Answer: B

Solution:

If we plug in the values into each answer choice, we get the following:

A: $\dfrac 6x = \dfrac 67$

B: $\dfrac 6{x+1} = \dfrac 68$

C: $\dfrac 6{x-1} = 1$

D: $\dfrac x6 = \dfrac 76$

E: $\dfrac {x+1}6 = \dfrac{4}{3}$

Thus, the correct answer is B.

Answer: E

Solution:

Plugging into the formula above, we get: $$3\cdot 2-1\cdot 4=2.$$

Thus, the correct answer is E.

Answer: B

Solution:

This evaluates to: $$\begin{align*} \dfrac{\frac{3}{8} + \frac{7}{8}}{\frac{4}{5}} &= \dfrac{\frac{5}{4}}{\frac{4}{5}} \\ &=\left(\dfrac 54\right)^2 \\ &= \dfrac{25}{16}.\end{align*}$$

Thus, the correct answer is B.

Answer: E

Solution:

First, we can clearly see the $3$ small triangles, and the triangle that encompasses the entire figue. Also, there is $1$ more triangle when combining the two rightmost smaller triangles.

This leads to us having $5$ triangles.

Thus, the correct answer is E.

Answer: B

Solution:

Each of them start with $9.1234.$ Thus, we need to look at the next digits.

For choices A and B, the next digit is $4$ while it is $3$ for C, $2$ for D, and $1$ for E.

This means we only have to look at A and B as possible solutions. After the $9.12344,$ choice A terminates while the next term for B is $4,$ so choice B is larger.

Thus, the correct answer is B.

Answer: B

Solution:

Consider the $2 \times 3$ rectangle on the bottom. The figure is the same as when we take some area out on the bottom and add the same area on the top. Thus, the area is the same as the the $2 \times 3$ rectangle, which is $6$

Thus, the correct answer is B.

Answer: D

Solution:

We will group the first two terms and the last two terms.

This will make the expression equal to $$(100\cdot 19.98)\cdot (1.998\cdot 1000)=$$ $$1998\cdot 1998=1998^2.$$

Thus, the correct answer is D.

Answer: C

Solution:

The amount lost is $0.5\cdot 30=15$ gallons. Therefore, the amount left is $$200-15=185.$$

Thus, the correct answer is C.

Answer: C

Solution:

After the $20\%$ reduction, the price is $10\cdot 0.8=$8.

Then, after halving the price, the price is \dfrac {$8}2 = $4.

Thus, the correct answer is C.

Answer: E

Solution:

The fractions $\dfrac{W}{X}$ and $\dfrac YZ$ must be integers as there is no other fractional part that can there twice.

Thus, they are both integers, making $1$ a denominator. The only other possible denominator could be $2$ with its numerator being $4.$

Thus, we could get $$\dfrac 31 - \dfrac 42=1,$$ making the sum equal to $$4+3=7.$$

Thus, the correct answer is E.

Answer: C

Solution:

Since Harry has 3 sisters and 5 brothers, the family has 3 girls and 6 boys. Then, Harriet would be one or the girls, so she had $2$ sisters and $6$ brothers. Thus, the product of the number of brothers and sisters is $2\cdot 6=12.$

Thus, the correct answer is C.

Answer: A

Solution:

Directly evaluating, we get that: $$\begin{align*}&2\left(1-\dfrac{1}{2}\right) + 3\left(1-\dfrac{1}{3}\right) +\\ &4\left(1-\dfrac{1}{4}\right) + \cdots + 10\left(1-\dfrac{1}{10}\right)\\ &=(2-1)+\cdots +(10-1)\\ &=1+2 \cdots 8+9 \\ &= 45\end{align*}$$

Thus, the correct answer is A.

Answer: C

Solution:

We could extend the figure to the following:

Then, we could look at the bottom triangle that makes up a quarter of the figure.

Half of that area is the shaded area, so the entire shaded area is $\dfrac 18.$

Thus, the correct answer is C.

Answer: E

Solution:

Since $80\%$ of people are in both clubs, the number of people in both clubs is $0.8\cdot 15=12.$ This is $30\%$ of the math club, so the number of people in the math club is $\dfrac{12}{0.3}=40.$

Thus, the correct answer is E.

Answer: D

Solution:

The population in $2048,$ which is $50$ years after $1998,$ is $3^2\cdot 200=1800.$

Therefore, as $2048\approx 2050,$ the population in $2050$ is approximately $1800,$ which is approximately equal to $2000$

Thus, the correct answer is D.

Answer: B

Solution:

This would be the year the population is $30$ times as much as in $1998.$ This means the population triples approximately $3$ times, making the year approximately $3\cdot 25=75$ years after $1998.$ This would be $2073,$ so $2075$ is the best approximation.

Thus, the correct answer is B.

Answer: C

Solution:

The maximal population is $\dfrac{24900}{1.5}=16600.$ This is $83$ times as much as the population in $1998,$ so it would be about $4$ triples from $1998.$ That would be $25\cdot 4=100$ years.

Thus, the correct answer is C.

Answer: B

Solution:

When we undo the fold, the rectangle with the punched hole in the upper left is in the bottom right. The only such answer choice is B.

Thus, the correct answer is B.

Answer: A

Solution:

Tamika, with equal probability, can get one of $17,18,19.$

Carlos, with equal probability, can get one of $15,18,30.$

There is a $\frac 13$ probability that Carlos gets $15,$ which would always have Tamika having a higher number.

There is a $\frac 13$ probability that Carlos gets $18,$ which would have a $\dfrac 13$ probability that Tamika gets a higer number of $19.$

There is a $\frac 13$ probability that Carlos gets $30,$ which would always have Tamika never a higher number.

Therefore, the total probability is: $$\dfrac 13 + \dfrac 13\cdot \dfrac 13 = \dfrac 49.$$

Thus, the correct answer is A.

Answer: D

Solution:

Let the side length be $s.$ Then, folding $P$ to $R$ would give an isoceles right triangle with area $\frac 12{s^2}.$ Here, $Q$ and $S$ are across from each other, so when it is folded, the area is $\frac14{s^2}=9,$ making $s=6.$

As such, the area is $4\cdot 6=24.$

Thus, the correct answer is D.

Answer: B

Solution:

The entire $4 \times 4 \times 1$ bottom with a volume of $16$ is counted. Then, there are $48$ cubes left. There are $3$ layers, with each of being a $4 \times 4$ square. Only the $2 \times 2$ interior doesn't touch the outside, so each layer has $$4^2-2^2=12$$ cubes. This makes the total number of cubes equal to $$12\cdot 3+16=52.$$

Thus, the correct answer is B.

Answer: D

Solution:

The sequence starts with the following: $$98,49,44,22,11,$$$$6,54,27,22, \cdots$$ Thus, after the first four, it has a cycle of length $5.$ This makes the $98^{th}$ term equal to the $8^{th}$ term of the sequence, which is $27.$

Thus, the correct answer is D.

Answer: C

Solution:

The total number of triangles in the $n^{th}$ triangle is $n^2.$

The total number of shaded triangles in the $n^{th}$ triangle is the $n-1^{th}$ triangular number, which is: $$\dfrac{n^2-n}2.$$

This makes the ratio equal to: $$\dfrac{n-1}{2n} = \dfrac{7}{16}.$$

Thus, the correct answer is C.

Answer: E

Solution:

The only shaded squares are the triangular numbers (the $n^{th}$ triangular number is the sum of the first $n$ numbers). As such, we have to find the first triangular number such that every possible value $\mod 8$ is accounted for.

Note that the formula for triangular numbers is $$\dfrac{n^2+n}2.$$

Thus, we must find a triangular number for each value modulo $8.$ For $0\mod 8,$ the first one is $120,$ meaning the answer is at least $120.$ $120$ is the greatest answer choice available, so we know the answer must be $120.$ By inspection, we can also see that every other value modulo $8$ comes before anyways, but that is omitted here.

Thus, the correct answer is E.

Answer: D

Solution:

Since Toy doubles his money after the first two turns, he has $144 in the end. Then, he gives away 144-36=$108. Since this doubles Any and Jan's money, they had $108 before Toy gives them money. Thus, the total amount of money is $144+$108=$252.

Thus, the correct answer is D.