1985 AMC 8 — Problems & Online Timed Exam

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

Small hint:

Multiply the numerators together and the denominators together before simplifying

Big hint:

The same factors

3,

5,

7,

9,

11

appear in both the numerator and the denominator

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

Small hint:

There are

10

terms, from

90

up to

99

Big hint:

The sum equals the average of the first and last term times the number of terms

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

Small hint:

\dfrac{10^7}{10^4} = 10^3

Big hint:

Then divide

10^3 = 1000

by

5

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

Small hint:

Extend sides

AF

and

DC

to complete a full

6 \times 9

rectangle

Big hint:

Subtract the area of the small rectangular notch that was cut from the lower-left corner

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

Small hint:

Add the counts for the satisfactory grades A, B, C, and D

Big hint:

Divide the satisfactory count by the total number of grades

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

Small hint:

The number of sheets is proportional to the height of the stack

Big hint:

A

7.5

cm stack is

1.5

times as tall as a

5

cm stack

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 shaded and unshaded squares in each row. Rows

1

through

4

are shown. All rows begin and end with an unshaded square. The number of shaded squares in the

37

th row is

34

35

36

37

38

Answer: C

Concepts:pattern recognition basic counting

Difficulty rating: 860

Small hint:

Row

n

has

2n - 1

squares, and they alternate starting and ending with an unshaded square

Big hint:

Since the ends are unshaded and the squares alternate, there is one more unshaded square than shaded square in each row

Solution:

Row

n

contains

2n - 1

squares. Because both ends are unshaded and the squares alternate, each row has one more unshaded square than shaded square, so the number of shaded 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

Small hint:

Substitute

a = -2

into each expression

Big hint:

Check signs first:

4a

and

\frac{24}{a}

are negative, while

-3a

and

a^2

are positive

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?

\begin{aligned} &\left(1 - \tfrac12\right)\left(1 - \tfrac13\right)\left(1 - \tfrac14\right) \cdots \\ &\quad {}\cdot \left(1 - \tfrac{1}{10}\right) \end{aligned}

\dfrac{1}{10}

\dfrac{1}{9}

\dfrac{1}{2}

\dfrac{10}{11}

\dfrac{11}{2}

Answer: A

Concepts:telescoping fraction

Difficulty rating: 860

Small hint:

Rewrite each factor:

1 - \dfrac1k = \dfrac{k-1}{k}

Big hint:

Write the product as

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

and watch consecutive terms cancel

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

Small hint:

The point halfway between two numbers is their average

Big hint:

\dfrac15 + \dfrac13 = \dfrac{8}{15};

then divide by

2

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

Small hint:

Pick one face to be the front and fold the rest of the net around it

Big hint:

If

X

becomes the bottom face, track which square ends up on top

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

Small hint:

Add the triangle’s three sides to get the common perimeter

Big hint:

The square’s side is one fourth of its perimeter

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

Small hint:

Convert each time to hours:

45

minutes

= \dfrac34

hour and

30

minutes

= \dfrac12

hour

Big hint:

Distance

=

rate

\times

time for each part, then add the two distances

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

\$0.01

\$0.10

\$0.50

\$1

\$10

Answer: B

Concepts:percentage

Difficulty rating: 730

Small hint:

The difference in tax rates is

6.5\% - 6\% = 0.5\%

Big hint:

Find

0.5\%

of

\$20

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

Small hint:

Every number from

200

to

299

already contains a

2

Big hint:

For

100

–

199

and

300

–

399,

count those with a

2

in the tens or units place, subtracting the ones counted twice

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

Small hint:

The class splits into

2 + 3 = 5

equal parts

Big hint:

Girls exceed boys by exactly one part, so find the size of one part

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

Small hint:

The total for seven tests is

7 \times 85,

and the total for six tests is

6 \times 84

Big hint:

The seventh score is the difference between those two totals

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

Small hint:

Let the price be

p;

then

9p \lt 10

and

10p \gt 11

Big hint:

Combine into

1.10 \lt p \lt 1.111\ldots

and find the price in whole cents

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

Small hint:

The perimeter is

2(\ell + w)

Big hint:

Multiplying both

\ell

and

w

by

1.1

multiplies the whole perimeter by

1.1

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

Small hint:

January has

31

days, which is

4

full weeks plus

3

extra days

Big hint:

The three weekdays that occur five times are the days of January

1,

2,

and

3;

Tuesday and Saturday must not be among them

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

Small hint:

Applying a

10\%

raise multiplies the salary by

1.1

Big hint:

After four raises the salary is multiplied by

1.1^4;

compare that to

1.45

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

Small hint:

The first digit can be any of

2

through

9,

which is

8

choices

Big hint:

Multiply the fraction of telephone numbers that begin with

9

by the fraction that end in

0

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

Small hint:

Count the total number of student-class attendances:

1200 \times 5

Big hint:

Divide by

30

to get the number of classes, then by

4

to get the number of teachers

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.

In a magic triangle, each of the six whole numbers

10

through

15

is placed in one of the circles so that the sum

S

of the three numbers on 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

Small hint:

Adding the three side sums counts each corner number twice and each midpoint number once

Big hint:

Let

C

be the sum of the corner numbers. Then

3S = 75 + C;

put the three largest numbers at the corners

Solution:

Let

C

be the sum of the corner numbers. Adding the three side sums gives

3S = 75 + C,

because the six numbers sum to

75

and every corner is counted one extra time. To maximize

S,

place the three largest numbers

13,

14,

15

at the corners, giving

C = 42.

Then

3S = 75 + 42 = 117,

so

S = 39.

This is achievable: put

12

between corners

13

and

14,

put

10

between

14

and

15,

and put

11

between

15

and

13.

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

Small hint:

To disprove the rule, find a card that could hide a vowel paired with an odd number

Big hint:

No vowel is visible (

P

and

Q

are consonants), so the card to check is the one showing an odd number

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 .

1985 AMC 8 Problems

Answer: A

Difficulty rating: 560

Small hint:

Big hint:

Solution:

Answer: B

Difficulty rating: 450

Small hint:

Big hint:

Solution:

Answer: D

Difficulty rating: 560

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 820

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 660

Small hint:

Big hint:

Solution:

Answer: D

Difficulty rating: 560

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 860

Small hint:

Big hint:

Solution:

Answer: A

Difficulty rating: 730

Small hint:

Big hint:

Solution:

Answer: A

Difficulty rating: 860

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 730

Small hint:

Big hint:

Solution:

Answer: E

Difficulty rating: 920

Small hint:

Big hint:

Solution:

Answer: B

Difficulty rating: 730

Small hint:

Big hint:

Solution:

Answer: B

Difficulty rating: 820

Small hint:

Big hint:

Solution:

Answer: B

Difficulty rating: 730

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 1050

Small hint:

Big hint:

Solution:

Answer: D

Difficulty rating: 730

Small hint:

Big hint: