Small Hint:

Find how many kilometers $1$ centimeter represents.

Big Hint:

Multiply that scale by $17.$

Solution:

Note that $1$ cm represents $72 \div 12 = 6$ kilometers. This means that $17$ cm represents $6 \cdot 17 = 102$ kilometers.

Thus, B is the correct answer.

Small Hint:

A four-digit number cannot start with $0.$

Big Hint:

Choose which nonzero digit comes first, then place the other nonzero digit.

Solution:

Note that there are $2$ non-zero digits that could be the thousands digit.

After choosing that, we need to arrange the other $3$ digits. There are $3$ spots for the other non-zero digit.

This gives us $2 \cdot 3 = 6$ possible numbers.

Thus, B is the correct answer.

Small Hint:

The $12$ meals made enough food for $18$ people.

Big Hint:

Find how many meals are needed per person, then scale to $12$ people.

Solution:

Note that $12$ meals feed $18$ people. This means that one meal feeds $\dfrac{18}{12} = \dfrac{3}{2}$ people.

This means that they need $12 \div \dfrac{3}{2} = 8$ meals for $12$ people.

Thus, A is the correct answer.

Small Hint:

Choosing the three starters is the same as choosing who sits out.

Big Hint:

There are $4$ possible players who could be the alternate.

Solution:

If there are $3$ starters, then one person must not be starting. Choosing the person who doesn’t start determines the starters.

There are $4$ choices for the person who doesn’t start.

Thus, B is the correct answer.

Small Hint:

Each game eliminates exactly one team.

Big Hint:

The tournament ends when all but one team have been eliminated.

Solution:

Note that after every game, one team gets eliminated. For there to be one team remaining, $15$ teams must have been eliminated.

This means that $15$ games had to have been played.

Thus, D is the correct answer.

Small Hint:

Find how many shots Sally had made after $20$ shots.

Big Hint:

Find how many total made shots she had after $25$ shots.

Solution:

Sally made $20 \times 0.55 = 11$ of her first $20$ shots. Then we get that $$\dfrac{11 + x}{25} = 0.56,$$ which tells us that $$11 + x = 14$$ and $x = 3.$

Thus, C is the correct answer.

Small Hint:

First compute $220-26.$

Big Hint:

Take $80\%$ of that maximum heart rate and round.

Solution:

The maximum heart rate for this athlete would be $220 - 26 = 194.$ Then the target heart rate would be $194 \times 0.8 \approx 155.$

Thus, B is the correct answer.

Small Hint:

The tens digit cannot be $0.$

Big Hint:

Once the tens digit is chosen, the ones digit is forced.

Solution:

Note that the tens digit can range from $1$ to $7,$ and this digit determines the units digit.

Therefore, there are $7$ numbers.

Thus, B is the correct answer.

Small Hint:

Turn each average into a sum.

Big Hint:

Subtract the sum of the first two numbers from the sum of all five.

Solution:

The sum of all $5$ numbers is $54 \cdot 5 = 270.$ The sum of the first $2$ numbers is $48 \cdot 2 = 96.$

The sum of the last $3$ numbers is $270 - 96 = 174.$ The average is therefore $174 \div 3 = 58.$

Thus, D is the correct answer.

Small Hint:

Convert all the work times to minutes.

Big Hint:

After finding total minutes, convert to hours before multiplying by $\$3.$

Solution:

Aaron worked $75$ minutes on Monday, $50$ minutes on Tuesday, $145$ minutes on Wednesday, and $30$ minutes on Friday.

The total is $75+50+145+30=300$ minutes, or $5$ hours.

At $\$3$ per hour, he earned $5\cdot3=15$ dollars.

Thus, E is the correct answer.

Small Hint:

Use the rules to locate the largest, smallest, and median numbers.

Big Hint:

The first and last positions are filled by the two remaining numbers.

Solution:

Note that the largest, smallest, and median numbers cannot be the first or last number.

This means that the first and last numbers are $4$ and $9$ in some order. Their average is $$(4 + 9) \div 2 = 13 \div 2 = 6.5.$$

Thus, C is the correct answer.

Small Hint:

One hour of talking uses as much battery as several idle hours.

Big Hint:

The phone was idle for $8$ hours and used for $1$ hour.

Solution:

When not in use, her cell phone uses up $\frac{1}{24}$ of its battery per hour. When it is in use, it uses up $\frac{1}{3}$ of its battery per hour.

Niki’s phone has been on for $9$ hours, with $8$ of those hours being idle and $1$ hour being used to talk on the phone.

This means that the phone has used up $\frac{2}{3}$ of its battery. In order to drain the remaining $\frac{1}{3}$ of the battery, the phone can last for $8$ more hours without being used.

Thus, B is the correct answer.

Small Hint:

If Bill were oldest, check how many statements would be true.

Big Hint:

After ruling out the oldest person, use the fact that exactly one statement is true.

Solution:

If Bill were oldest, then statements $\mathrm{I}$ and $\mathrm{II}$ would both be true, so Bill is not oldest.

If Celine were oldest, then statements $\mathrm{II}$ and $\mathrm{III}$ would both be true, so Celine is not oldest.

Therefore Amy is oldest. Statements $\mathrm{I}$ and $\mathrm{II}$ are false, so statement $\mathrm{III}$ must be the single true statement. Thus Celine is not youngest, leaving Bill youngest.

The order is Amy, Celine, Bill.

Thus, E is the correct answer.

Small Hint:

Put the quadrilateral inside an easy surrounding square.

Big Hint:

Subtract the outside rectangles and right triangles from the surrounding square.

Solution:

Place the quadrilateral inside the surrounding $10\times10$ square, whose area is $100.$

The five outside pieces have areas $15,$ $\frac12\cdot6\cdot7=21,$ $\frac12\cdot1\cdot3=\frac32,$ $\frac12\cdot4\cdot5=10,$ and $\frac12\cdot6\cdot10=30.$

The outside area is $15+21+\frac32+10+30=77\frac12,$ so the quadrilateral area is $100-77\frac12=22\frac12.$

Thus, C is the correct answer.

Small Hint:

The original figure has one center tile, a ring of $6,$ and a ring of $12.$

Big Hint:

The next hexagonal border has $18$ tiles.

Solution:

The original figure has $13$ shaded tiles and $6$ unshaded tiles.

The new border is the next hexagonal ring around the figure, which has $18$ unshaded tiles.

The new figure has $6+18=24$ unshaded tiles and $13$ shaded tiles, so the difference is $24-13=11.$

Thus, C is the correct answer.

Small Hint:

Compute the orange juice in each pitcher before water is added.

Big Hint:

After filling both pitchers, the total mixture volume is $1200$ mL.

Solution:

The first pitcher contains $600 \cdot \dfrac{1}{3} = 200$ mL of orange juice. The second one has $600 \cdot \dfrac{2}{5} = 240$ mL.

The large container then has $200 + 240 = 440$ mL of orange juice. The total amount of mixture is $2 \cdot 600 = 1200$ mL.

Then the fraction of orange juice is $\dfrac{440}{1200} = \dfrac{11}{30}.$

Thus, C is the correct answer.

Small Hint:

Give each friend one pencil first.

Big Hint:

Count the ways to distribute the remaining $3$ pencils among the $3$ friends.

Solution:

First give each friend one pencil. Then $3$ pencils remain to distribute among the $3$ friends.

If one friend gets all $3$ remaining pencils, there are $3$ ways. If the remaining pencils split as $2$ and $1,$ there are $3\cdot2=6$ ways. If each friend gets one more, there is $1$ way.

The total number of ways is $3+6+1=10.$

Thus, D is the correct answer.

Small Hint:

Ben’s score of $4$ has only one possible pair of distinct positive scores.

Big Hint:

Use each forced pair to eliminate numbers from the remaining scores.

Solution:

The only way to get Ben’s score of $4$ is with a $1$ and $3$ since he can’t hit $2$ twice.

Cindy can achieve her score with $$1 + 6,\quad 2 + 5,\quad 3 + 4.$$ Ben already hit $1$ and $3,$ so Cindy must have hit $2$ and $5.$

Similarly, Dave must have hit $4$ and $7.$ Finally, since $7$ is already used, Alice is forced to have hit $6$ and $10$ with Ellen hitting $8$ and $9.$

Thus, A is the correct answer.

Small Hint:

If the number leaves remainder $2,$ subtract $2.$

Big Hint:

The result must be divisible by $3,$ $4,$ $5,$ and $6.$

Solution:

Let $x$ be the number. Then $x - 2$ is divisible by $3,$ $4,$ $5,$ and $6.$

The least common multiple of these numbers is $60,$ which makes $x$ $62.$

Thus, B is the correct answer.

Small Hint:

The empty chairs are the remaining one-fourth of the chairs.

Big Hint:

The seated people are two-thirds of all the people.

Solution:

Since $\frac{3}{4}$ of the chairs are occupied, the $6$ empty chairs are $\frac{1}{4}$ of all the chairs. Thus there are $24$ chairs.

The number of seated people is $\frac{3}{4}$ of $24,$ which is $18.$

Those $18$ seated people are $\frac{2}{3}$ of all the people, so the total number of people is $18\div\frac23=27.$

Thus, D is the correct answer.

Small Hint:

It is easier to count the complement: an odd product.

Big Hint:

An odd product requires both spinner results to be odd.

Solution:

For the product to be even, at least one of the spinners must land on an even number.

We can use complementary counting and calculate the probability of both spinners landing on odds.

This happens with a probability of $$\dfrac{1}{2} \cdot \dfrac{2}{3} = \dfrac{1}{3}.$$

Then the probability of landing on at least one even is $$1 - \dfrac{1}{3} = \dfrac{2}{3}.$$

Thus, D is the correct answer.

Small Hint:

Use a convenient number of women matching the ratio $\frac{2}{5}.$

Big Hint:

Married women and married men come in equal numbers.

Solution:

For a convenient ratio model, let there be $5$ women in the room. Then there are $5 \cdot \dfrac{2}{5} = 2$ single women.

This means that there are $5 - 2 = 3$ married women, which is also the number of married men.

There are a total of $5 + 3 = 8$ people in the room. The fraction of married men is $\dfrac{3}{8}.$

Thus, B is the correct answer.

Small Hint:

Track the distance from corner $J$ at each side of the rectangle.

Big Hint:

The distance is greatest at the opposite corner $L.$

Solution:

As Tess runs from $J$ to $K,$ her distance from home increases.

From $K$ to $L,$ her distance continues to increase, but with a different shape, reaching its maximum at the opposite corner $L.$

From $L$ to $M$ and then from $M$ back to $J,$ her distance decreases back to $0,$ again changing behavior at $M.$

Graph $D$ is the only graph with this increase, changed-rate increase, changed-rate decrease, and final decrease to $0.$

Thus, D is the correct answer.

Small Hint:

Find the area of the rectangle and subtract the four corner triangles.

Big Hint:

Use the Pythagorean theorem to get the parallelogram base $HE.$

Solution:

The rectangle has side lengths $10$ and $8,$ so its area is $80.$

The four corner triangles have total area $$\begin{aligned} &2\cdot\frac12\cdot3\cdot4 \\ &\quad{}+2\cdot\frac12\cdot6\cdot5 \\ &=12+30=42. \end{aligned}$$ Thus parallelogram $EFGH$ has area $80-42=38.$

Segment $HE$ has length $5$ by the $3-4-5$ right triangle. Since the parallelogram area is $HE\cdot d,$ we have $5d=38,$ so $d=7.6.$

Thus, C is the correct answer.