1998 AMC 8

For $x = 7,$ which of the following is the smallest?

If $~\begin{array}{r|l}a&b \\ \hline c&d\end{array} = a\cdot d-b\cdot c,$ what is the value of $~\begin{array}{r|l}3&4 \\ \hline 1&2\end{array}~?$

What is the value of the following expression? $$
\dfrac{\frac{3}{8}+\frac{7}{8}}{\frac{4}{5}}
$$

How many triangles are in this figure? (Some triangles may overlap other triangles.)

Which of the following numbers is largest?

Dots are spaced one unit apart, horizontally and vertically. The number of square units enclosed by the polygon is

$100\times 19.98\times 1.998\times 1000=$

A child’s wading pool contains $200$ gallons of water. If water evaporates at the rate of $0.5$ gallons per day and no other water is added or removed, how many gallons of water will be in the pool after $30$ days?

For a sale, a store owner reduces the price of a $\$10$ scarf by $20\%.$ Later the price is lowered again, this time by one-half the reduced price. The price is now

Each of the letters $W,$ $X,$ $Y,$ and $Z$ represents a different integer in the set $\{1,2,3,4\},$ but not necessarily in that order. They satisfy $$
\dfrac{W}{X}-\dfrac{Y}{Z}=1
$$ What is the sum of $W$ and $Y?$

Harry has $3$ sisters and $5$ brothers. His sister Harriet has $S$ sisters and $B$ brothers. What is the product of $S$ and $B?$

What is the value of the following expression? $$
\begin{aligned} &2\left(1-\dfrac{1}{2}\right)+3\left(1-\dfrac{1}{3}\right) \\ &\quad {}+4\left(1-\dfrac{1}{4}\right)+\cdots \\ &\quad {}+10\left(1-\dfrac{1}{10}\right) \end{aligned}
$$

What is the ratio of the area of the shaded square to the area of the large square? (The figure is drawn to scale.)

At Annville Junior High School, $30\%$ of the students in the Math Club are in the Science Club, and $80\%$ of the students in the Science Club are in the Math Club. There are $15$ students in the Science Club. How many students are in the Math Club?

Problems $15,$ $16,$ and $17$ all refer to the following:

Don’t Crowd The Isles

In the very center of the Irenic Sea lie the beautiful Nisos Isles. In $1998$ the number of people on these islands is only $200,$ but the population triples every $25$ years. Queen Irene has decreed that there must be at least $1.5$ square miles for every person living in the Isles. The total area of the Nisos Isles is $24{,}900$ square miles.

Estimate the population of Nisos in the year $2050.$

Estimate the year in which the population of Nisos will be approximately $6000.$

In how many years, approximately, from $1998$ will the population of Nisos be as much as Queen Irene has proclaimed that the islands can support?

As indicated by the diagram below, a rectangular piece of paper is folded bottom to top, then left to right, and finally, a hole is punched at $X.$ What does the paper look like when unfolded?



**(A)**

**(B)**

**(C)**

**(D)**

**(E)**

Tamika selects two different numbers at random from the set $\{8,9,10\}$ and adds them. Carlos takes two different numbers at random from the set $\{3,5,6\}$ and multiplies them. What is the probability that Tamika’s result is greater than Carlos’ result?

Let $PQRS$ be a square piece of paper. $P$ is folded onto $R$ and then $Q$ is folded onto $S.$ The area of the resulting figure is $9$ square inches. Find the perimeter of square $PQRS.$