1995 AMC 8 — Problems & Online Timed Exam

1.

Walter has exactly one penny, one nickel, one dime and one quarter in his pocket. What percent of one dollar is in his pocket?

4\%

25\%

40\%

41\%

59\%

Answer: D

Concepts:money percentage

Difficulty rating: 370

Small hint:

Add up the values of the coins in cents

Big hint:

1 + 5 + 10 + 25

cents, and

100

cents make one dollar

Solution:

The coins total

1 + 5 + 10 + 25 = 41

cents.

Since one dollar is

100

cents, this is

41\%

of a dollar.

Thus, the correct answer is D .

2.

Jose is

4

years younger than Zack. Zack is

3

years older than Inez. Inez is

15

years old. How old is Jose?

8

11

14

16

22

Answer: C

Concepts:ages

Difficulty rating: 370

Small hint:

Work from Inez up to Zack, then down to Jose

Big hint:

Zack

= 15 + 3;

then Jose

=

Zack

- 4

Solution:

Zack is

15 + 3 = 18

years old.

Jose is

4

years younger, so Jose is

18 - 4 = 14

.

Thus, the correct answer is C .

3.

Which of the following operations has the same effect on a number as multiplying by

\dfrac34

and then dividing by

\dfrac35?

dividing by

\dfrac43

dividing by

\dfrac{9}{20}

multiplying by

\dfrac{9}{20}

dividing by

\dfrac54

multiplying by

\dfrac54

Answer: E

Concepts:fraction

Difficulty rating: 560

Small hint:

Dividing by

\dfrac35

is the same as multiplying by

\dfrac53

Big hint:

Combine

\dfrac34 \times \dfrac53

into a single fraction

Solution:

Dividing by

\dfrac35

is the same as multiplying by

\dfrac53,

so the two operations together multiply by

\dfrac34 \times \dfrac53 = \dfrac54.

So the combined effect is multiplying by

\dfrac54

.

Thus, the correct answer is E .

4.

A teacher tells the class, “Think of a number, add

1

to it, and double the result. Give the answer to your partner. Partner, subtract

1

from the number you are given and double the result to get your answer.” Ben thinks of

6,

and gives his answer to Sue. What should Sue’s answer be?

18

24

26

27

30

Answer: C

Concepts:order of operations

Difficulty rating: 450

Small hint:

First compute Ben’s answer: add

1

to

6,

then double

Big hint:

Sue takes Ben’s number, subtracts

1,

and doubles that

Solution:

Ben computes

(6 + 1) \times 2 = 14

and gives

14

to Sue.

Sue computes

(14 - 1) \times 2 = 26

.

Thus, the correct answer is C .

5.

Find the smallest whole number that is larger than the sum

2\tfrac12 + 3\tfrac13 + 4\tfrac14 + 5\tfrac15.

14

15

16

17

18

Answer: C

Concepts:fraction estimation

Difficulty rating: 660

Small hint:

Add the whole-number parts and the fraction parts separately

Big hint:

The fractions

\dfrac12 + \dfrac13 + \dfrac14 + \dfrac15

add to a little more than

1

Solution:

The whole-number parts sum to

2 + 3 + 4 + 5 = 14

.

The fractions

\dfrac12 + \dfrac13 + \dfrac14 + \dfrac15

add to about

1.28,

which is between

1

and

2.

So the total is between

15

and

16,

and the smallest whole number larger than it is

16

.

Thus, the correct answer is C .

6.

Figures

I,

II

and

III

are squares. The perimeter of

I

is

12

and the perimeter of

II

is

24.

The perimeter of

III

is

9

18

36

72

81

Answer: C

Concepts:perimeter square (geometry)

Difficulty rating: 770

Small hint:

The perimeter of a square is

4

times its side, so find the side of

I

and of

II

Big hint:

From the figure, the side of

III

equals the side of

I

plus the side of

II

Solution:

Square

I

has side

12 \div 4 = 3,

and square

II

has side

24 \div 4 = 6.

From the figure, the side of

III

is

3 + 6 = 9,

so its perimeter is

4 \times 9 = 36.

Thus, the correct answer is C .

7.

At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home?

\dfrac{1}{16}

\dfrac{3}{20}

\dfrac13

\dfrac{17}{20}

\dfrac{9}{10}

Answer: B

Concepts:fraction

Difficulty rating: 730

Small hint:

Give the fractions a common denominator to add them

Big hint:

Subtract the total who ride from

1

(the whole group)

Solution:

The students who ride make up

\begin{aligned} \dfrac12 + \dfrac14 + \dfrac{1}{10} &= \dfrac{10}{20} + \dfrac{5}{20} + \dfrac{2}{20} \\ &= \dfrac{17}{20}. \end{aligned}

So the fraction who walk is

1 - \dfrac{17}{20} = \dfrac{3}{20}.

Thus, the correct answer is B .

8.

An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If

3000

lire

= \$1.60,

how many lire will the traveler receive in exchange for

\$1.00?

180

480

1800

1875

4875

Answer: D

Concepts:ratio and proportion

Difficulty rating: 820

Small hint:

Find how many lire equal one dollar using the given rate

Big hint:

\$1.00

is

\dfrac{1.00}{1.60}

of

\$1.60,

so take that fraction of

3000

Solution:

Since

\$1.00

is

\dfrac{1.00}{1.60} = \dfrac58

of

\$1.60,

the traveler gets

\dfrac58

of

3000

lire.

That is

\dfrac58 \times 3000 = 1875

lire.

Thus, the correct answer is D .

9.

Three congruent circles with centers

P,

Q

and

R

are tangent to the sides of rectangle

ABCD

as shown. The circle centered at

Q

has diameter

4

and passes through points

P

and

R.

The area of the rectangle is

16

24

32

64

128

Answer: C

Concepts:tangent circles rectangle

Difficulty rating: 960

Small hint:

The diameter of each circle is

4,

so the short side of the rectangle equals

4

Big hint:

The long side spans two full diameters

Solution:

Each circle has diameter

4.

The short side of the rectangle equals one diameter, so it is

4.

Since the circle at

Q

passes through

P

and

R,

all three circles have radius

2,

and the long side spans two full diameters:

4 + 4 = 8.

The area is

8 \times 4 = 32.

Thus, the correct answer is C .

10.

A jacket and a shirt originally sold for

\$80

and

\$40,

respectively. During a sale Chris bought the

\$80

jacket at a

40\%

discount and the

\$40

shirt at a

55\%

discount. The total amount saved was what percent of the total of the original prices?

45\%

47\tfrac12\%

50\%

79\tfrac16\%

95\%

Answer: A

Concepts:percentage

Difficulty rating: 930

Small hint:

Find the dollars saved on each item separately

Big hint:

Divide the total saved by the total original price of

\$120

Solution:

The jacket discount saves

40\%

of

\$80 = \$32,

and the shirt discount saves

55\%

of

\$40 = \$22.

The total saved is

\$32 + \$22 = \$54.

The original total is

\$80 + \$40 = \$120,

so the percent saved is

\dfrac{54}{120} = 0.45 = 45\%.

Thus, the correct answer is A .

11.

Jane can walk any distance in half the time it takes Hector to walk the same distance. They set off in opposite directions around the outside of the

18

-block area as shown. When they meet for the first time, they will be closest to

A

B

C

D

E

Answer: D

Concepts:distance rate and time perimeter relative speed

Difficulty rating: 1030

Small hint:

When they first meet, together they have covered the full

18

-block loop

Big hint:

Jane walks twice as fast, so she covers

12

blocks while Hector covers

6

Solution:

The perimeter of the region is

18

blocks, so when Jane and Hector meet they have together walked

18

blocks. Since Jane walks twice as fast, she covers

12

blocks and Hector covers

6.

Starting from the middle of the bottom edge, Hector walks

6

blocks (to

E,

then up to

D

), and Jane walks

12

blocks (to

A,

up to

B,

then across the top to

D

). They meet at

D.

Thus, the correct answer is D .

12.

A lucky year is one in which at least one date, when written in the form month/day/year, has the following property: the product of the month times the day equals the last two digits of the year. For example,

1956

is a lucky year because it has the date

7/8/56

and

7 \times 8 = 56.

Which of the following is NOT a lucky year?

1990

1991

1992

1993

1994

Answer: E

Concepts:factor casework

Difficulty rating: 1120

Small hint:

For each year, try to write its last two digits as (month)

\times

(day) with a valid month and day

Big hint:

A year fails only when the last two digits cannot be a product of a number at most

12

and a number at most

31

Solution:

Each of the other years works:

90 = 9 \times 10,

91 = 7 \times 13,

92 = 4 \times 23,

and

93 = 3 \times 31,

each a valid month/day.

For

1994,

the last two digits factor only as

94 = 2 \times 47,

and

47

is too large for a day (and

94

or

47

is too large for a month). So

1994

has no lucky date.

Thus, the correct answer is E .

13.

In the figure,

\angle A,

\angle B

and

\angle C

are right angles. If

\angle AEB = 40^\circ

and

\angle BED = \angle BDE,

then

\angle CDE =

75^\circ

80^\circ

85^\circ

90^\circ

95^\circ

Answer: E

Concepts:angle chasing angle sum

Difficulty rating: 1150

Small hint:

In triangle

BDE,

the angles at

E

and

D

are equal and

\angle B = 90^\circ

Big hint:

\angle AED = \angle AEB + \angle BED;

then use quadrilateral

AEDC,

whose angles sum to

360^\circ

Solution:

In triangle

BDE,

the angles at

E

and

D

are equal and

\angle B = 90^\circ,

so

\angle BED = \angle BDE = 45^\circ.

Then

\angle AED = \angle AEB + \angle BED

= 40^\circ + 45^\circ

= 85^\circ.

In quadrilateral

AEDC,

the angles at

A

and

C

are

90^\circ,

so

\begin{aligned} \angle CDE &= 360^\circ - 90^\circ - 90^\circ \\ &\quad {}- 85^\circ \\ &= 95^\circ. \end{aligned}

Thus, the correct answer is E .

14.

A team won

40

of its first

50

games. How many of the remaining

40

games must this team win so it will have won exactly

70\%

of its games for the season?

20

23

28

30

35

Answer: B

Concepts:percentage

Difficulty rating: 930

Small hint:

The season has

50 + 40 = 90

games; find

70\%

of that

Big hint:

Subtract the

40

games already won from the needed total

Solution:

The season has

50 + 40 = 90

games, and

70\%

of

90

is

63

wins.

The team already has

40

wins, so it needs

63 - 40 = 23

more.

Thus, the correct answer is B .

15.

What is the

100

th digit to the right of the decimal point in the decimal form of

\dfrac{4}{37}?

0

1

2

7

8

Answer: B

Concepts:repeating decimal modular arithmetic

Difficulty rating: 1060

Small hint:

Write

\dfrac{4}{37}

as a repeating decimal and find the length of the repeating block

Big hint:

The block has length

3;

use the remainder of

100

divided by

3

Solution:

\dfrac{4}{37} = 0.\overline{108},

repeating with block length

3.

The digits in positions

3,

6,

9,

\ldots

(multiples of

3

) are

8.

Since

99

is a multiple of

3,

the

99

th digit is

8,

so the

100

th digit starts the next block: it is

1.

Thus, the correct answer is B .

16.

Students from three middle schools worked on a summer project. Seven students from Allen School worked for

3

days. Four students from Balboa School worked for

5

days. Five students from Carver School worked for

9

days. The total amount paid for the students’ work was

\$774.

Assuming each student received the same amount for a day’s work, how much did the students from Balboa School earn altogether?

\$9.00

\$48.38

\$180.00

\$193.50

\$258.00

Answer: C

Concepts:rate

Difficulty rating: 1090

Small hint:

Compute the total number of student-days worked across all three schools

Big hint:

Divide

\$774

by the total student-days to get the pay per student-day

Solution:

The total student-days are

7 \times 3 + 4 \times 5 + 5 \times 9

= 21 + 20 + 45

= 86.

So each student-day pays

\$774 \div 86 = \$9.

Balboa worked

20

student-days, earning

20 \times \$9 = \$180.

Thus, the correct answer is C .

17.

The table below gives the percent of students in each grade at Annville and Cleona elementary schools:

	$K$	$1$	$2$	$3$	$4$	$5$	$6$
Annville	$16\%$	$15\%$	$15\%$	$14\%$	$13\%$	$16\%$	$11\%$
Cleona	$12\%$	$15\%$	$14\%$	$13\%$	$15\%$	$14\%$	$17\%$

Annville has

100

students and Cleona has

200

students. In the two schools combined, what percent of the students are in grade

6?

12\%

13\%

14\%

15\%

28\%

Answer: D

Concepts:percentage mean

Difficulty rating: 980

Small hint:

Find the number of grade-

6

students at each school separately

Big hint:

Divide the combined grade-

6

count by the

300

total students

Solution:

Annville has

11\%

of

100 = 11

sixth graders, and Cleona has

17\%

of

200 = 34

sixth graders.

Combined, that is

11 + 34 = 45

out of

300

students, which is

\dfrac{45}{300} = 15\%.

Thus, the correct answer is D .

18.

The area of each of the four congruent L-shaped regions of this

100

-inch by

100

-inch square is

\dfrac{3}{16}

of the total area. How many inches long is the side of the center square?

25

44

50

62

75

Answer: C

Concepts:square (geometry)area

Difficulty rating: 980

Small hint:

The four L-shaped regions together make up

4 \times \dfrac{3}{16}

of the square

Big hint:

The center square is the remaining fraction of the area; its side is the square root of its area

Solution:

The four L-shaped regions cover

4 \times \dfrac{3}{16} = \dfrac34

of the square, so the center square is the remaining

\dfrac14

of the total area.

The total area is

100 \times 100 = 10000

square inches, so the center square has area

\dfrac14 \times 10000 = 2500,

and its side is

\sqrt{2500} = 50

inches.

Thus, the correct answer is C .

19.

The graph shows the distribution of the number of children in the families of the students in Ms. Jordan’s English class. The median number of children in the family for this distribution is

1

2

3

4

5

Answer: D

Concepts:median (data)data and graph interpretation

Difficulty rating: 960

Small hint:

The median is the middle value when all the family sizes are listed in order

Big hint:

Count the total number of families, then find which position is the middle one

Solution:

The graph gives

2

families with

1

child,

1

with

2,

2

with

3,

2

with

4,

and

6

with

5,

for

2 + 1 + 2 + 2 + 6 = 13

families.

The median is the

7

th value in order. Listing the family sizes, the

7

th value is

4.

Thus, the correct answer is D .

20.

Diana and Apollo each roll a standard die obtaining a number at random from

1

to

6.

What is the probability that Diana’s number is larger than Apollo’s number?

\dfrac13

\dfrac{5}{12}

\dfrac49

\dfrac{17}{36}

\dfrac12

Answer: B

Concepts:dice (probability)symmetry

Difficulty rating: 1120

Small hint:

Of the

36

equally likely outcomes, some are ties; remove those

Big hint:

By symmetry, Diana is larger in exactly half of the non-tie outcomes

Solution:

There are

6 \times 6 = 36

equally likely outcomes, of which

6

are ties, leaving

30

outcomes with different numbers.

By symmetry, Diana is larger in exactly half of those, or

15,

so the probability is

\dfrac{15}{36} = \dfrac{5}{12}.

Thus, the correct answer is B .

21.

A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?

3

4

5

6

8

Answer: B

Concepts:3D geometry

Difficulty rating: 1170

Small hint:

Each cube has one snap that must be hidden inside another cube’s hole

Big hint:

Try arranging the cubes in a small ring so every snap plugs into a neighbor

Solution:

Every cube’s single snap must be plugged into another cube’s hole to be hidden. Two cubes can share only one face, so two cubes cannot hide both snaps. With three cubes, hiding all three snaps would require a three-cube loop in which every pair shares a face, but three unit cubes cannot be pairwise face-adjacent. Thus one, two, or three cubes cannot work.

Four cubes can be arranged in a square ring, each snap fitting into the neighbor’s hole, so only receptacle holes show. The smallest number is

4.

Thus, the correct answer is B .

22.

The number

6545

can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers?

162

172

173

174

222

Answer: A

Concepts:prime factorization factor

Difficulty rating: 1170

Small hint:

Find the prime factorization of

6545

Big hint:

Group the prime factors into two numbers that are each between

10

and

99

Solution:

The prime factorization is

6545 = 5 \times 7 \times 11 \times 17.

To split into two two-digit factors, pair the primes:

5 \times 17 = 85

and

7 \times 11 = 77.

These are the only two-digit pair, and their sum is

85 + 77 = 162.

Thus, the correct answer is A .

23.

How many four-digit whole numbers are there such that the leftmost digit is odd, the second digit is even, and all four digits are different?

1120

1400

1800

2025

2500

Answer: B

Concepts:multiplication principle

Difficulty rating: 1220

Small hint:

Count choices for each digit position in turn: first, second, third, fourth

Big hint:

First digit:

5

odd options; second:

5

even options; then

8

and

7

for the distinct remaining digits

Solution:

The first digit is odd:

5

choices. The second is even:

5

choices (none of which repeats the odd first digit).

The third digit is any of the

8

unused digits, and the fourth is any of the

7

remaining. In total,

5 \times 5 \times 8 \times 7 = 1400.

Thus, the correct answer is B .

24.

In parallelogram

ABCD,

\overline{DE}

is the altitude to the base

\overline{AB}

and

\overline{DF}

is the altitude to the base

\overline{BC}.

(Both pictures represent the same parallelogram.) If

DC = 12,

EB = 4,

and

DE = 6,

then

DF =

6.4

7

7.2

8

10

Answer: C

Concepts:parallelogram Pythagorean Theorem

Difficulty rating: 1150

Small hint:

Opposite sides are equal, so

AB = 12;

find

AE,

then

AD

with the right triangle

ADE

Big hint:

The area equals base

\times

height for either base:

AB \cdot DE = BC \cdot DF

Solution:

Since

AB = DC = 12,

we get

AE = 12 - 4 = 8.

In right triangle

ADE,

AD = \sqrt{8^2 + 6^2} = 10,

so

BC = AD = 10.

The area is

AB \cdot DE = 12 \cdot 6 = 72,

and also

BC \cdot DF = 10 \cdot DF.

So

DF = \dfrac{72}{10} = 7.2.

Thus, the correct answer is C .

25.

Buses from Dallas to Houston leave every hour on the hour. Buses from Houston to Dallas leave every hour on the half hour. The trip from one city to the other takes

5

hours. Assuming the buses travel on the same highway, how many Dallas-bound buses does a Houston-bound bus pass on the highway (not in the station)?

5

6

9

10

11

Answer: D

Concepts:distance rate and time

Difficulty rating: 1260

Small hint:

Track one Houston-bound bus over its

5

-hour trip and see which oncoming buses share the road with it

Big hint:

Count the Dallas-bound buses already on the highway when it starts, plus those that leave before it arrives

Solution:

Consider a bus leaving Dallas at

6\!:\!00,

arriving in Houston at

11\!:\!00.

It meets every Dallas-bound bus that is on the highway during that window.

Dallas-bound buses leave Houston on the half hour and take

5

hours. The ones sharing the road (meeting away from a station) are those that left Houston at

1\!:\!30,

2\!:\!30,

\ldots,

10\!:\!30,

which is

10

buses.

Thus, the correct answer is D .

1995 AMC 8 Problems

Answer: D

Difficulty rating: 370

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 370

Small hint:

Big hint:

Solution:

Answer: E

Difficulty rating: 560

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 450

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 660

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 770

Small hint:

Big hint:

Solution:

Answer: B

Difficulty rating: 730

Small hint:

Big hint:

Solution:

Answer: D

Difficulty rating: 820

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 960

Small hint:

Big hint:

Solution:

Answer: A

Difficulty rating: 930

Small hint:

Big hint:

Solution:

Answer: D

Difficulty rating: 1030

Small hint:

Big hint:

Solution:

Answer: E

Difficulty rating: 1120

Small hint:

Big hint:

Solution:

Answer: E

Difficulty rating: 1150

Small hint:

Big hint:

Solution:

Answer: B

Difficulty rating: 930

Small hint:

Big hint:

Solution:

Answer: B

Difficulty rating: 1060

Small hint:

Big hint:

Solution:

Answer: C

Difficulty rating: 1090

Small hint:

Big hint: