from the isosceles condition, we know that $$BC = AB = 11.$$

Since $D$ is the midpoint of $\overline{BC},$ we know that $$BD = BC \div 2 = 5.5.$$

Thus, D is the correct answer.

Answer: C

Small Hint:

Count how many games are played in a six-player round robin.

Big Hint:

Every game has exactly one winner, so total wins equal total games.

Solution:

In every match, there was exactly one winner. There are $6 \cdot 5 \div 2 = 15$ games and therefore $15$ wins.

There are already $$4 + 3 + 2 + 2 + 2 = 13$$ wins accounted for, so Monica won $15 - 13 = 2$ games.

Thus, C is the correct answer.

Answer: A

Small Hint:

A submerged object raises the water by its own volume.

Big Hint:

Divide the rock’s volume by the base area of the aquarium.

Solution:

The base area of the aquarium is $100\cdot 40=4000\text{ cm}^2$.

The submerged rock displaces $1000\text{ cm}^3$ of water, so the water level rises by $1000\div 4000=0.25$ cm.

Thus, A is the correct answer.

Answer: D

Small Hint:

Write the three bottom entries as $a,$ $b,$ and $c$.

Big Hint:

The middle bottom entry is counted twice in the top number.

Solution:

If the lower cells contain $a,$ $b,$ and $c,$ the middle row will have $a+b$ and $b+c$.

This means that the top row will have $a+2b+c$. To minimize this, put $b=1$ in the middle and $2$ and $3$ in the outer cells. This yields a top number of $7$.

To maximize it, put $b=9$ in the middle and $7$ and $8$ in the outer cells. This yields a top number of $33$. The desired difference is $33-7=26$.

Thus, D is the correct answer.

Answer: A

Small Hint:

List small numbers that leave remainder $4$ when divided by $6$.

Big Hint:

Equivalently, adding two coins would make the number divisible by both $5$ and $6$.

Solution:

The positive integers that leave a remainder of $4$ when divided by $6$ are $$4, 10, 16, 22, 28, 34, 40, \cdots.$$ The positive integers that leave a remainder of $3$ when divided by $5$ are $$3, 8, 13, 18, 23, 28, 33, \cdots.$$

From this, we can see that the smallest number of coins that work is $28.$ This leaves a remainder of $0$ when divided by $7.$

Thus, A is the correct answer.