Small Hint:

Find the greatest whole number of sandwiches first

Big Hint:

Spend the remaining money on $\$1$ drinks

Video solution:

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Written solution:

If they buy $6$ sandwiches, they would spend $6 \times 4.5 = 27$ dollars. This means that they would not be able to buy any more sandwiches.

They would then have $30 - 27 = 3$ dollars left. With this, they could buy $3$ sodas. They would therefore buy a total of $6 + 3 = 9$ items.

Thus, the correct answer is D.

Small Hint:

The long side of each small rectangle is two short sides

Big Hint:

The big rectangle is $10$ feet by $15$ feet

Video solution:

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Written solution:

From the figure, we can see that the longer side has the same length as two of the shorter sides. This makes it $2 \cdot 5 = 10$ feet long.

This tells us that $BC = 10$ feet and $DC = 10 + 5 = 15$ feet. Therefore, the area is $10 \cdot 15 = 150$ square feet.

Thus, the correct answer is E.

Small Hint:

Subtract $1$ from each fraction

Big Hint:

Compare $\frac{4}{11},$ $\frac{4}{13},$ and $\frac{4}{15}$

Video solution:

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Written solution:

We can rewrite the fraction as follows: $$\begin{align*} \dfrac{15}{11} &= 1 + \dfrac{4}{11} \\ \dfrac{17}{13} &= 1 + \dfrac{4}{13} \\ \dfrac{19}{15} &= 1 + \dfrac{4}{15}. \end{align*}$$

Recall that if two fractions have the same numerator, then the fraction with the larger denominator is smaller.

Using this fact, we can see that the correct ordering is $$\dfrac{19}{15} \lt \dfrac{17}{13} \lt \dfrac{15}{11}.$$

Thus, the correct answer is E.

Small Hint:

Each rhombus side is $52\div4$

Big Hint:

The diagonals of a rhombus are perpendicular and bisect each other

Video solution:

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Written solution:

We can split the rhombus up into $4$ triangles, each of which looks like this.

We know that this is a right triangle, since the diagonals of a rhombus are perpendicular, and we know the hypotenuse is $52 \div 4 = 13.$

Using the Pythagorean theorem, we get the other leg to be $$\sqrt{13^2 - 12^2} = \sqrt{25} = 5.$$

This means that the area of the rhombus is $$4 \cdot \dfrac{5 \cdot 12}{2} = 4 \cdot 30 = 120.$$ Thus, the correct answer is D.

Small Hint:

The tortoise’s graph is one straight line

Big Hint:

The hare’s graph has a flat part during the nap

Video solution:

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Written solution:

The hare’s path is the one with the horizontal line in the middle, which represents the period when the hare took a nap.

We also know that the time it takes the hare to reach the finish line is longer than the time it took the tortoise.

This means that the end of the hare’s path should have a larger $x$-value. This is exactly graph $B.$

Thus, the correct answer is B.

Small Hint:

A square has $4$ symmetry lines

Big Hint:

Each symmetry line has $8$ possible points besides $P$

Video solution:

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Written solution:

Note the only lines of symmetry for a square are the two diagonals and the two lines connecting opposite midpoints.

On each symmetry line there are $8$ grid points other than $P.$ The four lines intersect only at $P,$ which is not eligible, so these give $4 \cdot 8 = 32$ distinct choices for $Q.$

The probability is therefore $\dfrac{32}{80} = \dfrac{2}{5}.$

Thus, the correct answer is C.

Small Hint:

The five scores must total $5\cdot81$

Big Hint:

Make one remaining score as large as possible

Video solution:

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Written solution:

To minimize one of the scores, we have to maximize the other score. Assume that Shauna gets a $100$ on her fourth test.

The sum of the $4$ tests is then $$76 + 94 + 87 + 100 = 357.$$ For an average of $81,$ Shauna’s test scores must add to $5 \cdot 81 = 405.$ This means that she needs to get a $405 - 357 = 48$ on her last test.

Thus, the correct answer is A.

Small Hint:

Start with $100$ marbles

Big Hint:

Successive leftovers are $80\%$, then $90\%$, then $75\%$

Video solution:

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Written solution:

Assume that Gilda starts off with $100$ marbles. After giving $20$ marbles to Pedro, she has $100 - 20 = 80$ marbles left.

She then gives $80 \times 0.1 = 8$ marbles to Ebony, with her having $80 - 8 = 72$ marbles left.

Finally, Gilda gives $72 \times 0.25 = 18$ marbles to Jimmy. She has a total of $72 - 18 = 54$ marbles left, which is $54\%$ of her original total.

Thus, the correct answer is E.

Small Hint:

Volume is $\pi r^2h$

Big Hint:

Felicia’s radius doubles while her height halves

Video solution:

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Written solution:

Felicia’s diameter is twice as much as Alex’s which means that her radius is also twice as much.

Recall that the formula for the area of a circle is $\pi r^2.$ If the radius is doubled, then the area is quadrupled.

This means that the base of Felicia’s can has $4$ times the area as the base of Alex’s can.

We also see that Alex’s can is twice as tall as Felicia’s can. The formula for the volume of a can is base times height.

Using this formula, we see that Felicia’s can will have twice the volume of Alex’s can since $4 \div 2 = 2.$

Thus, the correct answer is B.

Small Hint:

The correction adds $5$ students total

Big Hint:

Compare the ordered lists before and after replacing Wednesday’s value

Video solution:

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Written solution:

There are $5$ more participants, which means the mean is increased by $5 \div 5 = 1.$

Right now, we can see that the median is $20.$ If $16$ gets replaced with $21,$ then the median becomes $21.$

This shows that the median is also increased by $1.$

Thus, the correct answer is B.

Small Hint:

First find how many students are counted twice

Big Hint:

Only math = math total minus both classes

Video solution:

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Written solution:

The number of kids that are taking both classes is $$70 + 54 - 93 = 31.$$ Subtracting this from the number of kids taking a math class will give us the number of kids taking only a math class.

The desired number is $70 - 31 = 39.$

Thus, the correct answer is D.

Small Hint:

Find two opposite pairs from the shared adjacent faces

Big Hint:

After two opposite pairs are known, the remaining two colors are opposite

Video solution:

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Written solution:

Using the first and third cubes, we can see that brown is opposite purple.

Similarly, with the first and second cubes, we see that green and white are opposites.

The only color left to pair with aqua is red, so aqua and red are opposite faces.

Thus, the correct answer is A.

Small Hint:

Every two-digit palindrome is a multiple of $11$

Big Hint:

Check the first three-digit multiple of $11$ that is not a palindrome

Video solution:

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Written solution:

Note that $2$-digit palindromes have the tens and units digits the same. This means that they are all multiples of $11.$

If a $3$-digit number is the sum of $3$ $2$-digit palindromes, then it itself must also be a multiple of $11.$

The smallest $3$-digit multiple of $11$ that is not a palindrome is $110.$ This can be achieved by adding $11 + 22 + 77.$

Therefore, $N = 110.$ The sum of its digits is $1 + 1 + 0 = 2.$

Thus, the correct answer is A.

Small Hint:

Every $10$ days advances the weekday by $3$

Big Hint:

The six days hit every weekday except one

Video solution:

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Written solution:

Let us go through the answer choices.

Each redemption is $10$ days later, which advances the weekday by $3$ days.

Starting on day $x,$ the six redemption days are $x, x+3, x+6, x+2, x+5,$ and $x+1$ modulo $7.$

The only missed weekday is $x+4,$ so Sunday must be four days after the first redemption day. Four days before Sunday is Wednesday.

Thus, the correct answer is C.

Small Hint:

Use the cap group to find the number wearing both

Big Hint:

Then divide that same overlap by the number wearing sunglasses

Video solution:

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Written solution:

If the probability is $\dfrac{2}{5},$ that means $\dfrac{2}{5}$ of the people wearing caps are also wearing sunglasses.

This means that $35 \cdot \dfrac{2}{5} = 14$ people are wearing both caps and sunglasses.

The probability of a person wearing sunglasses also wearing a cap is $\dfrac{14}{50} = \dfrac{7}{25}.$

Thus, the correct answer is B.

Small Hint:

The first $15$ miles took $\frac12$ hour

Big Hint:

Set total time equal to total distance divided by $50$

Video solution:

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Written solution:

For the first $15$ miles, Qiang drove for $\dfrac{15}{30} = \dfrac{1}{2}$ an hour.

Let $x$ be the distance that Qiang must drive to satisfy the condition. It will take him $\dfrac{x}{55}$ hours to drive this distance.

The total trip is now $15 + x$ miles. We know the average speed is $50$ miles per hour, so this will take him $\dfrac{x + 15}{50}$ hours.

Setting the two times equal to each other, we get $$\dfrac{1}{2} + \dfrac{x}{55} = \dfrac{x + 15}{50}.$$

Cross-multiplying and simplifying yields $$15 + x = 25 + \dfrac{10x}{11} \Rightarrow x = 110.$$

Thus, the correct answer is D.

Small Hint:

Rewrite each factor to line up cancellations

Big Hint:

Only the first $\frac12$ and the last $\frac{100}{99}$ remain

Video solution:

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Written solution:

We can regroup all the factors as follows. $$\begin{align*} & \dfrac{1}{2} \cdot \left(\dfrac{3 \cdot 2}{2\cdot 3}\right)\left(\dfrac{4\cdot 3}{3\cdot 4}\right) \\ & \dots\left(\dfrac{99\cdot 98}{98\cdot 99}\right) \cdot \dfrac{100}{99} \end{align*}$$

From this representation, we can see that all the middle terms cancel leaving only $$\dfrac{1}{2} \cdot \dfrac{100}{99} = \dfrac{50}{99}.$$

Thus, the correct answer is B.

Small Hint:

An even sum means both dice have the same parity

Big Hint:

Each die has $2$ even faces and $4$ odd faces

Video solution:

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Written solution:

The only two ways that the sum can be even is if both rolls are even or both are odd.

The probability that a roll is even is $\dfrac{2}{6} = \dfrac{1}{3},$ and the probability that it is odd is $\dfrac{4}{6} = \dfrac{2}{3}.$

Therefore, the desired probability is $$\dfrac{1}{3} \cdot \dfrac{1}{3} + \dfrac{2}{3} \cdot \dfrac{2}{3} = \dfrac{5}{9}.$$

Thus, the correct answer is C.

Small Hint:

The top three should win all games against the bottom three

Big Hint:

Among the top three, keep all points while splitting each pair evenly

Video solution:

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Written solution:

Each top team plays $6$ games against the bottom three teams, so it can earn at most $6 \cdot 3 = 18$ points from those games.

Among the top three teams, there are $3$ pairs and hence $6$ games. Each game awards at most $3$ points total, so these games award at most $18$ points among the three teams. If their final scores are equal, each can therefore receive at most $\frac{18}{3}=6$ of those points.

Thus each top team has at most $18+6=24$ points. This bound is attainable: let every top team win all its games against the bottom teams, and for each pair of top teams let each team win one of their two games. Then each top team earns $18+6=24$ points.

Thus, the correct answer is C.

Small Hint:

Let $x^2 - 5$ be the quantity being squared

Big Hint:

Solve $x^2 - 5 = 4$ and $x^2 - 5 = -4$

Video solution:

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Written solution:

Recall that $$a^2 - b^2 = (a + b)(a - b).$$ We can do the following rearrangement. $$\begin{aligned} (x^2 - 5)^2 - 4^2 &= 0 \\ (x^2 - 1)(x^2 - 9) &= 0 \end{aligned}$$

From this we can see that there are $4$ possible values for $x.$

Thus, the correct answer is D.

Small Hint:

Find the three pairwise intersections

Big Hint:

Use the horizontal side on $y = 5$ as the base

Video solution:

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Written solution:

First, let us find the vertices of the triangle. The intersections between $y = 5$ and the other two lines are $(4, 5)$ and $(-4, 5)$.

To find the third intersection, we can equate the two equations to get $$1 + x = 1 - x \Rightarrow x = 0.$$

This gives us the point $(0, 1).$

Note that the first two intersection points are mirrored across the $y$-axis. This tells us that the triangle is isosceles.

The base is therefore $2 \cdot 4 = 8,$ and the height is $5 - 1 = 4.$ The area of the triangle is therefore $$\dfrac{1}{2} \cdot 8 \cdot 4 = 16.$$

Thus, the correct answer is E.

Small Hint:

An increase by $x$ and decrease by $x$ multiplies by $(1 + x)(1 - x)$

Big Hint:

Set $1 - x^2 = 0.84$

Video solution:

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Written solution:

Let $x$ be the percent converted to a real number. Then the percent increase would multiply the original by $1 + x.$

The percent decrease would multiply the new price by $1 - x.$ The final price will then be the original price multiplied by $$(1 + x)(1 - x) = 1 - x^2 = 0.84.$$

This tells us that $$x^2 = 0.16 \Rightarrow x = 0.4.$$

This tells us that the percent would be $40\%.$

Thus, the correct answer is E.