1985 AMC 8 Exam Problems

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1.

What is the value of the following product?

$\dfrac{3 \times 5}{9 \times 11} \times \dfrac{7 \times 9 \times 11}{3 \times 5 \times 7}$

$1$

$0$

$49$

$\dfrac{1}{49}$

$50$

Answer: A

Concepts:fraction

Difficulty rating: 560

Solution:

Combining the two fractions, the numerator is $3 \cdot 5 \cdot 7 \cdot 9 \cdot 11$ and the denominator is $9 \cdot 11 \cdot 3 \cdot 5 \cdot 7.$ These are the same product.

So the value is $1.$

Thus, the correct answer is A .

2.

What is the value of $90 + 91 + 92 + \cdots + 98 + 99?$

$845$

$945$

$1005$

$1025$

$1045$

Answer: B

Concepts:arithmetic sequence pairing and grouping

Difficulty rating: 450

Solution:

There are $10$ terms. The average of the first and last is $\dfrac{90 + 99}{2} = 94.5.$

So the sum is $10 \cdot 94.5 = 945.$

Thus, the correct answer is B .

3.

What is the value of the following expression?

$\dfrac{10^7}{5 \times 10^4}$

$0.002$

$0.2$

$20$

$200$

$2000$

Answer: D

Concepts:exponent

Difficulty rating: 560

Solution:

Since $\dfrac{10^7}{10^4} = 10^3 = 1000,$ the expression is $\dfrac{1000}{5} = 200.$

Thus, the correct answer is D .

4.

The area of polygon $ABCDEF,$ in square units, is

$24$

$30$

$46$

$66$

$74$

Answer: C

Concepts:area area decomposition

Difficulty rating: 820

Solution:

Completing the figure to the full $6 \times 9$ rectangle gives an area of $54.$ The piece removed from the lower-left corner is a rectangle measuring $2$ by $4,$ with area $8.$

So the polygon's area is $54 - 8 = 46.$

Thus, the correct answer is C .

5.

The grades in a mathematics class for the last grading period were: $5$ students earned an A, $4$ earned a B, $3$ earned a C, $3$ earned a D, and $5$ earned an F. If A, B, C, and D are satisfactory grades, what fraction of the grades are satisfactory?

$\dfrac{1}{2}$

$\dfrac{2}{3}$

$\dfrac{3}{4}$

$\dfrac{4}{5}$

$\dfrac{9}{10}$

Answer: C

Concepts:data and graph interpretation fraction

Difficulty rating: 660

Solution:

The satisfactory grades number $5 + 4 + 3 + 3 = 15,$ and the total number of grades is $15 + 5 = 20.$

So the fraction that is satisfactory is $\dfrac{15}{20} = \dfrac{3}{4}.$

Thus, the correct answer is C .

6.

A ream of paper containing $500$ sheets is $5$ cm thick. Approximately how many sheets of this type of paper would there be in a stack $7.5$ cm high?

$250$

$550$

$667$

$750$

$1250$

Answer: D

Concepts:ratio and proportion

Difficulty rating: 560

Solution:

Since $7.5$ cm is $1.5$ times $5$ cm, the stack holds $1.5$ times as many sheets.

That is $1.5 \times 500 = 750$ sheets.

Thus, the correct answer is D .

7.

A "stair-step" figure is made up of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. The number of black squares in the $37$ th row is

$34$

$35$

$36$

$37$

$38$

Answer: C

Concepts:pattern recognition basic counting

Difficulty rating: 860

Solution:

Row $n$ contains $2n - 1$ squares. Because both ends are white and the colors alternate, each row has one more white square than black square, so the number of black squares is $n - 1.$

For the $37$ th row, this is $37 - 1 = 36.$

Thus, the correct answer is C .

8.

If $a = -2,$ the largest number in the set

$\left\{ -3a,\ 4a,\ \dfrac{24}{a},\ a^2,\ 1 \right\}$

is

$-3a$

$4a$

$\dfrac{24}{a}$

$a^2$

$1$

Answer: A

Concepts:substitution

Difficulty rating: 730

Solution:

Substituting $a = -2$ gives the values $-3a = 6,$ $4a = -8,$ $\dfrac{24}{a} = -12,$ $a^2 = 4,$ and $1.$

The largest of these is $6,$ which is $-3a.$

Thus, the correct answer is A .

9.

What is the value of the product of the $9$ factors below?

$\left(1 - \tfrac12\right)\left(1 - \tfrac13\right)\left(1 - \tfrac14\right) \cdots \left(1 - \tfrac{1}{10}\right)$

$\dfrac{1}{10}$

$\dfrac{1}{9}$

$\dfrac{1}{2}$

$\dfrac{10}{11}$

$\dfrac{11}{2}$

Answer: A

Concepts:telescoping fraction

Difficulty rating: 860

Solution:

Each factor $1 - \dfrac1k$ equals $\dfrac{k-1}{k},$ so the product is

$\dfrac12 \cdot \dfrac23 \cdot \dfrac34 \cdots \dfrac{9}{10}.$

Every numerator cancels the previous denominator, leaving $\dfrac{1}{10}.$

Thus, the correct answer is A .

10.

What fraction lies halfway between $\dfrac15$ and $\dfrac13$ on the number line?

$\dfrac{1}{4}$

$\dfrac{2}{15}$

$\dfrac{4}{15}$

$\dfrac{53}{200}$

$\dfrac{8}{15}$

Answer: C

Concepts:mean fraction

Difficulty rating: 730

Solution:

The midpoint of $\dfrac15$ and $\dfrac13$ is their average.

$\dfrac{\frac15 + \frac13}{2} = \dfrac{\frac{8}{15}}{2} = \dfrac{4}{15}.$

Thus, the correct answer is C .

11.

A piece of paper containing six joined squares labeled as shown is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $X$ is

$Z$

$U$

$V$

$W$

$Y$

Answer: E

Concepts:net (3D geometry)cube geometry

Difficulty rating: 920

Solution:

Fold the net so that $X$ is the bottom face. Then $U,$ $V,$ $W,$ and $Z$ wrap around to become the four side faces.

The only remaining square, $Y,$ becomes the top face, which is opposite the bottom face $X.$

Thus, the correct answer is E .

12.

A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $6.2$ cm, $8.3$ cm, and $9.5$ cm. The area of the square, in square centimeters, is

$24 \text{ cm}^2$

$36 \text{ cm}^2$

$48 \text{ cm}^2$

$64 \text{ cm}^2$

$144 \text{ cm}^2$

Answer: B

Concepts:perimeter square (geometry)

Difficulty rating: 730

Solution:

The triangle's perimeter is $6.2 + 8.3 + 9.5 = 24$ cm, so the square also has perimeter $24$ cm.

The square's side is $\dfrac{24}{4} = 6$ cm, so its area is $6^2 = 36 \text{ cm}^2.$

Thus, the correct answer is B .

13.

If you walk for $45$ minutes at a rate of $4$ mph and then run for $30$ minutes at a rate of $10$ mph, how many miles have you gone at the end of one hour and $15$ minutes?

$3.5$ miles

$8$ miles

$9$ miles

$25\dfrac13$ miles

$480$ miles

Answer: B

Concepts:distance rate and time unit conversion

Difficulty rating: 820

Solution:

Walking: $4 \times \dfrac34 = 3$ miles. Running: $10 \times \dfrac12 = 5$ miles.

The total distance is $3 + 5 = 8$ miles.

Thus, the correct answer is B .

14.

The difference between a $6.5\%$ sales tax and a $6\%$ sales tax on an item priced at $ $20$ before tax is

$.01

$.10

$.50

$1

$10

Answer: B

Concepts:percentage

Difficulty rating: 730

Solution:

The two taxes differ by $0.5\%$ of the price.

That is $0.005 \times \$20 = \$0.10.$

Thus, the correct answer is B .

15.

How many whole numbers between $100$ and $400$ contain the digit $2?$

$100$

$120$

$138$

$140$

$148$

Answer: C

Concepts:digits inclusion-exclusion casework

Difficulty rating: 1050

Solution:

All $100$ numbers from $200$ to $299$ contain a $2.$

Among $100$ to $199,$ there are $10$ with a $2$ in the tens place and $10$ with a $2$ in the units place, but $122$ is counted twice, giving $10 + 10 - 1 = 19.$ The range $300$ to $399$ similarly contributes $19.$

The total is $100 + 19 + 19 = 138.$

Thus, the correct answer is C .

16.

The ratio of boys to girls in Mr. Brown's math class is $2 : 3.$ If there are $30$ students in the class, how many more girls than boys are in the class?

$1$

$3$

$5$

$6$

$10$

Answer: D

Concepts:ratio and proportion

Difficulty rating: 730

Solution:

The $30$ students split into $5$ equal parts of $6.$ Boys make up $2$ parts ( $12$ ) and girls $3$ parts ( $18$ ).

So there are $18 - 12 = 6$ more girls than boys.

Thus, the correct answer is D .

17.

If your average score on your first six mathematics tests was $84$ and your average score on your first seven mathematics tests was $85,$ then your score on the seventh test was

$86$

$88$

$90$

$91$

$92$

Answer: D

Concepts:mean

Difficulty rating: 820

Solution:

Seven tests averaging $85$ total $7 \times 85 = 595$ points; six tests averaging $84$ total $6 \times 84 = 504$ points.

The seventh score is $595 - 504 = 91.$

Thus, the correct answer is D .

18.

Nine copies of a certain pamphlet cost less than $ $10.00$ while ten copies of the same pamphlet (at the same price) cost more than $ $11.00.$ How much does one copy of this pamphlet cost?

$1.07

$1.08

$1.09

$1.10

$1.11

Answer: E

Concepts:inequality bounding to limit cases

Difficulty rating: 950

Solution:

From $9p \lt 10$ we get $p \lt 1.111\ldots,$ and from $10p \gt 11$ we get $p \gt 1.10.$

The only price in whole cents between $\$1.10$ and $\$1.111\ldots$ is $\$1.11.$

Thus, the correct answer is E .

19.

If the length and width of a rectangle are each increased by $10\%,$ then the perimeter of the rectangle is increased by

$1\%$

$10\%$

$20\%$

$21\%$

$40\%$

Answer: B

Concepts:percentage perimeter

Difficulty rating: 800

Solution:

The new perimeter is $2(1.1\ell + 1.1w) = 1.1 \cdot 2(\ell + w),$ which is $1.1$ times the old perimeter.

That is a $10\%$ increase.

Thus, the correct answer is B .

20.

In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January $1$ fall that year?

Monday

Tuesday

Wednesday

Friday

Saturday

Answer: C

Concepts:date and time modular arithmetic

Difficulty rating: 1090

Solution:

Since $31 = 4 \cdot 7 + 3,$ the weekdays falling on January $1, 2,$ and $3$ each occur five times that month, and every other weekday occurs four times.

For Tuesday and Saturday to occur only four times, neither may be January $1, 2,$ or $3.$ The only starting day that works is Wednesday: then Wednesday, Thursday, and Friday occur five times, while Tuesday and Saturday each occur four times.

Thus, the correct answer is C .

21.

Mr. Green receives a $10\%$ raise every year. His salary after four such raises has gone up by what percent?

less than $40\%$

$40\%$

$44\%$

$45\%$

more than $45\%$

Answer: E

Concepts:percentage exponent

Difficulty rating: 950

Solution:

After four raises the salary is multiplied by $1.1^4 = 1.4641.$

That is an increase of $46.41\%,$ which is more than $45\%.$

Thus, the correct answer is E .

22.

Assume every $7$ -digit whole number is a possible telephone number except those that begin with $0$ or $1.$ What fraction of telephone numbers begin with $9$ and end with $0?$

$\dfrac{1}{63}$

$\dfrac{1}{80}$

$\dfrac{1}{81}$

$\dfrac{1}{90}$

$\dfrac{1}{100}$

Answer: B

Concepts:basic probability multiplication principle

Difficulty rating: 1000

Solution:

The first digit is one of $8$ allowed digits ( $2$ through $9$ ), so $\dfrac18$ of the numbers begin with $9.$ The last digit is any of $10$ digits, so $\dfrac{1}{10}$ end in $0.$

These conditions are independent, so the fraction is $\dfrac18 \cdot \dfrac{1}{10} = \dfrac{1}{80}.$

Thus, the correct answer is B .

23.

King Middle School has $1200$ students. Each student takes $5$ classes a day. Each teacher teaches $4$ classes. Each class has $30$ students and $1$ teacher. How many teachers are there at King Middle School?

$30$

$32$

$40$

$45$

$50$

Answer: E

Concepts:double counting

Difficulty rating: 950

Solution:

Each day there are $1200 \times 5 = 6000$ student-class attendances. Since each class holds $30$ students, there are $\dfrac{6000}{30} = 200$ classes.

Each teacher teaches $4$ classes, so there are $\dfrac{200}{4} = 50$ teachers.

Thus, the correct answer is E .

24.

The six whole numbers $10, 11, 12, 13, 14,$ and $15$ are placed in six circles — one at each of the three corners of a triangle and one at the midpoint of each side — so that the sum $S$ of the three numbers along each side of the triangle is the same. The largest possible value for $S$ is

$36$

$37$

$38$

$39$

$40$

Answer: D

Concepts:magic square optimization

Difficulty rating: 1140

Solution:

Adding the three side sums gives $3S = (10 + 11 + \cdots + 15) + (\text{sum of corners}) = 75 + (\text{sum of corners}).$ To maximize $S,$ place the three largest numbers $13, 14, 15$ at the corners, giving corner sum $42.$

Then $3S = 75 + 42 = 117,$ so $S = 39.$ This is achievable: with corners $13, 14, 15$ and midpoints $12, 10, 11,$ each side sums to $39.$

Thus, the correct answer is D .

25.

Five cards are lying on a table. Each card has a letter on one side and a whole number on the other side. The visible faces show $P, Q, 3, 4,$ and $6.$ Jane said, "If a vowel is on one side of any card, then an even number is on the other side." Mary showed Jane was wrong by turning over one card. Which card did Mary turn over?

$3$

$4$

$6$

$P$

$Q$

Answer: A

Concepts:logical deduction counterexample

Difficulty rating: 1140

Solution:

To show the rule "a vowel forces an even number" is false, Mary needs a card with a vowel on one side and an odd number on the other. No card shows a vowel, since $P$ and $Q$ are consonants.

The only card showing an odd number is $3.$ Turning it over reveals a vowel, which contradicts Jane's claim.

Thus, the correct answer is A .