1997 AMC 8 Exam Problems
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Used with permission of the Mathematical Association of America.
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1.
Answer: C
Solution(s):
Converting all the fractions to decimals, we get
Thus, C is the correct answer.
2.
Ahn chooses a two-digit integer, subtracts it from and doubles the result. What is the largest number Ahn can get?
Answer: D
Solution(s):
To get the largest number, we would want to subtract the smallest number possible from
The smallest two-digit number is Having Ahn choose this number will give us as the final result.
Thus, D is the correct answer.
3.
Which of the following numbers is the largest?
Answer: B
Solution(s):
We have that all the tenths place digits are the same, so we then look at the hundredths digit.
The largest hundredths digit is so we can limit the answer choices to the ones with this value.
The largest thousandths digit is which is achieved by
Thus, B is the correct answer.
4.
Julie is preparing a speech for her class. Her speech must last between one-half hour and three-quarters of an hour. The ideal rate of speech is words per minute. If Julie speaks at the ideal rate, which of the following number of words would be an appropriate length for her speech?
Answer: E
Solution(s):
One-half hour is minutes, in which Julie can speak words. Three-quarters of an hour is minutes, in which Julie can speak words. The only answer choice in between these two values is
Thus, E is the correct answer.
5.
There are many two-digit multiples of but only two of the multiples have a digit sum of The sum of these two multiples of is
Answer: A
Solution(s):
Listing out all the two-digit multiples of we get We have that and are the two multiples whose digits add to
The sum of these numbers is
Thus, A is the correct answer.
6.
In the number the value of the place occupied by the digit is how many times as great as the value of the place occupied by the digit
Answer: C
Solution(s):
Note that each digit has ten times the value of the digit to its right.
The place occupied by is spaces to the right of the place occupied by
This means that it is times as a great.
Thus, C is the correct answer.
7.
The area of the smallest square that will contain a circle of radius is
Answer: D
Solution(s):
We can inscribe the circle inside the square so that it is tangent to the midpoints of each side of the square.
This means that the side length of the square is two times the radius of the circle, making it
Then the area of the square is
Thus, D is the correct answer.
8.
Walter gets up at a.m., catches the school bus at a.m., has classes that last minutes each, has minutes for lunch, and has hours additional time at school. He takes the bus home and arrives at p.m. How many minutes has he spent on the bus?
Answer: B
Solution(s):
There are hours between the time Walter catches the school bus and arrives at home.
This is a total of minutes.
The total time Walter spends at school is minutes.
This means that Walter spends minutes on the bus.
Thus, B is the correct answer.
9.
Three students, with different names, line up single file. What is the probability that they are in alphabetical order from front-to-back?
Answer: C
Solution(s):
There are options for the person in front. Then, there are options for the person in the middle.
This leaves choice for the person at the end. There are ways for the people to line up.
Only one of these lines is correct, and the probability it occurs is
Thus, C is the correct answer.
10.
What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.
Answer: C
Solution(s):
We can split the square into unit squares. The number of black squares is
The fraction of the square that is shaded is
Thus, C is the correct answer.
11.
Let mean the number of whole number divisors of For example, because has two divisors, and Find the value of
Answer: A
Solution(s):
We know that is prime, which means that it only has divisors.
The prime factorization of is Recall that the number of divisors a number has is the product of all the exponents plus one in the prime factorization.
Here, that product would be
Then We have the prime factorization of is This also has divisors.
Thus, A is the correct answer.
12.
Find
Answer: D
Solution(s):
We have that using the fact that the interior angles of a triangle add to
This tells us that since and are supplementary.
Finally,
Thus, D is the correct answer.
13.
Three bags of jelly beans contain and beans. The ratios of yellow beans to all beans in each of these bags are and respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?
Answer: A
Solution(s):
There are yellow jelly beans in the first bag, yellow jelly beans in the second bag, and yellow jelly beans in the third bag.
The total number of yellow jelly beans is and the total number of jelly beans is The ratio of yellow jelly beans to all the beans is
Thus, A is the correct answer.
14.
There is a set of five positive integers whose average (mean) is whose median is and whose only mode is What is the difference between the largest and smallest integers in the set?
Answer: D
Solution(s):
The sum of all the numbers in the list is
The only mode is which means that there are guaranteed two s.
The sum of the numbers is up to
The other two numbers must then add to They are both positive and not equal, leaving the only possible two numbers as
The desired difference is then
Thus, D is the correct answer.
15.
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
Answer: B
Solution(s):
Let be the side length of the large square. Then we can find the side length of the inner square via from the Pythagorean Theorem.
The area of the larger square is and that of the inner square is
The ratio of the areas is then
Thus, B is the correct answer.
16.
Penni Precisely buys $ 100 worth of stock in each of three companies: Alabama Almonds, Boston Beans, and California Cauliflower. After one year, AA was up BB was down and CC was unchanged. For the second year, AA was down from the previous year, BB was up from the previous year, and CC was unchanged. If and are the final values of the stock, then
Answer: E
Solution(s):
After the first year, AA's stock's worth goes up to $ 100 \cdot 1.2 = $ 120 and BB's goes down to $ 100 \cdot .75 = $ 75.
After the second year, AA's stock is worth $ 120 \cdot .8 = $ 96 and BB's is worth $ 75 \cdot 1.25 = $ 93.75.
CC's stock worth remains the same, so the ordering of the stock worths is now
Thus, E is the correct answer.
17.
A cube has eight vertices (corners) and twelve edges. A segment, such as which joins two vertices not joined by an edge is called a diagonal. Segment is also a diagonal. How many diagonals does a cube have?
Answer: E
Solution(s):
Each face has two diagonals connecting each of the two pairs of opposite vertices.
Also, for each vertex, there is one corresponding vertex that lies opposite it on the cube.
There are then interior space diagonals in the cube.
The total number of diagonals is then
Thus, E is the correct answer.
18.
At the grocery store last week, small boxes of facial tissue were priced at boxes for $5. This week they are on sale at boxes for $4. The percent decrease in the price per box during the sale was closest to
Answer: B
Solution(s):
Originally, each box is worth $ 5 \div 4 = $ 1.25.
Now, each box is worth $ 4 \div 5 = $ .8.
The percent decrease is then
Thus, B is the correct answer.
19.
If the product what is the sum of and
Answer: D
Solution(s):
Note that the numerator of each fraction cancels with the denominator of the fraction to its right.
We can then cancel out all these terms to get a final equation of
The desired sum is then
Thus, D is the correct answer.
20.
A pair of -sided dice have sides numbered through Each side has the same probability (chance) of landing face up. The probability that the product of the two numbers that land face-up exceeds is
Answer: A
Solution(s):
We can case on the value of the first die. If its value is then it is impossible for the product be greater than
If it is a then the other dice has to roll an otherwise the product is less than
If the first die is a or then other die has to be at least a giving possibilities.
Finally, if the first roll is an the other die must roll at least a giving us more possibilities.
The total number of working pairs is and the total number of pairs is The desired probability is then
Thus, A is the correct answer.
21.
Each corner cube is removed from this cube. The surface area of the remaining figure is
Answer: D
Solution(s):
Note that there is one unit cube between any pair of corner cubes, so the removal of each does not affect the others.
When we remove a corner, we are losing three unit squares. We, however, gain these back from the three faces that get uncovered.
This means that removing a corner cube does not change the surface area. The surface area of the original cube is square centimeters.
Thus, D is the correct answer.
22.
A two-inch cube of silver weighs pounds and is worth $ 200. How much is a three-inch cube of silver worth?
$300
$375
$450
$560
$675
Answer: E
Solution(s):
The two-inch cube consists of unit cubes. Each of these unit cubes is worth dollars. To form a three-inch cube, you need unit cubes. This means that it is worth dollars.
Thus, E is the correct answer.
23.
There are positive integers that have these properties:
I. the sum of the squares of their digits is and
II. each digit is larger than the one to its left.
The product of the digits of the largest integer with both properties is
Answer: C
Solution(s):
Note that if the number has digits, then the sum of the squares of the digits is at least which violates the first condition.
This means that we should aim to find the largest four digit number that satisfies the properties.
Let the four digit number be where We must have that since otherwise
If then which means that the sum of the digits is greater than
Then if we have that and work. If then The only option is which causes us to end up with a -digit number.
If then which does not satisfy the first condition. The only viable option is then The product of the digits is then
Thus, C is the correct answer.
24.
Diameter is divided at in the ratio The two semicircles, and divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is
Answer: C
Solution(s):
WLOG, let Then and Then the area of the semicircle is
We also have that the area of semicircle is
Then the area of the upper shaded region is
Subtracting this from the area of the total circle gives us that the area of the lower region is
The desired ratio is then
Thus, C is the correct answer.
25.
All of the even numbers from to inclusive, excluding those ending in are multiplied together. What is the rightmost digit (the units digit) of the product?
Answer: D
Solution(s):
We only care about the units digit, which means that the tens digits don't matter.
Then we have groups of
The units digit of each group is We now need to find the units digit of
The units digit of is This means we only need to find the units digit of
Note that every power of always ends in a (e.g. ).
Thus, D is the correct answer.