1995 AMC 8 Exam Problems

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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

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

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

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

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

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

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

Solution:

The students who ride make up

$\dfrac12 + \dfrac14 + \dfrac{1}{10} = \dfrac{10}{20} + \dfrac{5}{20} + \dfrac{2}{20} = \dfrac{17}{20}.$

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

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

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

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

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

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

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

$\angle CDE = 360^\circ - 90^\circ - 90^\circ - 85^\circ = 95^\circ.$

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

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

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

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. The percentages for grades K, $1, 2, 3, 4, 5, 6$ are: 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

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

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

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

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

Solution:

Every cube's single snap must be plugged into another cube's hole to be hidden. With one, two, or three cubes, at least one snap is always left exposed.

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

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

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 $\overline{AB}$ (with $E$ on $\overline{AB}$ ) and $\overline{DF}$ is the altitude to the base $\overline{BC}.$ 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

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

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 .