1986 AMC 8 Exam Problems

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1.

In July $1861,$ $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month?

$\dfrac{366}{31 \times 24}$

$\dfrac{366 \times 31}{24}$

$\dfrac{366 \times 24}{31}$

$\dfrac{31 \times 24}{366}$

$366 \times 31 \times 24$

Answer: A

Concepts:rate unit conversion

Difficulty rating: 660

Solution:

The average rainfall per hour equals the total rainfall divided by the total number of hours. July has $31$ days, or $31 \times 24$ hours, so the average is $\dfrac{366}{31 \times 24}$ inches per hour.

Thus, the correct answer is A .

2.

Which of the following numbers has the largest reciprocal?

$\dfrac13$

$\dfrac25$

$1$

$5$

$1986$

Answer: A

Concepts:fraction

Difficulty rating: 560

Solution:

A large positive number has a small reciprocal, and a small positive number has a large reciprocal. The smallest number listed is $\dfrac13,$ whose reciprocal $3$ is the largest.

Thus, the correct answer is A .

3.

The smallest sum one could get by adding three different numbers from the set $\{7, 25, -1, 12, -3\}$ is

$-3$

$-1$

$3$

$5$

$21$

Answer: C

Concepts:optimization

Difficulty rating: 660

Solution:

The three smallest numbers in the set are $-3, -1,$ and $7.$ Their sum is $-3 + (-1) + 7 = 3.$

Thus, the correct answer is C .

4.

The product $(1.8)(40.3 + .07)$ is closest to

$7$

$42$

$74$

$84$

$737$

Answer: C

Concepts:estimation

Difficulty rating: 730

Solution:

The second factor $40.3 + .07 = 40.37$ is about $40,$ and $1.8$ is about $2.$ A quick estimate is $2(40) - .2(40) = 80 - 8 = 72,$ so the product is closest to $74.$

Thus, the correct answer is C .

5.

A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end?

$10\!:\!00$ p.m.

midnight

$2\!:\!30$ a.m.

$4\!:\!40$ a.m.

$6\!:\!40$ a.m.

Answer: D

Concepts:date and time unit conversion

Difficulty rating: 730

Solution:

Since $1000$ minutes $= \dfrac{1000}{60}$ hours $= 16\dfrac23$ hours $= 16$ hours $40$ minutes, the contest ended $16$ hours $40$ minutes past noon, which is $4\!:\!40$ a.m.

Thus, the correct answer is D .

6.

What is the value of the following expression?

$\frac{2}{1 - \frac{2}{3}}$

$-3$

$-\dfrac43$

$\dfrac23$

$2$

$6$

Answer: E

Concepts:fraction order of operations

Difficulty rating: 660

Solution:

The denominator is $1 - \dfrac23 = \dfrac13,$ so the expression is $\dfrac{2}{1/3} = 2 \times 3 = 6.$

Thus, the correct answer is E .

7.

How many whole numbers are between $\sqrt{8}$ and $\sqrt{80}?$

$5$

$6$

$7$

$8$

$9$

Answer: B

Concepts:counting integers in a range perfect square

Difficulty rating: 820

Solution:

Since $\sqrt{8} \lt \sqrt{9} = 3$ and $\sqrt{80} \lt \sqrt{81} = 9,$ the whole numbers strictly between $\sqrt{8}$ and $\sqrt{80}$ are $3, 4, 5, 6, 7, 8.$ There are six of them.

Thus, the correct answer is B .

8.

In the multiplication shown, $B$ represents a digit. What is the value of $B?$

$\begin{array}{r} B2 \\ \times\, 7B \\ \hline 6396 \end{array}$

$3$

$5$

$6$

$7$

$8$

Answer: E

Concepts:cryptarithm units digit

Difficulty rating: 930

Solution:

The units digit of the product comes from $B \times 2,$ which ends in $6$ only when $B = 3$ or $B = 8.$ Since the product $6396$ exceeds $6000,$ $B$ must be $8:$ indeed $82 \times 78 = 6396.$

Thus, the correct answer is E .

9.

Using only the paths and the directions shown, how many different routes are there from $M$ to $N?$

$2$

$3$

$4$

$5$

$6$

Answer: E

Concepts:basic counting systematic listing

Difficulty rating: 960

Solution:

Following the arrows, the possible routes are $MADCN,$ $MACN,$ $MBADCN,$ $MBACN,$ $MBCN,$ and $MBN.$ There are six of them.

Thus, the correct answer is E .

10.

A picture $3$ feet across is hung in the center of a wall that is $19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture?

$1\dfrac12$

$8$

$9\dfrac12$

$16$

$22$

Answer: B

Concepts:symmetry

Difficulty rating: 730

Solution:

The picture leaves $19 - 3 = 16$ feet of wall, split equally into two gaps of $8$ feet each. So the nearest edge of the picture is $8$ feet from the end of the wall.

Thus, the correct answer is B .

11.

If $A * B$ means $\dfrac{A + B}{2},$ then $(3 * 5) * 8$ is

$6$

$8$

$12$

$16$

$30$

Answer: A

Concepts:custom operation

Difficulty rating: 820

Solution:

First, $3 * 5 = \dfrac{3 + 5}{2} = 4.$ Then $4 * 8 = \dfrac{4 + 8}{2} = 6.$

Thus, the correct answer is A .

12.

The table shown displays the grade distribution of the $30$ students in a mathematics class on the last two tests. For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2 (the circled entry). What percent of the students received the same grade on both tests?

$12\%$

$25\%$

$33\dfrac13\%$

$40\%$

$50\%$

Answer: D

Concepts:data and graph interpretation percentage

Difficulty rating: 860

Solution:

A student received the same grade on both tests exactly when counted on the main diagonal. Those entries are $2 + 4 + 5 + 1 + 0 = 12.$

So the fraction is $\dfrac{12}{30} = \dfrac{4}{10} = 40\%.$

Thus, the correct answer is D .

13.

The perimeter of the polygon shown is

$14$

$20$

$28$

$48$

cannot be determined from the information given

Answer: C

Concepts:perimeter rectangle

Difficulty rating: 960

Solution:

Because every angle is a right angle, sliding the edges shows that the horizontal edges together traverse the width twice and the vertical edges traverse the height twice. So the perimeter equals that of the full $8$ by $6$ rectangle, $2(8 + 6) = 28.$

The answer does not depend on exactly where the notch is cut.

Thus, the correct answer is C .

14.

If $200 \le a \le 400$ and $600 \le b \le 1200,$ then the largest value of the quotient $\dfrac{b}{a}$ is

$\dfrac32$

$3$

$6$

$300$

$600$

Answer: C

Concepts:inequality optimization

Difficulty rating: 820

Solution:

The quotient $\dfrac{b}{a}$ is largest with the biggest $b$ and smallest $a,$ giving $\dfrac{1200}{200} = 6.$

Thus, the correct answer is C .

15.

Sale prices at the Ajax Outlet Store are $50\%$ below original prices. On Saturdays an additional discount of $20\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $180?

$54

$72

$90

$108

$110

Answer: B

Concepts:percentage

Difficulty rating: 860

Solution:

The sale price is $50\%$ of $\$180,$ or $\$90.$ The Saturday price takes another $20\%$ off, leaving $80\%$ of $\$90,$ which is $\$72.$

Thus, the correct answer is B .

16.

A fast food chain sold $4.5$ million hamburgers in the spring, $5$ million in the summer, and $4$ million in the fall; the number sold in the winter is unknown. If exactly $25\%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter?

$2.5$

$3$

$3.5$

$4$

$4.5$

Answer: A

Concepts:percentage

Difficulty rating: 960

Solution:

If the fall sales of $4$ million are $25\%$ of the yearly total, then the yearly total is $16$ million.

The winter sales are $16 - (4.5 + 5 + 4) = 2.5$ million.

Thus, the correct answer is A .

17.

Let $o$ be an odd whole number and let $n$ be any whole number. Which of the following statements about the whole number $o^2 + no$ is always true?

it is always odd

it is always even

it is even only if $n$ is even

it is odd only if $n$ is odd

it is odd only if $n$ is even

Answer: E

Concepts:parity factoring

Difficulty rating: 1000

Solution:

Factor $o^2 + no = o(o + n).$ Because $o$ is odd, the product is odd exactly when $o + n$ is odd, which happens only when $n$ is even. When $n$ is odd, $o + n$ is even and the product is even.

So the number is odd only if $n$ is even.

Thus, the correct answer is E .

18.

A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ meters along the fence, including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area $36$ m by $60$ m?

$11$

$12$

$13$

$14$

$16$

Answer: B

Concepts:perimeter optimization fencepost counting

Difficulty rating: 1030

Solution:

The fewest posts are used when the wall serves as the longer $60$ meter side, so the fence covers two $36$ meter sides and one $60$ meter side, a path of $36 + 60 + 36 = 132$ meters.

Placing a post every $12$ meters, including both ends, uses $\dfrac{132}{12} + 1 = 12$ posts.

Thus, the correct answer is B .

19.

At the beginning of a trip, the mileage odometer read $56{,}200$ miles. The driver filled the gas tank with $6$ gallons of gasoline. During the trip, the driver filled the tank again with $12$ gallons of gasoline when the odometer read $56{,}560.$ At the end of the trip, the driver filled the tank again with $20$ gallons of gasoline; the odometer read $57{,}060.$ To the nearest tenth, what was the car's average miles-per-gallon for the entire trip?

$22.5$

$22.6$

$24.0$

$26.9$

$27.5$

Answer: D

Concepts:rate

Difficulty rating: 1060

Solution:

The trip was $57{,}060 - 56{,}200 = 860$ miles. The initial $6$ gallons only topped off the tank before the trip; the gas actually used during the trip is the $12 + 20 = 32$ gallons later added to replace it.

So the average is $\dfrac{860}{32} \approx 26.9$ miles per gallon.

Thus, the correct answer is D .

20.

The value of the expression

$\frac{(304)^5}{(29.7)(399)^4}$

is closest to

$.003$

$.03$

$.3$

$3$

$30$

Answer: D

Concepts:estimation exponent

Difficulty rating: 980

Solution:

Estimating, $\dfrac{300^5}{30 \cdot 400^4} = 10 \left(\dfrac{300}{400}\right)^4 = 10 \left(\dfrac34\right)^4 = 10 \cdot \dfrac{81}{256} \approx 3.$

Thus, the correct answer is D .

21.

Suppose one of the eight lettered identical squares is included with the four shaded squares in the T-shaped figure shown. How many of the resulting figures can be folded into a topless cubical box?

$2$

$3$

$4$

$5$

$6$

Answer: E

Concepts:net (3D geometry)casework

Difficulty rating: 1220

Solution:

The four shaded squares fold into four faces of an open box, and the added square supplies the fifth face. Picturing the folds, the squares $A, E, H, B, D,$ and $F$ each complete a valid topless box.

The squares $C$ and $G$ do not work, because folding would force four faces to meet at a single corner. That leaves $6$ valid figures.

Thus, the correct answer is E .

22.

Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's?

Alan, Beth

Beth, Carlos

Carlos, Diana

Alan, Diana

Beth, Diana

Answer: C

Concepts:logical deduction

Difficulty rating: 1060

Solution:

The statements form a chain: Alan's A forces Beth's, Beth's forces Carlos's, and Carlos's forces Diana's. If Alan got an A, all four would; if Beth got an A, three would.

The only way to have exactly two A's is for them to be the last two in the chain, Carlos and Diana.

Thus, the correct answer is C .

23.

The large circle has diameter $AC.$ The two small circles have their centers on $AC$ and just touch at $O,$ the center of the large circle. If each small circle has radius $1,$ what is the value of the ratio of the area of the shaded region to the area of one of the small circles?

between $\dfrac12$ and $1$

$1$

between $1$ and $\dfrac32$

between $\dfrac32$ and $2$

cannot be determined from the information given

Answer: B

Concepts:circle area area decomposition

Difficulty rating: 1090

Solution:

The large circle has radius $2,$ so its area is $\pi(2)^2 = 4\pi,$ and each small circle has area $\pi.$

By symmetry the shaded region is half the difference of the areas: $\dfrac12(4\pi - 2\pi) = \pi.$ The ratio of this to one small circle's area $\pi$ is $1.$

Thus, the correct answer is B .

24.

The $600$ students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of the three lunch groups. The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately

$\dfrac{1}{27}$

$\dfrac19$

$\dfrac18$

$\dfrac16$

$\dfrac13$

Answer: B

Concepts:basic probability independent events

Difficulty rating: 1120

Solution:

Whatever group Al is in, Bob joins that same group with probability about $\dfrac13,$ and Carol joins it with probability about $\dfrac13.$

So all three share a group with probability about $\dfrac13 \times \dfrac13 = \dfrac19.$

Thus, the correct answer is B .

25.

Which of the following sets of whole numbers has the largest average?

multiples of $2$ between $1$ and $101$

multiples of $3$ between $1$ and $101$

multiples of $4$ between $1$ and $101$

multiples of $5$ between $1$ and $101$

multiples of $6$ between $1$ and $101$

Answer: D

Concepts:arithmetic sequence mean

Difficulty rating: 1060

Solution:

For a set of evenly spaced whole numbers, the average is the average of the smallest and largest. The averages are: A: $\dfrac{2 + 100}{2} = 51,$ B: $\dfrac{3 + 99}{2} = 51,$ C: $\dfrac{4 + 100}{2} = 52,$ D: $\dfrac{5 + 100}{2} = 52.5,$ E: $\dfrac{6 + 96}{2} = 51.$

The largest average is $52.5,$ from the multiples of $5.$

Thus, the correct answer is D .