Difference between revisions of "1986 AJHSME Problems"
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+ | {{AJHSME Problems | ||
+ | |year = 1986 | ||
+ | }} | ||
== Problem 1 == | == Problem 1 == | ||
In July 1861, <math>366</math> inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? | In July 1861, <math>366</math> inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? | ||
− | <math>\text{(A)}\ \frac{366}{31\times 24} | + | <math>\text{(A)}\ \frac{366}{31\times 24} \qquad \text{(B)}\ \frac{366\times 31}{24}\qquad \text{(C)}\ \frac{366\times 24}{31}\qquad \text{(D)}\ \frac{31\times 24}{366}\qquad \text{(E)}\ 366\times 31\times 24</math> |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
[[1986 AJHSME Problems/Problem 1|Solution]] | [[1986 AJHSME Problems/Problem 1|Solution]] | ||
Line 64: | Line 59: | ||
== Problem 8 == | == Problem 8 == | ||
+ | |||
+ | In the product shown, <math>\text{B}</math> is a digit. The value of <math>\text{B}</math> is | ||
+ | |||
+ | <cmath>\begin{array}{rr} | ||
+ | &\text{B}2 \\ | ||
+ | \times &7\text{B} \\ \hline | ||
+ | &6396 \\ | ||
+ | \end{array}</cmath> | ||
+ | |||
+ | <math>\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8</math> | ||
[[1986 AJHSME Problems/Problem 8|Solution]] | [[1986 AJHSME Problems/Problem 8|Solution]] | ||
Line 95: | Line 100: | ||
== Problem 10 == | == Problem 10 == | ||
− | A picture <math>3</math> feet across is hung in the center of a wall that is <math>19</math> feet | + | A picture <math>3</math> feet across is hung in the center of a wall that is <math>19</math> feet wide. How many feet from the end of the wall is the nearest edge of the picture? |
<math>\text{(A)}\ 1\frac{1}{2} \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9\frac{1}{2} \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 22</math> | <math>\text{(A)}\ 1\frac{1}{2} \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9\frac{1}{2} \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 22</math> | ||
Line 102: | Line 107: | ||
== Problem 11 == | == Problem 11 == | ||
+ | |||
+ | If <math>\text{A}*\text{B}</math> means <math>\frac{\text{A}+\text{B}}{2}</math>, then <math>(3*5)*8</math> is | ||
+ | |||
+ | <math>\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16\qquad \text{(E)}\ 30</math> | ||
[[1986 AJHSME Problems/Problem 11|Solution]] | [[1986 AJHSME Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
+ | |||
+ | The table below displays the grade distribution of the <math>30</math> students in a mathematics class on the last two tests. For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2 (see circled entry). What percent of the students received the same grade on both tests? | ||
+ | |||
+ | <asy> | ||
+ | draw((2,0)--(7,0)--(7,5)--(2,5)--cycle); | ||
+ | draw((3,0)--(3,5)); | ||
+ | draw((4,0)--(4,5)); | ||
+ | draw((5,0)--(5,5)); | ||
+ | draw((6,0)--(6,5)); | ||
+ | draw((2,1)--(7,1)); | ||
+ | draw((2,2)--(7,2)); | ||
+ | draw((2,3)--(7,3)); | ||
+ | draw((2,4)--(7,4)); | ||
+ | draw((.2,6.8)--(1.8,5.2)); | ||
+ | draw(circle((4.5,1.5),.5),linewidth(.6 mm)); | ||
+ | label("0",(2.5,.2),N); | ||
+ | label("0",(3.5,.2),N); | ||
+ | label("2",(4.5,.2),N); | ||
+ | label("1",(5.5,.2),N); | ||
+ | label("0",(6.5,.2),N); | ||
+ | label("0",(2.5,1.2),N); | ||
+ | label("0",(3.5,1.2),N); | ||
+ | label("1",(4.5,1.2),N); | ||
+ | label("1",(5.5,1.2),N); | ||
+ | label("1",(6.5,1.2),N); | ||
+ | label("1",(2.5,2.2),N); | ||
+ | label("3",(3.5,2.2),N); | ||
+ | label("5",(4.5,2.2),N); | ||
+ | label("2",(5.5,2.2),N); | ||
+ | label("0",(6.5,2.2),N); | ||
+ | label("1",(2.5,3.2),N); | ||
+ | label("4",(3.5,3.2),N); | ||
+ | label("3",(4.5,3.2),N); | ||
+ | label("0",(5.5,3.2),N); | ||
+ | label("0",(6.5,3.2),N); | ||
+ | label("2",(2.5,4.2),N); | ||
+ | label("2",(3.5,4.2),N); | ||
+ | label("1",(4.5,4.2),N); | ||
+ | label("0",(5.5,4.2),N); | ||
+ | label("0",(6.5,4.2),N); | ||
+ | label("F",(1.5,.2),N); | ||
+ | label("D",(1.5,1.2),N); | ||
+ | label("C",(1.5,2.2),N); | ||
+ | label("B",(1.5,3.2),N); | ||
+ | label("A",(1.5,4.2),N); | ||
+ | label("A",(2.5,5.2),N); | ||
+ | label("B",(3.5,5.2),N); | ||
+ | label("C",(4.5,5.2),N); | ||
+ | label("D",(5.5,5.2),N); | ||
+ | label("F",(6.5,5.2),N); | ||
+ | label("Test 1",(-.5,5.2),N); | ||
+ | label("Test 2",(2.6,6),N); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 12\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 33\frac{1}{3}\% \qquad \text{(D)}\ 40\% \qquad \text{(E)}\ 50\% </math> | ||
[[1986 AJHSME Problems/Problem 12|Solution]] | [[1986 AJHSME Problems/Problem 12|Solution]] | ||
== Problem 13 == | == Problem 13 == | ||
+ | |||
+ | The perimeter of the polygon shown is | ||
+ | |||
+ | <center> | ||
+ | <asy> | ||
+ | draw((0,0)--(0,6)--(8,6)--(8,3)--(2.7,3)--(2.7,0)--cycle); | ||
+ | label("$6$",(0,3),W); | ||
+ | label("$8$",(4,6),N); | ||
+ | draw((0.5,0)--(0.5,0.5)--(0,0.5)); | ||
+ | draw((0.5,6)--(0.5,5.5)--(0,5.5)); | ||
+ | draw((7.5,6)--(7.5,5.5)--(8,5.5)); | ||
+ | draw((7.5,3)--(7.5,3.5)--(8,3.5)); | ||
+ | draw((2.2,0)--(2.2,0.5)--(2.7,0.5)); | ||
+ | draw((2.7,2.5)--(3.2,2.5)--(3.2,3)); | ||
+ | </asy> | ||
+ | </center> | ||
+ | |||
+ | <math>\text{(A)}\ 14 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 48</math> | ||
+ | |||
+ | <math>\text{(E)}\ \text{cannot be determined from the information given}</math> | ||
[[1986 AJHSME Problems/Problem 13|Solution]] | [[1986 AJHSME Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | |||
+ | If <math>200\leq a \leq 400</math> and <math>600\leq b\leq 1200</math>, then the largest value of the quotient <math>\frac{b}{a}</math> is | ||
+ | |||
+ | <math>\text{(A)}\ \frac{3}{2} \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 300 \qquad \text{(E)}\ 600</math> | ||
[[1986 AJHSME Problems/Problem 14|Solution]] | [[1986 AJHSME Problems/Problem 14|Solution]] | ||
== Problem 15 == | == Problem 15 == | ||
+ | |||
+ | Sale prices at the Ajax Outlet Store are <math>50\% </math> below original prices. On Saturdays an additional discount of <math>20\% </math> off the sale price is given. What is the Saturday price of a coat whose original price is \$<math>180</math>? | ||
+ | |||
+ | <math>\text{(A)}</math> \$<math>54</math> | ||
+ | |||
+ | <math>\text{(B)}</math> \$<math>72</math> | ||
+ | |||
+ | <math>\text{(C)}</math> \$<math>90</math> | ||
+ | |||
+ | <math>\text{(D)}</math> \$<math>108</math> | ||
+ | |||
+ | <math>\text{(D)}</math> \$<math>110</math> | ||
[[1986 AJHSME Problems/Problem 15|Solution]] | [[1986 AJHSME Problems/Problem 15|Solution]] | ||
== Problem 16 == | == Problem 16 == | ||
+ | |||
+ | A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly <math>25\% </math> of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter? | ||
+ | |||
+ | <asy> | ||
+ | size(250); | ||
+ | |||
+ | void bargraph(real X, real Y, real ymin, real ymax, real ystep, real tickwidth, | ||
+ | string yformat, Label LX, Label LY, Label[] LLX, real[] height,pen p=nullpen) | ||
+ | { | ||
+ | draw((0,0)--(0,Y),EndArrow); | ||
+ | draw((0,0)--(X,0),EndArrow); | ||
+ | label(LX,(X,0),plain.SE,fontsize(9)); | ||
+ | label(LY,(0,Y),plain.NW,fontsize(9)); | ||
+ | real yscale=Y/(ymax+ystep); | ||
+ | |||
+ | for(real y=ymin; y<ymax; y+=ystep) | ||
+ | { | ||
+ | draw((-tickwidth,yscale*y)--(0,yscale*y)); | ||
+ | label(format(yformat,y),(-tickwidth,yscale*y),plain.W,fontsize(9)); | ||
+ | } | ||
+ | |||
+ | int n=LLX.length; | ||
+ | real xscale=X/(2*n+2); | ||
+ | for(int i=0;i<n;++i) | ||
+ | { | ||
+ | real x=xscale*(2*i+1); | ||
+ | path P=(x,0)--(x,height[i]*yscale)--(x+xscale,height[i]*yscale)--(x+xscale,0)--cycle; | ||
+ | fill(P,p); | ||
+ | draw(P); | ||
+ | label(LLX[i],(x+xscale/2),plain.S,fontsize(10)); | ||
+ | } | ||
+ | for(int i=0;i<n;++i) draw((0,height[i]*yscale)--(X,height[i]*yscale),dashed); | ||
+ | } | ||
+ | |||
+ | string yf="%#.1f"; | ||
+ | Label[] LX={"Spring","Summer","Fall","Winter"}; | ||
+ | for(int i=0;i<LX.length;++i) LX[i]=rotate(90)*LX[i]; | ||
+ | real[] H={4.5,5,4,4}; | ||
+ | |||
+ | bargraph(60,50,1,5.1,0.5,2,yf,"season","hamburgers (millions)",LX,H,yellow); | ||
+ | fill(ellipse((45,30),7,10),brown); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 2.5 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 3.5 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 4.5</math> | ||
[[1986 AJHSME Problems/Problem 16|Solution]] | [[1986 AJHSME Problems/Problem 16|Solution]] | ||
== Problem 17 == | == Problem 17 == | ||
+ | |||
+ | Let <math>\text{o}</math> be an odd whole number and let <math>\text{n}</math> be any whole number. Which of the following statements about the whole number <math>(\text{o}^2+\text{no})</math> is always true? | ||
+ | |||
+ | <math>\text{(A)}\ \text{it is always odd} \qquad \text{(B)}\ \text{it is always even}</math> | ||
+ | |||
+ | <math>\text{(C)}\ \text{it is even only if n is even} \qquad \text{(D)}\ \text{it is odd only if n is odd}</math> | ||
+ | |||
+ | <math>\text{(E)}\ \text{it is odd only if n is even}</math> | ||
[[1986 AJHSME Problems/Problem 17|Solution]] | [[1986 AJHSME Problems/Problem 17|Solution]] | ||
== Problem 18 == | == Problem 18 == | ||
+ | |||
+ | A rectangular grazing area is to be fenced off on three sides using part of a <math>100</math> meter rock wall as the fourth side. Fence posts are to be placed every <math>12</math> meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area <math>36</math> m by <math>60</math> m? | ||
+ | |||
+ | <asy> | ||
+ | draw((0,0)--(16,12)); | ||
+ | draw((5.33333,4)--(10.66666,8)--(6.66666,13.33333)--(1.33333,9.33333)--cycle); | ||
+ | label("WALL",(7,4),SE); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16</math> | ||
[[1986 AJHSME Problems/Problem 18|Solution]] | [[1986 AJHSME Problems/Problem 18|Solution]] | ||
== Problem 19 == | == Problem 19 == | ||
+ | |||
+ | At the beginning of a trip, the mileage odometer read <math>56,200</math> miles. The driver filled the gas tank with <math>6</math> gallons of gasoline. During the trip, the driver filled his tank again with <math>12</math> gallons of gasoline when the odometer read <math>56,560</math>. At the end of the trip, the driver filled his tank again with <math>20</math> gallons of gasoline. The odometer read <math>57,060</math>. To the nearest tenth, what was the car's average miles-per-gallon for the entire trip? | ||
+ | |||
+ | <math>\text{(A)}\ 22.5 \qquad \text{(B)}\ 22.6 \qquad \text{(C)}\ 24.0 \qquad \text{(D)}\ 26.9 \qquad \text{(E)}\ 27.5</math> | ||
[[1986 AJHSME Problems/Problem 19|Solution]] | [[1986 AJHSME Problems/Problem 19|Solution]] | ||
== Problem 20 == | == Problem 20 == | ||
+ | |||
+ | The value of the expression <math>\frac{(304)^5}{(29.7)(399)^4}</math> is closest to | ||
+ | |||
+ | <math>\text{(A)}\ .003 \qquad \text{(B)}\ .03 \qquad \text{(C)}\ .3 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 30</math> | ||
[[1986 AJHSME Problems/Problem 20|Solution]] | [[1986 AJHSME Problems/Problem 20|Solution]] | ||
== Problem 21 == | == Problem 21 == | ||
+ | |||
+ | Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box? | ||
+ | |||
+ | <asy> | ||
+ | draw((1,0)--(2,0)--(2,5)--(1,5)--cycle); | ||
+ | draw((0,1)--(3,1)--(3,4)--(0,4)--cycle); | ||
+ | draw((0,2)--(4,2)--(4,3)--(0,3)--cycle); | ||
+ | draw((1,1)--(2,1)--(2,2)--(3,2)--(3,3)--(2,3)--(2,4)--(1,4)--cycle,linewidth(.7 mm)); | ||
+ | label("A",(1.5,4.2),N); | ||
+ | label("B",(.5,3.2),N); | ||
+ | label("C",(2.5,3.2),N); | ||
+ | label("D",(.5,2.2),N); | ||
+ | label("E",(3.5,2.2),N); | ||
+ | label("F",(.5,1.2),N); | ||
+ | label("G",(2.5,1.2),N); | ||
+ | label("H",(1.5,.2),N); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math> | ||
[[1986 AJHSME Problems/Problem 21|Solution]] | [[1986 AJHSME Problems/Problem 21|Solution]] | ||
== Problem 22 == | == Problem 22 == | ||
+ | |||
+ | Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's? | ||
+ | |||
+ | <math>\text{(A)}\ \text{Alan, Beth} \qquad \text{(B)}\ \text{Beth, Carlos} \qquad \text{(C)}\ \text{Carlos, Diana}</math> | ||
+ | |||
+ | <math>\text{(D)}\ \text{Alan, Diana} \qquad \text{(E)}\ \text{Beth, Diana}</math> | ||
[[1986 AJHSME Problems/Problem 22|Solution]] | [[1986 AJHSME Problems/Problem 22|Solution]] | ||
== Problem 23 == | == Problem 23 == | ||
+ | |||
+ | The large circle has diameter <math>\text{AC}</math>. The two small circles have their centers on <math>\text{AC}</math> and just touch at <math>\text{O}</math>, the center of the large circle. If each small circle has radius <math>1</math>, what is the value of the ratio of the area of the shaded region to the area of one of the small circles? | ||
+ | |||
+ | <asy> | ||
+ | pair A=(-2,0), O=origin, C=(2,0); | ||
+ | path X=Arc(O,2,0,180), Y=Arc((-1,0),1,180,0), Z=Arc((1,0),1,180,0), M=X..Y..Z..cycle; | ||
+ | filldraw(M, black, black); | ||
+ | draw(reflect(A,C)*M); | ||
+ | draw(A--C, dashed); | ||
+ | |||
+ | label("A",A,W); | ||
+ | label("C",C,E); | ||
+ | label("O",O,SE); | ||
+ | dot((-1,0)); | ||
+ | dot(O); | ||
+ | dot((1,0)); | ||
+ | label("$1$",(-.5,0),N); | ||
+ | label("$1$",(1.5,0),N); | ||
+ | </asy> | ||
+ | |||
+ | <math>\text{(A)}\ \text{between }\frac{1}{2}\text{ and 1} \qquad \text{(B)}\ 1 \qquad \text{(C)}\ \text{between 1 and }\frac{3}{2}</math> | ||
+ | |||
+ | <math>\text{(D)}\ \text{between }\frac{3}{2}\text{ and 2} \qquad \text{(E)}\ \text{cannot be determined from the information given}</math> | ||
[[1986 AJHSME Problems/Problem 23|Solution]] | [[1986 AJHSME Problems/Problem 23|Solution]] | ||
== Problem 24 == | == Problem 24 == | ||
+ | |||
+ | The <math>600</math> students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately | ||
+ | |||
+ | <math>\text{(A)}\ \frac{1}{27} \qquad \text{(B)}\ \frac{1}{9} \qquad \text{(C)}\ \frac{1}{8} \qquad \text{(D)}\ \frac{1}{6} \qquad \text{(E)}\ \frac{1}{3}</math> | ||
[[1986 AJHSME Problems/Problem 24|Solution]] | [[1986 AJHSME Problems/Problem 24|Solution]] | ||
== Problem 25 == | == Problem 25 == | ||
+ | |||
+ | Which of the following sets of whole numbers has the largest average? | ||
+ | |||
+ | <math>\text{(A)}\ \text{multiples of 2 between 1 and 101} \qquad \text{(B)}\ \text{multiples of 3 between 1 and 101}</math> | ||
+ | |||
+ | <math>\text{(C)}\ \text{multiples of 4 between 1 and 101} \qquad \text{(D)}\ \text{multiples of 5 between 1 and 101}</math> | ||
+ | |||
+ | <math>\text{(E)}\ \text{multiples of 6 between 1 and 101}</math> | ||
[[1986 AJHSME Problems/Problem 25|Solution]] | [[1986 AJHSME Problems/Problem 25|Solution]] | ||
− | == See | + | == See Also == |
+ | {{AJHSME box|year=1986|before=[[1985 AJHSME Problems|1985 AJHSME]]|after=[[1987 AJHSME Problems|1987 AJHSME]]}} | ||
* [[AJHSME]] | * [[AJHSME]] | ||
* [[AJHSME Problems and Solutions]] | * [[AJHSME Problems and Solutions]] | ||
− | |||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
+ | |||
+ | {{MAA Notice}} |
Latest revision as of 15:41, 8 August 2020
1986 AJHSME (Answer Key) Printable versions: • AoPS Resources • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 |
Contents
- 1 Problem 1
- 2 Problem 2
- 3 Problem 3
- 4 Problem 4
- 5 Problem 5
- 6 Problem 6
- 7 Problem 7
- 8 Problem 8
- 9 Problem 9
- 10 Problem 10
- 11 Problem 11
- 12 Problem 12
- 13 Problem 13
- 14 Problem 14
- 15 Problem 15
- 16 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 See Also
Problem 1
In July 1861, inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month?
Problem 2
Which of the following numbers has the largest reciprocal?
Problem 3
The smallest sum one could get by adding three different numbers from the set is
Problem 4
The product is closest to
Problem 5
A contest began at noon one day and ended minutes later. At what time did the contest end?
Problem 6
Problem 7
How many whole numbers are between and
?
Problem 8
In the product shown, is a digit. The value of
is
Problem 9
Using only the paths and the directions shown, how many different routes are there from to
?
Problem 10
A picture feet across is hung in the center of a wall that is
feet wide. How many feet from the end of the wall is the nearest edge of the picture?
Problem 11
If means
, then
is
Problem 12
The table below displays the grade distribution of the students in a mathematics class on the last two tests. For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2 (see circled entry). What percent of the students received the same grade on both tests?
Problem 13
The perimeter of the polygon shown is
Problem 14
If and
, then the largest value of the quotient
is
Problem 15
Sale prices at the Ajax Outlet Store are below original prices. On Saturdays an additional discount of
off the sale price is given. What is the Saturday price of a coat whose original price is $
?
$
$
$
$
$
Problem 16
A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter?
Problem 17
Let be an odd whole number and let
be any whole number. Which of the following statements about the whole number
is always true?
Problem 18
A rectangular grazing area is to be fenced off on three sides using part of a meter rock wall as the fourth side. Fence posts are to be placed every
meters along the fence including the two posts where the fence meets the rock wall. What is the fewest number of posts required to fence an area
m by
m?
Problem 19
At the beginning of a trip, the mileage odometer read miles. The driver filled the gas tank with
gallons of gasoline. During the trip, the driver filled his tank again with
gallons of gasoline when the odometer read
. At the end of the trip, the driver filled his tank again with
gallons of gasoline. The odometer read
. To the nearest tenth, what was the car's average miles-per-gallon for the entire trip?
Problem 20
The value of the expression is closest to
Problem 21
Suppose one of the eight lettered identical squares is included with the four squares in the T-shaped figure outlined. How many of the resulting figures can be folded into a topless cubical box?
Problem 22
Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's?
Problem 23
The large circle has diameter . The two small circles have their centers on
and just touch at
, the center of the large circle. If each small circle has radius
, what is the value of the ratio of the area of the shaded region to the area of one of the small circles?
Problem 24
The students at King Middle School are divided into three groups of equal size for lunch. Each group has lunch at a different time. A computer randomly assigns each student to one of three lunch groups. The probability that three friends, Al, Bob, and Carol, will be assigned to the same lunch group is approximately
Problem 25
Which of the following sets of whole numbers has the largest average?
See Also
1986 AJHSME (Problems • Answer Key • Resources) | ||
Preceded by 1985 AJHSME |
Followed by 1987 AJHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.