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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>
+
<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>
−
 
 
−
<math>\text{(B)}\ \frac{366\times 31}{24}</math>
 
−
 
 
−
<math>\text{(C)}\ \frac{366\times 24}{31}</math>
 
−
 
 
−
<math>\text{(D)}\ \frac{31\times 24}{366}</math>
 
−
 
 
−
<math>\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 wall.  How many feet from the end of the wall is the nearest edge of the picture?
+
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 110: Line 115:
  
 
== 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]]
Line 117: Line 177:
 
The perimeter of the polygon shown is  
 
The perimeter of the polygon shown is  
  
 +
<center>
 
<asy>
 
<asy>
 
draw((0,0)--(0,6)--(8,6)--(8,3)--(2.7,3)--(2.7,0)--cycle);
 
draw((0,0)--(0,6)--(8,6)--(8,3)--(2.7,3)--(2.7,0)--cycle);
 
label("$6$",(0,3),W);
 
label("$6$",(0,3),W);
 
label("$8$",(4,6),N);
 
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>
 
</asy>
 +
</center>
  
 
<math>\text{(A)}\ 14 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 48</math>
 
<math>\text{(A)}\ 14 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 48</math>
Line 138: Line 206:
  
 
== 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{(E)}</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 also ==
+
== 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]]
−
* [[1986 AJHSME]]
 
 
* [[Mathematics competition resources]]
 
* [[Mathematics competition resources]]
 +
 +
{{MAA Notice}}

Latest revision as of 22:48, 18 November 2024

1986 AJHSME (Answer Key)
Printable versions: Wiki • AoPS Resources • PDF

Instructions

  1. This is a 25-question, multiple choice test. Each question is followed by answers marked A, B, C, D and E. Only one of these is correct.
  2. You will receive ? points for each correct answer, ? points for each problem left unanswered, and ? points for each incorrect answer.
  3. No aids are permitted other than scratch paper, graph paper, ruler, compass, protractor and erasers.
  4. Figures are not necessarily drawn to scale.
  5. You will have ? minutes working time to complete the test.
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

Problem 1

In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month?

$\text{(A)}\ \frac{366}{31\times 24} \qquad \text{(B)}\ \frac{366\times 31}{24}\qquad \text{(C)}\ \frac{366\times 24}{31}\qquad \text{(D)}\ \frac{31\times 24}{366}\qquad \text{(E)}\ 366\times 31\times 24$

Solution

Problem 2

Which of the following numbers has the largest reciprocal?

$\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{2}{5} \qquad \text{(C)}\ 1 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 1986$

Solution

Problem 3

The smallest sum one could get by adding three different numbers from the set $\{ 7,25,-1,12,-3 \}$ is

$\text{(A)}\ -3 \qquad \text{(B)}\ -1 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 21$

Solution

Problem 4

The product $(1.8)(40.3+.07)$ is closest to

$\text{(A)}\ 7 \qquad \text{(B)}\ 42 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 84 \qquad \text{(E)}\ 737$

Solution

Problem 5

A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end?

$\text{(A)}\ \text{10:00 p.m.} \qquad \text{(B)}\ \text{midnight} \qquad \text{(C)}\ \text{2:30 a.m.} \qquad \text{(D)}\ \text{4:40 a.m.} \qquad \text{(E)}\ \text{6:40 a.m.}$

Solution

Problem 6

$\frac{2}{1-\frac{2}{3}}=$

$\text{(A)}\ -3 \qquad \text{(B)}\ -\frac{4}{3} \qquad \text{(C)}\ \frac{2}{3} \qquad \text{(D)}\ 2 \qquad \text{(E)}\ 6$

Solution

Problem 7

How many whole numbers are between $\sqrt{8}$ and $\sqrt{80}$?

$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 9$

Solution

Problem 8

In the product shown, $\text{B}$ is a digit. The value of $\text{B}$ is

\[\begin{array}{rr} &\text{B}2 \\ \times &7\text{B} \\ \hline &6396 \\ \end{array}\]

$\text{(A)}\ 3 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 8$

Solution

Problem 9

Using only the paths and the directions shown, how many different routes are there from $\text{M}$ to $\text{N}$?

[asy] draw((0,0)--(3,0),MidArrow); draw((3,0)--(6,0),MidArrow); draw(6*dir(60)--3*dir(60),MidArrow); draw(3*dir(60)--(0,0),MidArrow); draw(3*dir(60)--(3,0),MidArrow); draw(5.1961524227066318805823390245176*dir(30)--(6,0),MidArrow); draw(6*dir(60)--5.1961524227066318805823390245176*dir(30),MidArrow); draw(5.1961524227066318805823390245176*dir(30)--3*dir(60),MidArrow); draw(5.1961524227066318805823390245176*dir(30)--(3,0),MidArrow); label("M",6*dir(60),N); label("N",(6,0),SE); label("A",3*dir(60),NW); label("B",5.1961524227066318805823390245176*dir(30),NE); label("C",(3,0),S); label("D",(0,0),SW); [/asy]

$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$

Solution

Problem 10

A picture $3$ feet across is hung in the center of a wall that is $19$ feet wide. How many feet from the end of the wall is the nearest edge of the picture?

$\text{(A)}\ 1\frac{1}{2} \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 9\frac{1}{2} \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 22$

Solution

Problem 11

If $\text{A}*\text{B}$ means $\frac{\text{A}+\text{B}}{2}$, then $(3*5)*8$ is

$\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16\qquad \text{(E)}\ 30$

Solution

Problem 12

The table below displays the grade distribution of the $30$ 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]

$\text{(A)}\ 12\% \qquad \text{(B)}\ 25\% \qquad \text{(C)}\ 33\frac{1}{3}\% \qquad \text{(D)}\ 40\% \qquad \text{(E)}\ 50\%$

Solution

Problem 13

The perimeter of the polygon shown is

[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]

$\text{(A)}\ 14 \qquad \text{(B)}\ 20 \qquad \text{(C)}\ 28 \qquad \text{(D)}\ 48$

$\text{(E)}\ \text{cannot be determined from the information given}$

Solution

Problem 14

If $200\leq a \leq 400$ and $600\leq b\leq 1200$, then the largest value of the quotient $\frac{b}{a}$ is

$\text{(A)}\ \frac{3}{2} \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 300 \qquad \text{(E)}\ 600$

Solution

Problem 15

Sale prices at the Ajax Outlet Store are $50\%$ below original prices. On Saturdays an additional discount of $20\%$ off the sale price is given. What is the Saturday price of a coat whose original price is $$180$?

$\text{(A)}$ $$54$

$\text{(B)}$ $$72$

$\text{(C)}$ $$90$

$\text{(D)}$ $$108$

$\text{(E)}$ $$110$

Solution

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 $25\%$ 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]

$\text{(A)}\ 2.5 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 3.5 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 4.5$

Solution

Problem 17

Let $\text{o}$ be an odd whole number and let $\text{n}$ be any whole number. Which of the following statements about the whole number $(\text{o}^2+\text{no})$ is always true?

$\text{(A)}\ \text{it is always odd} \qquad \text{(B)}\ \text{it is always even}$

$\text{(C)}\ \text{it is even only if n is even} \qquad \text{(D)}\ \text{it is odd only if n is odd}$

$\text{(E)}\ \text{it is odd only if n is even}$

Solution

Problem 18

A rectangular grazing area is to be fenced off on three sides using part of a $100$ meter rock wall as the fourth side. Fence posts are to be placed every $12$ 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 $36$ m by $60$ 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]

$\text{(A)}\ 11 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 13 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$

Solution

Problem 19

At the beginning of a trip, the mileage odometer read $56,200$ miles. The driver filled the gas tank with $6$ gallons of gasoline. During the trip, the driver filled his tank again with $12$ gallons of gasoline when the odometer read $56,560$. At the end of the trip, the driver filled his tank again with $20$ gallons of gasoline. The odometer read $57,060$. To the nearest tenth, what was the car's average miles-per-gallon for the entire trip?

$\text{(A)}\ 22.5 \qquad \text{(B)}\ 22.6 \qquad \text{(C)}\ 24.0 \qquad \text{(D)}\ 26.9 \qquad \text{(E)}\ 27.5$

Solution

Problem 20

The value of the expression $\frac{(304)^5}{(29.7)(399)^4}$ is closest to

$\text{(A)}\ .003 \qquad \text{(B)}\ .03 \qquad \text{(C)}\ .3 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ 30$

Solution

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]

$\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6$

Solution

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?

$\text{(A)}\ \text{Alan, Beth} \qquad \text{(B)}\ \text{Beth, Carlos} \qquad \text{(C)}\ \text{Carlos, Diana}$

$\text{(D)}\ \text{Alan, Diana} \qquad \text{(E)}\ \text{Beth, Diana}$

Solution

Problem 23

The large circle has diameter $\text{AC}$. The two small circles have their centers on $\text{AC}$ and just touch at $\text{O}$, the center of the large circle. If each small circle has radius $1$, 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]

$\text{(A)}\ \text{between }\frac{1}{2}\text{ and 1} \qquad \text{(B)}\ 1 \qquad \text{(C)}\ \text{between 1 and }\frac{3}{2}$

$\text{(D)}\ \text{between }\frac{3}{2}\text{ and 2} \qquad \text{(E)}\ \text{cannot be determined from the information given}$

Solution

Problem 24

The $600$ 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

$\text{(A)}\ \frac{1}{27} \qquad \text{(B)}\ \frac{1}{9} \qquad \text{(C)}\ \frac{1}{8} \qquad \text{(D)}\ \frac{1}{6} \qquad \text{(E)}\ \frac{1}{3}$

Solution

Problem 25

Which of the following sets of whole numbers has the largest average?

$\text{(A)}\ \text{multiples of 2 between 1 and 101} \qquad \text{(B)}\ \text{multiples of 3 between 1 and 101}$

$\text{(C)}\ \text{multiples of 4 between 1 and 101} \qquad \text{(D)}\ \text{multiples of 5 between 1 and 101}$

$\text{(E)}\ \text{multiples of 6 between 1 and 101}$

Solution

See Also

1986 AJHSME (Problems • Answer Key • Resources)
Preceded by
1985 AJHSME 		Followed by
1987 AJHSME
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All AJHSME/AMC 8 Problems and Solutions

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