Small Hint:

Find the next multiple of $6$ after $23$.

Big Hint:

Only one more car is needed to reach that multiple.

Video solution:

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Written solution:

In order for Danica to arrange her model cars in rows of exactly six, the number of model cars she has must be a multiple of six. The smallest multiple of $6$ greater than $23$ is $24,$ so she must buy $1$ more car to attain this amount.

Thus, A is the correct answer.

Small Hint:

The quoted price is for a half pound after the discount.

Big Hint:

First undo the $50\%$ discount, then double to get one pound.

Video solution:

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Written solution:

During the sale, the price for half a pound is $\$3,$ and therefore, the price for a full pound is $2 \cdot \$3 = \$6.$

The sale price is $50\%$ off the regular price, and so the regular price is twice the sale price. Therefore, the regular price for a full pound of fish is $2 \cdot \$6 = \$12.$

Thus, D is the correct answer.

Small Hint:

Pair the terms inside the parentheses.

Big Hint:

Each pair $(-1+2),(-3+4),\ldots$ has the same sum.

Video solution:

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Written solution:

Within the parentheses, pair consecutive terms: $$\begin{aligned} &(-1+2)+(-3+4) \\ &\quad {}+\cdots+(-999+1000). \end{aligned}$$

Each pair has sum $1$, and there are $500$ pairs, so the value inside the parentheses is $500$. Multiplying by $4$ gives $2000$.

Thus, E is the correct answer.

Small Hint:

The seven friends together paid Judi’s share.

Big Hint:

Judi’s share is $7$ times the extra amount each friend paid.

Video solution:

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Written solution:

Since Judi’s portion was fully covered by everyone’s contributions, the portion of the bill she was responsible for was equal to $$7 \cdot \$2.50 = \$17.50.$$

As everyone paid the same amount, we can conclude that the total bill is $$8 \cdot \$17.50 = \$140.$$

Thus, C is the correct answer.

Small Hint:

Order the five weights before finding the median.

Big Hint:

Compare the median with the total weight divided by $5$.

Video solution:

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Written solution:

The median weight is the middle value, which we can find as: $$\cancel{ 5 },\cancel{ 5 },\underline{6},\cancel{8},\cancel{106}$$ Therefore, the median is $6$ pounds.

The average (mean) can be calculated to be: $$\begin{align*} &\dfrac{106 + 5 + 5 + 6 + 8}{5} \\&= \dfrac{130}{5}\\ &= 26. \end{align*}$$ This is greater than the median, specifically, the mean is $20$ pounds greater than the median.

Thus, E is the correct answer.

Small Hint:

Use the product $600$ to find the missing number in the second row.

Big Hint:

Then use $5$ times the top missing number to get that second-row value.

Video solution:

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Written solution:

The product of the numbers in the second row is $600,$ so the missing number in the middle right box is $600 \div 30 = 20.$

Now we know that the product of $5$ and the missing number in the first row is $20.$ Therefore, the missing number is $20 \div 5 = 4.$

Thus, C is the correct answer.

Small Hint:

Convert the total crossing time to seconds.

Big Hint:

Use a proportion from $6$ cars in $10$ seconds.

Video solution:

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Written solution:

Converting $2$ minutes and $45$ seconds into seconds, we get $2 \cdot 60 + 45 = 165$ seconds. Since $6$ cars passed in $10$ seconds, at the same rate the number of cars $x$ passing in $165$ seconds satisfies $$\dfrac{6}{10}=\dfrac{x}{165}.$$

Solving gives $$x=\dfrac{165\cdot6}{10}=99.$$ The closest answer choice is $100$.

Thus, C is the correct answer.

Small Hint:

List the $8$ possible outcomes of three coin tosses.

Big Hint:

Count only outcomes containing $HH$.

Video solution:

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Written solution:

The outcomes with at least two consecutive heads are $HHT$, $THH$, and $HHH$. There are $2^3=8$ equally likely outcomes from three fair coin tosses, so the probability is $\dfrac{3}{8}$.

Thus, C is the correct answer.

Small Hint:

The $n$th jump has length $2^{n-1}$.

Big Hint:

Compare powers of $2$ with $1000$.

Video solution:

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Written solution:

On the $n$th jump, the Hulk jumps $2^{n-1}$ meters. We need the smallest $n$ for which $2^{n-1}>1000$.

Since $2^9=512$ and $2^{10}=1024$, the first jump longer than $1000$ meters has $n-1=10$, so $n=11$.

Thus, C is the correct answer.

Small Hint:

Prime-factorize both numbers.

Big Hint:

The LCM uses the larger prime powers, while the GCF uses the smaller ones.

Video solution:

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Written solution:

Prime-factorize the two numbers: $$180=2^2\cdot3^2\cdot5,$$ $$594=2\cdot3^3\cdot11.$$

The greatest common factor uses the smaller powers, so it is $2\cdot3^2=18$. The least common multiple uses the larger powers, so it is $$2^2\cdot3^3\cdot5\cdot11=5940.$$ Therefore the desired ratio is $\dfrac{5940}{18}=330$.

Thus, C is the correct answer.

Small Hint:

Time equals distance divided by rate.

Big Hint:

Compare the actual total minutes with $6$ miles at $4$ mph.

Video solution:

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Written solution:

Grandfather spent $\frac{2}{5}$ hours jogging on Monday. He spent $\frac{2}{3}$ hours walking on Wednesday and $\frac{2}{4}=\frac{1}{2}$ hours on Friday. Converting these times to minutes, we get $$\dfrac{2}{5} \cdot 60 = 24$$ minutes for Monday, $$\dfrac{2}{3} \cdot 60 = 40$$ minutes for Wednesday, and $$\dfrac{1}{2} \cdot 60 = 30$$ minutes for Friday. Therefore, Grandfather totaled $94$ minutes of exercise throughout the week.

If he walked at a pace of $4$ miles per hour each day, he would have spent $\frac{2}{4}=\frac{1}{2}$ hours each day walking. This equals $\frac{2}{4} \cdot 60 = 30$ minutes every day. If he did this for $3$ days, he would have totaled $3 \cdot 30 = 90$ minutes of exercise.

Therefore, Grandfather would have spent $94-90=4$ fewer minutes on the treadmill.

Thus, D is the correct answer.

Small Hint:

Compute the price paid for each pair of sandals.

Big Hint:

Compare the total savings with the regular total of $\$150$.

Video solution:

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Written solution:

Javier saved $0.40 \cdot 50 = 20$ dollars on the second pair. He also saved $\frac{50}{2} = 25$ dollars on the third pair. This shows that he saved a total of $45$ dollars.

As $$\frac{45}{150} = \frac{3}{10} = 0.30,$$ we can conclude that Javier saved $30 \%$ compared to the regular price.

Thus, B is the correct answer.

Small Hint:

Reversing tens and units changes a number by a multiple of $9$.

Big Hint:

Check which answer choice is divisible by $9$.

Video solution:

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Written solution:

If the last two digits of the score are $10a+b$, then reversing those digits gives $10b+a$. The difference is $$\begin{aligned} &(10a+b)-(10b+a) \\ &= 9(a-b), \end{aligned}$$ so the incorrect sum must differ from the correct sum by a multiple of $9$.

Among the answer choices, only $45$ is divisible by $9$.

Thus, A is the correct answer.

Small Hint:

There are $2\cdot4$ equally likely jelly-bean pairings.

Big Hint:

Count the green match and the two red matches.

Video solution:

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Written solution:

There are $2\cdot4=8$ equally likely ways for Abe and Bea to choose their jelly beans. The colors match if they both choose green, which can happen in $1$ way, or if they both choose red, which can happen in $1\cdot2=2$ ways.

Thus $3$ of the $8$ outcomes match, for probability $\dfrac{3}{8}$.

Thus, C is the correct answer.

Small Hint:

Subtract the known number from each equation.

Big Hint:

Identify $3^p$, $2^r$, and $6^s$ as familiar powers.

Video solution:

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Written solution:

Subtract the known number from each equation: $$3^p=90-81=9,$$ $$2^r=76-44=32,$$ $$6^s=1421-125=1296.$$

Therefore $p=2$, $r=5$, and $s=4$. Their product is $2\cdot5\cdot4=40$.

Thus, B is the correct answer.

Small Hint:

The number of eighth-graders must fit both ratios.

Big Hint:

Use the least common multiple of $5$ and $8$.

Video solution:

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Written solution:

The number of $8$th-graders must be a multiple of $5$ and $8,$ which means that it is at least $40.$ Using this number, we get that the number of $6$th-graders is $40 \cdot \dfrac{3}{5} = 24.$ Similarly, the number of $7$th-graders is $40 \cdot \dfrac{5}{8} = 25.$

The total number of students is therefore $40 + 24 + 25 = 89.$

Thus, E is the correct answer.

Small Hint:

The average of six consecutive integers is halfway between the middle two.

Big Hint:

The largest is $2.5$ more than the average.

Video solution:

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Written solution:

Let the six consecutive integers be $x,$ $x+1,$ $x+2,$ $x+3,$ $x+4,$ $x+5$. Their sum is $$6x+15=2013.$$

Thus $6x=1998$, so $x=333$. The largest integer is $x+5=338$.

Thus, B is the correct answer.

Small Hint:

Subtract the empty inside volume from the outside rectangular prism.

Big Hint:

The inside dimensions are reduced by two feet in length/width and one foot in height.

Video solution:

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Written solution:

The rectangular prism determined by the outside faces has volume $12\cdot10\cdot5=600$ cubic feet. The empty inside is smaller by $2$ feet in length, $2$ feet in width, and $1$ foot in height, so its dimensions are $10\times8\times4$.

The empty volume is $10\cdot8\cdot4=320$ cubic feet, so the fort contains $600-320=280$ one-foot cubical blocks.

Thus, B is the correct answer.

Small Hint:

Cassie can compare herself with Hannah because she sees Hannah’s test.

Big Hint:

Cassie is above Hannah, while Bridget is below Hannah.

Video solution:

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Written solution:

Since Cassie deduced she didn’t get the lowest score, she must have got a higher score than Hannah. Similarly, Bridget’s statement reveals that she must have got a lower score than Hannah.

Therefore, the ranking from highest to lowest is Cassie, Hannah, and then Bridget.

Thus, D is the correct answer.

Small Hint:

The radius goes from the midpoint of the diameter to a top corner of the rectangle.

Big Hint:

That radius is the hypotenuse of a $1$ by $1$ right triangle.

Video solution:

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Written solution:

Using the Pythagorean theorem, we get that the radius of the semicircle is $\sqrt{2}.$ This means that the area of the semicircle is $$\dfrac{1}{2} \cdot \pi \cdot \sqrt{2}^2 = \pi.$$ Thus, C is the correct answer.

Small Hint:

Count shortest street routes before and after the park separately.

Big Hint:

There are $3$ ways before the park and $6$ ways after it.

Video solution:

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Written solution:

To reach the southwest corner of the park as quickly as possible, Samantha must go $2$ blocks east and $1$ block north, which can be arranged in $3$ ways. After the diagonal path through the park, she must go $2$ blocks east and $2$ blocks north to reach school, which can be arranged in $\binom{4}{2}=6$ ways.

The choices before and after the park are independent, so the total number of shortest routes is $3\cdot6=18$.

Thus, E is the correct answer.

Small Hint:

Count vertical and horizontal toothpicks separately.

Big Hint:

A $60$ by $32$ grid has $61$ vertical lines and $33$ horizontal lines.

Video solution:

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Written solution:

A $60$-by-$32$ grid has $61$ vertical grid lines, each made of $32$ toothpicks, and $33$ horizontal grid lines, each made of $60$ toothpicks.

Therefore the total number of toothpicks is $$\begin{aligned} 61\cdot32+33\cdot60 &= 1952+1980 \\ &= 3932. \end{aligned}$$

Thus, E is the correct answer.