1987 AMC 8 Exam Solutions

1.

What is $.4 + .02 + .006?$

$.012$

$.066$

$.12$

$.24$

$.426$

Concepts:decimal place value

Difficulty rating: 450

Solution:

Adding place by place, $.4 + .02 + .006 = .426.$

Thus, the correct answer is E .

2.

What is $\dfrac{2}{25}$ written as a decimal?

$.008$

$.08$

$.8$

$1.25$

$12.5$

Concepts:fraction decimal

Difficulty rating: 560

Solution:

Multiplying top and bottom by $4$ gives $\dfrac{2}{25} = \dfrac{8}{100} = .08.$

Thus, the correct answer is B .

3.

What is the value of

$2(81 + 83 + 85 + 87 + 89 + 91 + 93 + 95 + 97 + 99)?$

$1600$

$1650$

$1700$

$1750$

$1800$

Concepts:arithmetic sequence whole number operations

Difficulty rating: 660

Solution:

The ten evenly spaced numbers have average $90,$ so their sum is $10 \times 90 = 900.$

Doubling gives $2 \times 900 = 1800.$

Thus, the correct answer is E .

4.

Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle?

$90$

$100$

$125$

$180$

$250$

Concepts:ratio and proportion

Difficulty rating: 560

Solution:

A right angle is a quarter of a full circle, so it contains $\dfrac14 \times 500 = 125$ clerts.

Thus, the correct answer is C .

5.

A rectangular region is $0.4$ m long and $0.22$ m wide. What is its area, in square meters?

$0.088 \text{ m}^2$

$0.62 \text{ m}^2$

$0.88 \text{ m}^2$

$1.24 \text{ m}^2$

$4.22 \text{ m}^2$

Concepts:area decimal

Difficulty rating: 660

Solution:

The area is $0.4 \times 0.22 = 0.088$ square meters.

Thus, the correct answer is A .

6.

The smallest product one could obtain by multiplying two numbers in the set $\{ -7, -5, -1, 1, 3 \}$ is

$-35$

$-21$

$-15$

$-1$

$3$

Concepts:optimization

Difficulty rating: 800

Solution:

A negative product comes from multiplying a negative number by a positive number, and it is most negative when both factors are largest in size.

The most negative product is $(-7)(3) = -21.$

Thus, the correct answer is B .

7.

The large cube shown is made up of $27$ identical sized smaller cubes. For each face of the large cube, the opposite face is shaded the same way. The total number of smaller cubes that must have at least one face shaded is

$10$

$16$

$20$

$22$

$24$

Concepts:cube geometry complementary counting

Difficulty rating: 1200

Solution:

Count the cubes with no shaded face and subtract from $27.$ A small cube is unshaded exactly when every one of its exposed squares is blank.

The three patterns are: the top and bottom faces show only their center square shaded; one pair of opposite side faces shows the four corners and the center shaded; the remaining pair shows the four edge-midpoints shaded.

Exactly $7$ small cubes avoid every shaded square: the one hidden cube at the very center of the block; the two cubes at the centers of the edge-midpoints faces, whose only exposed square is that blank center; and the four cubes at the midpoints of the edges where a corners-and-center face meets the top or bottom face, since there each exposed square is a blank edge cell.

Hence $27 - 7 = 20$ cubes have at least one shaded face.

Thus, the correct answer is C .

8.

In the addition problem below, $A$ and $B$ are nonzero digits:

$\begin{array}{r} 9876 \\ A32 \\ +B1 \\ \hline \end{array}$

How many digits (not necessarily different) are in the sum of the three whole numbers?

$4$

$5$

$6$

$9$

depends on the values of $A$ and $B$

Concepts:place value bounding to limit cases

Difficulty rating: 930

Solution:

Since $A$ and $B$ are at least $1,$ the sum is at least $9876 + 132 + 11 = 10{,}019,$ which has $5$ digits.

At the other extreme, with $A = B = 9$ the sum is $9876 + 932 + 91 = 10{,}899,$ still $5$ digits. So the sum always has exactly $5$ digits.

Thus, the correct answer is B .

9.

When finding the sum

$\dfrac12 + \dfrac13 + \dfrac14 + \dfrac15 + \dfrac16 + \dfrac17,$

what is the least common denominator used?

$120$

$210$

$420$

$840$

$5040$

Concepts:least common multiple

Difficulty rating: 800

Solution:

The least common multiple must include $4 = 2^2,$ $3,$ $5,$ and $7$ (the factor $6 = 2 \times 3$ is already covered).

So it is $4 \times 3 \times 5 \times 7 = 420.$

Thus, the correct answer is C .

10.

What is the value of

$4(299) + 3(299) + 2(299) + 298?$

$2889$

$2989$

$2991$

$2999$

$3009$

Concepts:distributive property

Difficulty rating: 800

Solution:

The first three terms combine to $(4 + 3 + 2)(299) = 9 \times 299 = 2691.$

Adding the last term gives $2691 + 298 = 2989.$

Thus, the correct answer is B .

11.

The sum $2\dfrac17 + 3\dfrac12 + 5\dfrac{1}{19}$ is between which two values?

$10$ and $10\dfrac12$

$10\dfrac12$ and $11$

$11$ and $11\dfrac12$

$11\dfrac12$ and $12$

$12$ and $12\dfrac12$

Concepts:fraction estimation

Difficulty rating: 860

Solution:

The whole-number parts sum to $10.$ The fractional parts are $\dfrac12 + \dfrac17 + \dfrac{1}{19}.$

Since $\dfrac17 + \dfrac{1}{19}$ is a small positive amount, the fractional total is more than $\dfrac12$ but well under $1.$ So the sum lies between $10\dfrac12$ and $11.$

Thus, the correct answer is B .

12.

What fraction of the large $12$ by $18$ rectangular region is shaded?

$\dfrac{1}{108}$

$\dfrac{1}{18}$

$\dfrac{1}{12}$

$\dfrac{2}{9}$

$\dfrac{1}{3}$

Concepts:area fraction

Difficulty rating: 930

Solution:

The whole region has area $12 \times 18 = 216.$ The shaded region is $3$ wide and $6$ tall, with area $3 \times 6 = 18.$

So the shaded fraction is $\dfrac{18}{216} = \dfrac{1}{12}.$ Equivalently, it is $\dfrac13$ of $\dfrac14$ of the whole.

Thus, the correct answer is C .

13.

Which of the following fractions has the largest value?

$\dfrac{3}{7}$

$\dfrac{4}{9}$

$\dfrac{17}{35}$

$\dfrac{100}{201}$

$\dfrac{151}{301}$

Concepts:fraction

Difficulty rating: 930

Solution:

For $\dfrac37, \dfrac49, \dfrac{17}{35},$ and $\dfrac{100}{201},$ each numerator is less than half its denominator, so each is less than $\dfrac12.$

For $\dfrac{151}{301},$ the numerator $151$ is more than half of $301,$ so this fraction exceeds $\dfrac12$ and is the largest.

Thus, the correct answer is E .

14.

A computer can do $10{,}000$ additions per second. How many additions can it do in one hour?

$6$ million

$36$ million

$60$ million

$216$ million

$360$ million

Concepts:rate unit conversion

Difficulty rating: 730

Solution:

One hour has $3600$ seconds, so the computer does $10{,}000 \times 3600 = 36{,}000{,}000$ additions.

That is $36$ million.

Thus, the correct answer is B .

15.

A sale ad read: "Buy three tires at the regular price and get the fourth tire for $3." Sam paid $240 for a set of four tires at the sale. What was the regular price of one tire?

$59.25

$60

$70

$79

$80

Concepts:work backwards

Difficulty rating: 860

Solution:

The three regular-price tires cost $\$240 - \$3 = \$237.$

So one tire costs $\dfrac{\$237}{3} = \$79.$

Thus, the correct answer is D .

16.

Joyce made $12$ of her first $30$ shots in the first three games of the basketball season, so her seasonal shooting average was $40\%.$ In her next game, she took $10$ shots and raised her seasonal shooting average to $50\%.$ How many of these $10$ shots did she make?

$2$

$3$

$5$

$6$

$8$

Concepts:percentage mean

Difficulty rating: 960

Solution:

After the fourth game she has taken $40$ shots. A $50\%$ average means she made $\tfrac12 \times 40 = 20$ shots in total.

She had already made $12,$ so she made $20 - 12 = 8$ of the last $10.$

Thus, the correct answer is E .

17.

Abby, Bret, Carl, and Dana are seated in a row of four seats numbered $1$ to $4.$ Joe looks at them and says: "Bret is next to Carl." "Abby is between Bret and Carl." However, each one of Joe's statements is false. Bret is actually sitting in seat $3.$ Who is sitting in seat $2?$

Abby

Bret

Carl

Dana

There is not enough information to be sure

Concepts:logical deduction

Difficulty rating: 1060

Solution:

Bret is in seat $3.$ Since "Bret is next to Carl" is false, Carl is not in seat $2$ or seat $4,$ so Carl must be in seat $1.$

The seat between Bret (seat $3$ ) and Carl (seat $1$ ) is seat $2.$ Since "Abby is between Bret and Carl" is false, Abby is not in seat $2,$ so Abby must be in seat $4.$

That leaves Dana in seat $2.$

Thus, the correct answer is D .

18.

Half the people in a room left. One third of those remaining started to dance. There were then $12$ people who were not dancing. What was the original number of people in the room?

$24$

$30$

$36$

$42$

$72$

Concepts:fraction work backwards

Difficulty rating: 960

Solution:

Of those remaining, $\dfrac13$ danced, so $\dfrac23$ did not. Thus $\dfrac23$ of the remaining people equals $12,$ giving $18$ remaining.

Those $18$ are half the original group, so the room started with $2 \times 18 = 36$ people.

Thus, the correct answer is C .

19.

A calculator has a squaring key that replaces the number currently displayed with its square. For example, if the display reads $3$ and the squaring key is pressed, the display becomes $9.$ If the display reads $2,$ how many times must the squaring key be pressed to produce a displayed number greater than $500?$

$4$

$5$

$8$

$9$

$250$

Concepts:exponent

Difficulty rating: 1030

Solution:

Pressing the key repeatedly gives $2 \to 4 \to 16 \to 256 \to 65536.$

Since $256 \lt 500 \lt 65536,$ the display first exceeds $500$ on the fourth press.

Thus, the correct answer is A .

20.

Consider the statement: "If a whole number $n$ is not prime, then the whole number $n - 2$ is not prime." Which of the following values of $n$ shows this statement to be false?

$9$

$12$

$13$

$16$

$23$

Concepts:counterexample prime

Difficulty rating: 1010

Solution:

A counterexample needs $n$ not prime but $n - 2$ prime. For $n = 9,$ the number $9$ is not prime while $9 - 2 = 7$ is prime, so it breaks the statement.

The others fail to be counterexamples: $12 - 2 = 10$ and $16 - 2 = 14$ are not prime, while $13$ and $23$ are themselves prime.

Thus, the correct answer is A .

21.

Suppose $n^\ast$ means $\dfrac1n,$ the reciprocal of $n.$ For example, $5^\ast = \dfrac15.$ How many of the following four statements are true?

(i) $3^\ast + 6^\ast = 9^\ast$ ; (ii) $6^\ast - 4^\ast = 2^\ast$ ; (iii) $2^\ast \cdot 6^\ast = 12^\ast$ ; (iv) $10^\ast \div 2^\ast = 5^\ast$

$0$

$1$

$2$

$3$

$4$

Concepts:custom operation fraction

Difficulty rating: 1030

Solution:

(i) $\dfrac13 + \dfrac16 = \dfrac12,$ but $9^\ast = \dfrac19,$ so false. (ii) $\dfrac16 - \dfrac14 = -\dfrac{1}{12},$ but $2^\ast = \dfrac12,$ so false.

(iii) $\dfrac12 \cdot \dfrac16 = \dfrac{1}{12} = 12^\ast,$ true. (iv) $\dfrac{1}{10} \div \dfrac12 = \dfrac15 = 5^\ast,$ true.

So exactly $2$ of the statements are true.

Thus, the correct answer is C .

22.

$ABCD$ is a rectangle, $D$ is the center of the circle, and $B$ is on the circle. If $AD = 4$ and $CD = 3,$ then the area of the shaded region is between which two values?

$4$ and $5$

$5$ and $6$

$6$ and $7$

$7$ and $8$

$8$ and $9$

Concepts:Pythagorean Theorem sector

Difficulty rating: 1120

Solution:

Since $AD = 4$ and $CD = 3,$ the diagonal $DB = \sqrt{3^2 + 4^2} = 5,$ which is the radius.

The shaded region is the quarter circle at $D$ minus the rectangle: $\dfrac14 \pi (5)^2 - (3)(4) = \dfrac{25\pi}{4} - 12 \approx 19.6 - 12 = 7.6.$

This lies between $7$ and $8.$

Thus, the correct answer is D .

23.

In $1980,$ the U.S. Black population (in millions) was $5$ in the Northeast, $5$ in the Midwest, $15$ in the South, and $2$ in the West. To the nearest percent, what percent of the U.S. Black population lived in the South?

$20\%$

$25\%$

$40\%$

$56\%$

$80\%$

Concepts:percentage

Difficulty rating: 820

Solution:

The total Black population is $5 + 5 + 15 + 2 = 27$ million.

The South's share is $\dfrac{15}{27} = \dfrac59 \approx 55.6\%,$ which rounds to $56\%.$

Thus, the correct answer is D .

24.

A multiple choice examination consists of $20$ questions. The scoring is $+5$ for each correct answer, $-2$ for each incorrect answer, and $0$ for each unanswered question. John's score on the examination is $48.$ What is the maximum number of questions he could have answered correctly?

$9$

$10$

$11$

$12$

$16$

Concepts:linear equation optimization parity

Difficulty rating: 1150

Solution:

Let $c$ be the number correct and $w$ the number wrong, so $5c - 2w = 48.$ Then $2w = 5c - 48,$ which requires $c$ to be even.

Trying $c = 14$ gives $w = 11,$ but $c + w = 25 \gt 20,$ too many. Trying $c = 12$ gives $w = 6$ and $c + w = 18 \le 20,$ which works. So the maximum is $12.$

Thus, the correct answer is D .

25.

Ten balls numbered $1$ to $10$ are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. What is the probability that the sum of the two numbers on the balls removed is even?

$\dfrac{4}{9}$

$\dfrac{9}{19}$

$\dfrac{1}{2}$

$\dfrac{10}{19}$

$\dfrac{5}{9}$

Concepts:basic probability parity

Difficulty rating: 1090

Solution:

The sum is even when both balls are odd or both are even. There are $\binom{5}{2} = 10$ all-odd pairs and $\binom{5}{2} = 10$ all-even pairs, for $20$ favorable pairs.

The total number of pairs is $\binom{10}{2} = 45,$ so the probability is $\dfrac{20}{45} = \dfrac49.$

Thus, the correct answer is A .