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Difference between revisions of "1985 AJHSME Problems"

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{{AJHSME Problems
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|year = 1985
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}}
 
== Problem 1 ==
 
== Problem 1 ==
  
 
<math>\frac{3\times 5}{9\times 11}\times \frac{7\times 9\times 11}{3\times 5\times 7}=</math>
 
<math>\frac{3\times 5}{9\times 11}\times \frac{7\times 9\times 11}{3\times 5\times 7}=</math>
  
−
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 0 \qquad \text{(C)}\ 49 \qquad \text{(D)}\ \frac{1}{49} \qquad \text{(E)}\ 50</math>
+
<math>\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)} \frac{1}{49}\ \qquad \textbf{(E)}\ 50</math>
 +
 
 +
[[1985 AJHSME Problem 1 | Solution]]
  
 
== Problem 2 ==
 
== Problem 2 ==
Line 10: Line 15:
  
  
−
<math>\text{(A)}\ 845 \qquad \text{(B)}\ 945 \qquad \text{(C)}\ 1005 \qquad \text{(D)}\ 1025 \qquad \text{(E)}\ 1045</math>
+
<math>\textbf{(A)}\ 845 \qquad \textbf{(B)}\ 945 \qquad \textbf{(C)}\ 1005 \qquad \textbf{(D)}\ 1025 \qquad \textbf{(E)}\ 1045</math>
 +
 
 +
[[1985 AJHSME Problem 2 | Solution]]
  
 
== Problem 3 ==
 
== Problem 3 ==
Line 17: Line 24:
  
  
−
<math>\text{(A)}\ .002 \qquad \text{(B)}\ .2 \qquad \text{(C)}\ 20 \qquad \text{(D)}\ 200 \qquad \text{(E)}\ 2000</math>
+
<math>\textbf{(A)}\ .002 \qquad \textbf{(B)}\ .2 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 200 \qquad \textbf{(E)}\ 2000</math>
 +
 
 +
[[1985 AJHSME Problem 3 | Solution]]
  
 
== Problem 4 ==
 
== Problem 4 ==
Line 23: Line 32:
 
The area of polygon <math>ABCDEF</math>, in square units, is
 
The area of polygon <math>ABCDEF</math>, in square units, is
  
−
<math>\text{(A)}\ 24 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 46 \qquad \text{(D)}\ 66 \qquad \text{(E)}\ 74</math>
+
<math>\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 46 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 74</math>
  
 
<asy>
 
<asy>
 
draw((0,9)--(6,9)--(6,0)--(2,0)--(2,4)--(0,4)--cycle);
 
draw((0,9)--(6,9)--(6,0)--(2,0)--(2,4)--(0,4)--cycle);
−
label("A",(0,9),NW);
+
label("$A$",(0,9),NW);
−
label("B",(6,9),NE);
+
label("$B$",(6,9),NE);
−
label("C",(6,0),SE);
+
label("$C$",(6,0),SE);
−
label("D",(2,0),SW);
+
label("$D$",(2,0),SW);
−
label("E",(2,4),NE);
+
label("$E$",(2,4),NE);
−
label("F",(0,4),SW);
+
label("$F$",(0,4),SW);
 
label("6",(3,9),N);
 
label("6",(3,9),N);
 
label("9",(6,4.5),E);
 
label("9",(6,4.5),E);
Line 38: Line 47:
 
label("5",(0,6.5),W);
 
label("5",(0,6.5),W);
 
</asy>
 
</asy>
 +
 +
[[1985 AJHSME Problem 4 | Solution]]
  
 
== Problem 5 ==
 
== Problem 5 ==
Line 70: Line 81:
 
The bar graph shows the grades in a mathematics class for the last grading period.  If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?
 
The bar graph shows the grades in a mathematics class for the last grading period.  If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?
  
−
<math>\text{(A)}\ \frac{1}{2} \qquad \text{(B)}\ \frac{2}{3} \qquad \text{(C)}\ \frac{3}{4} \qquad \text{(D)}\ \frac{4}{5} \qquad \text{(E)}\ \frac{9}{10}</math>
+
<math>\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{2}{3} \qquad \textbf{(C)}\ \frac{3}{4} \qquad \textbf{(D)}\ \frac{4}{5} \qquad \textbf{(E)}\ \frac{9}{10}</math>
 +
 
 +
[[1985 AJHSME Problem 5 | Solution]]
  
 
== Problem 6 ==
 
== Problem 6 ==
  
−
A ream of paper containing <math>500</math> sheets is <math>5</math> cm thick. Approximately how many sheets of this type of paper would there be in a stack <math>7.5</math> cm high?
+
A stack of paper containing <math>500</math> sheets is <math>5</math> cm thick. Approximately how many sheets of this type of paper would there be in a stack <math>7.5</math> cm high?
  
−
<math>\text{(A)}\ 250 \qquad \text{(B)}\ 550 \qquad \text{(C)}\ 667 \qquad \text{(D)}\ 750 \qquad \text{(E)}\ 1250</math>
+
<math>\textbf{(A)}\ 250 \qquad \textbf{(B)}\ 550 \qquad \textbf{(C)}\ 667 \qquad \textbf{(D)}\ 750 \qquad \textbf{(E)}\ 1250</math>
 +
 
 +
[[1985 AJHSME Problem 6 | Solution]]
  
 
== Problem 7 ==
 
== Problem 7 ==
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</asy>
 
</asy>
  
−
<math>\text{(A)}\ 34 \qquad \text{(B)}\ 35 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 37 \qquad \text{(E)}\ 38</math>
+
<math>\textbf{(A)}\ 34 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 38</math>
 +
 
 +
[[1985 AJHSME Problem 7 | Solution]]
  
 
== Problem 8 ==
 
== Problem 8 ==
Line 101: Line 118:
 
If <math>a = - 2</math>, the largest number in the set <math> - 3a, 4a, \frac {24}{a}, a^2, 1</math> is
 
If <math>a = - 2</math>, the largest number in the set <math> - 3a, 4a, \frac {24}{a}, a^2, 1</math> is
  
−
<math>\text{(A)}\ -3a \qquad \text{(B)}\ 4a \qquad \text{(C)}\ \frac {24}{a} \qquad \text{(D)}\ a^2 \qquad \text{(E)}\ 1</math>
+
<math>\textbf{(A)}\ -3a \qquad \textbf{(B)}\ 4a \qquad \textbf{(C)}\ \frac {24}{a} \qquad \textbf{(D)}\ a^2 \qquad \textbf{(E)}\ 1</math>
 +
 
 +
[[1985 AJHSME Problem 8 | Solution]]
  
 
== Problem 9 ==
 
== Problem 9 ==
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The product of the 9 factors <math>\Big(1 - \frac12\Big)\Big(1 - \frac13\Big)\Big(1 - \frac14\Big)\cdots\Big(1 - \frac {1}{10}\Big) =</math>
 
The product of the 9 factors <math>\Big(1 - \frac12\Big)\Big(1 - \frac13\Big)\Big(1 - \frac14\Big)\cdots\Big(1 - \frac {1}{10}\Big) =</math>
  
−
<math>\text{(A)}\ \frac {1}{10} \qquad \text{(B)}\ \frac {1}{9} \qquad \text{(C)}\ \frac {1}{2} \qquad \text{(D)}\ \frac {10}{11} \qquad \text{(E)}\ \frac {11}{2}</math>
+
<math>\textbf{(A)}\ \frac {1}{10} \qquad \textbf{(B)}\ \frac {1}{9} \qquad \textbf{(C)}\ \frac {1}{2} \qquad \textbf{(D)}\ \frac {10}{11} \qquad \textbf{(E)}\ \frac {11}{2}</math>
 +
 
 +
[[1985 AJHSME Problem 9 | Solution]]
  
 
== Problem 10 ==
 
== Problem 10 ==
Line 124: Line 145:
 
</asy>
 
</asy>
  
−
<math>\text{(A)}\ \frac{1}{4} \qquad \text{(B)}\ \frac{2}{15} \qquad \text{(C)}\ \frac{4}{15} \qquad \text{(D)}\ \frac{53}{200} \qquad \text{(E)}\ \frac{8}{15}</math>
+
<math>\textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{2}{15} \qquad \textbf{(C)}\ \frac{4}{15} \qquad \textbf{(D)}\ \frac{53}{200} \qquad \textbf{(E)}\ \frac{8}{15}</math>
 +
 
 +
[[1985 AJHSME Problem 10 | Solution ]]
  
 
== Problem 11 ==
 
== Problem 11 ==
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</asy>
 
</asy>
  
−
<math>\text{(A)}\ \text{Z} \qquad \text{(B)}\ \text{U} \qquad \text{(C)}\ \text{V} \qquad \text{(D)}\ \ \text{W} \qquad \text{(E)}\ \text{Y}</math>
+
<math>\textbf{(A)}\ \text{Z} \qquad \textbf{(B)}\ \text{U} \qquad \textbf{(C)}\ \text{V} \qquad \textbf{(D)}\ \ \text{W} \qquad \textbf{(E)}\ \text{Y}</math>
 +
 
 +
[[1985 AJHSME Problem 11 | Solution ]]
  
 
== Problem 12 ==
 
== Problem 12 ==
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A square and a triangle have equal perimeters.  The lengths of the three sides of the triangle are <math>6.2 \text{ cm}</math>, <math>8.3 \text{ cm}</math> and <math>9.5 \text{ cm}</math>.  The area of the square is
 
A square and a triangle have equal perimeters.  The lengths of the three sides of the triangle are <math>6.2 \text{ cm}</math>, <math>8.3 \text{ cm}</math> and <math>9.5 \text{ cm}</math>.  The area of the square is
  
−
<math>\text{(A)}\ 24\text{ cm}^2 \qquad \text{(B)}\ 36\text{ cm}^2 \qquad \text{(C)}\ 48\text{ cm}^2 \qquad \text{(D)}\ 64\text{ cm}^2 \qquad \text{(E)}\ 144\text{ cm}^2</math>
+
<math>\textbf{(A)}\ 24\text{ cm}^2 \qquad \textbf{(B)}\ 36\text{ cm}^2 \qquad \textbf{(C)}\ 48\text{ cm}^2 \qquad \textbf{(D)}\ 64\text{ cm}^2 \qquad \textbf{(E)}\ 144\text{ cm}^2</math>
 +
 
 +
[[1985 AJHSME Problem 12 | Solution ]]
  
 
== Problem 13 ==
 
== Problem 13 ==
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If you walk for <math>45</math> minutes at a rate of <math>4 \text{ mph}</math> and then run for <math>30</math> minutes at a rate of <math>10\text{ mph,}</math> how many miles will you have gone at the end of one hour and <math>15</math> minutes?
 
If you walk for <math>45</math> minutes at a rate of <math>4 \text{ mph}</math> and then run for <math>30</math> minutes at a rate of <math>10\text{ mph,}</math> how many miles will you have gone at the end of one hour and <math>15</math> minutes?
  
−
<math>\text{(A)}\ 3.5\text{ miles} \qquad \text{(B)}\ 8\text{ miles} \qquad \text{(C)}\ 9\text{ miles} \qquad \text{(D)}\ 25\frac{1}{3}\text{ miles} \qquad \text{(E)}\ 480\text{ miles}</math>
+
<math>\textbf{(A)}\ 3.5\text{ miles} \qquad \textbf{(B)}\ 8\text{ miles} \qquad \textbf{(C)}\ 9\text{ miles} \qquad \textbf{(D)}\ 25\frac{1}{3}\text{ miles} \qquad \textbf{(E)}\ 480\text{ miles}</math>
 +
 
 +
[[1985 AJHSME Problem 13 | Solution ]]
  
 
== Problem 14 ==
 
== Problem 14 ==
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The difference between a <math>6.5\% </math> sales tax and a <math>6\% </math> sales tax on an item priced at <math>\$20</math> before tax is
 
The difference between a <math>6.5\% </math> sales tax and a <math>6\% </math> sales tax on an item priced at <math>\$20</math> before tax is
  
−
<math>\text{(A)}</math> <math>\$.01</math>  
+
<math>\textbf{(A)}</math> <math>\$.01</math>  
  
−
<math>\text{(B)}</math> <math>\$.10</math>
+
<math>\textbf{(B)}</math> <math>\$.10</math>
 
   
 
   
−
<math>\text{(C)}</math> <math>\$ .50</math>  
+
<math>\textbf{(C)}</math> <math>\$ .50</math>
 +
 
 +
<math>\textbf{(D)}</math> <math>\$ 1</math>  
  
−
<math>\text{(D)}</math> <math>\$ 1</math>  
+
<math>\textbf{(E)}</math> <math>\$10</math>
  
−
<math>\text{(E)}</math> <math>\$10</math>
+
[[1985 AJHSME Problem 14 | Solution ]]
  
 
== Problem 15 ==
 
== Problem 15 ==
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How many whole numbers between <math>100</math> and <math>400</math> contain the digit <math>2</math>?
 
How many whole numbers between <math>100</math> and <math>400</math> contain the digit <math>2</math>?
  
−
<math>\text{(A)}\ 100 \qquad \text{(B)}\ 120 \qquad \text{(C)}\ 138 \qquad \text{(D)}\ 140 \qquad \text{(E)}\ 148</math>
+
<math>\textbf{(A)}\ 100 \qquad \textbf{(B)}\ 120 \qquad \textbf{(C)}\ 138 \qquad \textbf{(D)}\ 140 \qquad \textbf{(E)}\ 148</math>
 +
 
 +
[[1985 AJHSME Problem 15 | Solution ]]
 +
 
  
 
== Problem 16 ==
 
== Problem 16 ==
Line 183: Line 217:
 
The ratio of boys to girls in Mr. Brown's math class is <math>2:3</math>.  If there are <math>30</math> students in the class, how many more girls than boys are in the class?
 
The ratio of boys to girls in Mr. Brown's math class is <math>2:3</math>.  If there are <math>30</math> students in the class, how many more girls than boys are in the class?
  
−
<math>\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 6 \qquad \text{(E)}\ 10</math>
+
<math>\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 2</math>
 +
 
 +
[[1985 AJHSME Problem 16 | Solution ]]
  
 
== Problem 17 ==
 
== Problem 17 ==
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If your average score on your first six mathematics tests was <math>84</math> and your average score on your first seven mathematics tests was <math>85</math>, then your score on the seventh test was
 
If your average score on your first six mathematics tests was <math>84</math> and your average score on your first seven mathematics tests was <math>85</math>, then your score on the seventh test was
  
−
<math>\text{(A)}\ 86 \qquad \text{(B)}\ 88 \qquad \text{(C)}\ 90 \qquad \text{(D)}\ 91 \qquad \text{(E)}\ 92</math>
+
<math>\textbf{(A)}\ 86 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 91 \qquad \textbf{(E)}\ 92</math>
 +
 
 +
[[1985 AJHSME Problem 17 | Solution ]]
  
 
== Problem 18 ==
 
== Problem 18 ==
Line 195: Line 233:
 
Nine copies of a certain pamphlet cost less than <math>\$10.00</math> while ten copies of the same pamphlet (at the same price) cost more than <math>\$11.00</math>.  How much does one copy of this pamphlet cost?
 
Nine copies of a certain pamphlet cost less than <math>\$10.00</math> while ten copies of the same pamphlet (at the same price) cost more than <math>\$11.00</math>.  How much does one copy of this pamphlet cost?
  
−
<math>\text{(A)}</math> <math>\$1.07</math>  
+
<math>\textbf{(A)}</math> <math>\$1.07</math>  
  
−
<math>\text{(B)}</math> <math>\$1.08</math>
+
<math>\textbf{(B)}</math> <math>\$1.08</math>
  
−
<math>\text{(C)}</math> <math>\$1.09</math>  
+
<math>\textbf{(C)}</math> <math>\$1.09</math>  
  
−
<math>\text{(D)}</math> <math>\$1.10</math>  
+
<math>\textbf{(D)}</math> <math>\$1.10</math>  
  
−
<math>\text{(E)}</math> <math>\$1.11</math>
+
<math>\textbf{(E)}</math> <math>\$1.11</math>
 +
 
 +
[[1985 AJHSME Problem 18 | Solution ]]
  
 
== Problem 19 ==
 
== Problem 19 ==
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If the length and width of a rectangle are each increased by <math>10\% </math>, then the perimeter of the rectangle is increased by
 
If the length and width of a rectangle are each increased by <math>10\% </math>, then the perimeter of the rectangle is increased by
  
−
<math>\text{(A)}\ 1\% \qquad \text{(B)}\ 10\% \qquad \text{(C)}\ 20\% \qquad \text{(D)}\ 21\% \qquad \text{(E)}\ 40\% </math>
+
<math>\textbf{(A)}\ 1\% \qquad \textbf{(B)}\ 10\% \qquad \textbf{(C)}\ 20\% \qquad \textbf{(D)}\ 21\% \qquad \textbf{(E)}\ 40\% </math>
 +
 
 +
[[1985 AJHSME Problem 19 | Solution ]]
  
 
== Problem 20 ==
 
== Problem 20 ==
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In a certain year, January had exactly four Tuesdays and four Saturdays.  On what day did January <math>1</math> fall that year?
 
In a certain year, January had exactly four Tuesdays and four Saturdays.  On what day did January <math>1</math> fall that year?
  
−
<math>\text{(A)}\ \text{Monday} \qquad \text{(B)}\ \text{Tuesday} \qquad \text{(C)}\ \text{Wednesday} \qquad \text{(D)}\ \text{Friday} \qquad \text{(E)}\ \text{Saturday}</math>
+
<math>\textbf{(A)}\ \text{Monday} \qquad \textbf{(B)}\ \text{Tuesday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Friday} \qquad \textbf{(E)}\ \text{Saturday}</math>
 +
 
 +
[[1985 AJHSME Problem 20 | Solution ]]
  
 
== Problem 21 ==
 
== Problem 21 ==
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Mr. Green receives a <math>10\% </math> raise every year.  His salary after four such raises has gone up by what percent?
 
Mr. Green receives a <math>10\% </math> raise every year.  His salary after four such raises has gone up by what percent?
  
−
<math>\text{(A)}\ \text{less than }40\% \qquad \text{(B)}\ 40\% \qquad \text{(C)}\ 44\% \qquad \text{(D)}\ 45\% \qquad \text{(E)}\ \text{more than }45\% </math>
+
<math>\textbf{(A)}\ \text{less than }40\% \qquad \textbf{(B)}\ 40\% \qquad \textbf{(C)}\ 44\% \qquad \textbf{(D)}\ 45\% \qquad \textbf{(E)}\ \text{more than }45\% </math>
 +
 
 +
[[1985 AJHSME Problem 21 | Solution ]]
  
 
== Problem 22 ==
 
== Problem 22 ==
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Assume every 7-digit whole number is a possible telephone number except those that begin with <math>0</math> or <math>1</math>.  What fraction of telephone numbers begin with <math>9</math> and end with <math>0</math>?
 
Assume every 7-digit whole number is a possible telephone number except those that begin with <math>0</math> or <math>1</math>.  What fraction of telephone numbers begin with <math>9</math> and end with <math>0</math>?
  
−
<math>\text{(A)}\ \frac{1}{63} \qquad \text{(B)}\ \frac{1}{80} \qquad \text{(C)}\ \frac{1}{81} \qquad \text{(D)}\ \frac{1}{90} \qquad \text{(E)}\ \frac{1}{100}</math>
+
<math>\textbf{(A)}\ \frac{1}{63} \qquad \textbf{(B)}\ \frac{1}{80} \qquad \textbf{(C)}\ \frac{1}{81} \qquad \textbf{(D)}\ \frac{1}{90} \qquad \textbf{(E)}\ \frac{1}{100}</math>
  
 
''Note: All telephone numbers are 7-digit whole numbers.''
 
''Note: All telephone numbers are 7-digit whole numbers.''
 +
 +
[[1985 AJHSME Problem 22 | Solution ]]
 +
  
 
== Problem 23 ==
 
== Problem 23 ==
  
−
King Middle School has <math>1200</math> students.  Each student takes <math>5</math> classes a day.  Each teacher teaches <math>4</math> classes.  Each class has <math>30</math> students and <math>1</math> teacher.  How many teachers are there at King Middle School?   
+
King Middle School has <math>1200</math> students.  Each pupil takes <math>5</math> classes a day.  Each teacher teaches <math>4</math> classes.  Each class has <math>30</math> students and <math>1</math> teacher.  How many teachers are there at King Middle School?   
 +
 
 +
<math>\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 50</math>
 +
 
 +
[[1985 AJHSME Problem 23 | Solution ]]
  
−
<math>\text{(A)}\ 30 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 40 \qquad \text{(D)}\ 45 \qquad \text{(E)}\ 50</math>
 
  
 
== Problem 24 ==
 
== Problem 24 ==
  
−
In a magic triangle, each of the six whole numbers <math>10-15</math> is placed in one of the circles so that the sum, <math>S</math>, of the three numbers on each side of the triangle is the same.  The largest possible value for <math>S</math> is
+
In a magic triangle, each of the six whole numbers <math>10\text{--}15</math> is placed in one of the circles so that the sum, <math>S</math>, of the three numbers on each side of the triangle is the same.  The largest possible value for <math>S</math> is
  
 
<asy>
 
<asy>
Line 256: Line 308:
 
</asy>
 
</asy>
  
−
<math>\text{(A)}\ 36 \qquad \text{(B)}\ 37 \qquad \text{(C)}\ 38 \qquad \text{(D)}\ 39 \qquad \text{(E)}\ 40</math>
+
<math>\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 38 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 40</math>
 +
 
 +
[[1985 AJHSME Problem 24 | Solution ]]
 +
 
  
−
[[1985 AJHSME Problems/Problem 24|Solution]]
+
== Problem 25  ==
  
−
== Problem 25 ==
 
  
 
Five cards are lying on a table as shown.
 
Five cards are lying on a table as shown.
Line 270: Line 324:
 
Each card has a letter on one side and a whole number on the other side.  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?
 
Each card has a letter on one side and a whole number on the other side.  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?
  
−
<math>\text{(A)}\ 3 \qquad \text{(B)}\ 4 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ \text{P} \qquad \text{(E)}\ \text{Q}</math>
+
<math>\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ \text{P} \qquad \textbf{(E)}\ \text{Q}</math>
  
−
[[1985 AJHSME Problems/Problem 25|Solution]]
+
[[1985 AJHSME Problem 25 | Solution ]]
  
 
== See Also ==
 
== See Also ==

Latest revision as of 17:42, 24 November 2021

1985 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

$\frac{3\times 5}{9\times 11}\times \frac{7\times 9\times 11}{3\times 5\times 7}=$

$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 0 \qquad \textbf{(C)}\ 49 \qquad \textbf{(D)} \frac{1}{49}\ \qquad \textbf{(E)}\ 50$

Solution

Problem 2

$90+91+92+93+94+95+96+97+98+99=$


$\textbf{(A)}\ 845 \qquad \textbf{(B)}\ 945 \qquad \textbf{(C)}\ 1005 \qquad \textbf{(D)}\ 1025 \qquad \textbf{(E)}\ 1045$

Solution

Problem 3

$\frac{10^7}{5\times 10^4}=$


$\textbf{(A)}\ .002 \qquad \textbf{(B)}\ .2 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 200 \qquad \textbf{(E)}\ 2000$

Solution

Problem 4

The area of polygon $ABCDEF$, in square units, is

$\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 46 \qquad \textbf{(D)}\ 66 \qquad \textbf{(E)}\ 74$

[asy] draw((0,9)--(6,9)--(6,0)--(2,0)--(2,4)--(0,4)--cycle); label("$A$",(0,9),NW); label("$B$",(6,9),NE); label("$C$",(6,0),SE); label("$D$",(2,0),SW); label("$E$",(2,4),NE); label("$F$",(0,4),SW); label("6",(3,9),N); label("9",(6,4.5),E); label("4",(4,0),S); label("5",(0,6.5),W); [/asy]

Solution

Problem 5

[asy] unitsize(13); draw((0,0)--(20,0)); draw((0,0)--(0,15)); draw((0,3)--(-1,3)); draw((0,6)--(-1,6)); draw((0,9)--(-1,9)); draw((0,12)--(-1,12)); draw((0,15)--(-1,15)); fill((2,0)--(2,15)--(3,15)--(3,0)--cycle,black); fill((4,0)--(4,12)--(5,12)--(5,0)--cycle,black); fill((6,0)--(6,9)--(7,9)--(7,0)--cycle,black); fill((8,0)--(8,9)--(9,9)--(9,0)--cycle,black); fill((10,0)--(10,15)--(11,15)--(11,0)--cycle,black); label("A",(2.5,-.5),S); label("B",(4.5,-.5),S); label("C",(6.5,-.5),S); label("D",(8.5,-.5),S); label("F",(10.5,-.5),S); label("Grade",(15,-.5),S); label("$1$",(-1,3),W); label("$2$",(-1,6),W); label("$3$",(-1,9),W); label("$4$",(-1,12),W); label("$5$",(-1,15),W); [/asy]

The bar graph shows the grades in a mathematics class for the last grading period. If A, B, C, and D are satisfactory grades, what fraction of the grades shown in the graph are satisfactory?

$\textbf{(A)}\ \frac{1}{2} \qquad \textbf{(B)}\ \frac{2}{3} \qquad \textbf{(C)}\ \frac{3}{4} \qquad \textbf{(D)}\ \frac{4}{5} \qquad \textbf{(E)}\ \frac{9}{10}$

Solution

Problem 6

A stack 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?

$\textbf{(A)}\ 250 \qquad \textbf{(B)}\ 550 \qquad \textbf{(C)}\ 667 \qquad \textbf{(D)}\ 750 \qquad \textbf{(E)}\ 1250$

Solution

Problem 7

A "stair-step" figure is made of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. The number of black squares in the $37\text{th}$ row is

[asy] draw((0,0)--(7,0)--(7,1)--(0,1)--cycle); draw((1,0)--(6,0)--(6,2)--(1,2)--cycle); draw((2,0)--(5,0)--(5,3)--(2,3)--cycle); draw((3,0)--(4,0)--(4,4)--(3,4)--cycle); fill((1,0)--(2,0)--(2,1)--(1,1)--cycle,black); fill((3,0)--(4,0)--(4,1)--(3,1)--cycle,black); fill((5,0)--(6,0)--(6,1)--(5,1)--cycle,black); fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,black); fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,black); fill((3,2)--(4,2)--(4,3)--(3,3)--cycle,black); [/asy]

$\textbf{(A)}\ 34 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 37 \qquad \textbf{(E)}\ 38$

Solution

Problem 8

If $a = - 2$, the largest number in the set $- 3a, 4a, \frac {24}{a}, a^2, 1$ is

$\textbf{(A)}\ -3a \qquad \textbf{(B)}\ 4a \qquad \textbf{(C)}\ \frac {24}{a} \qquad \textbf{(D)}\ a^2 \qquad \textbf{(E)}\ 1$

Solution

Problem 9

The product of the 9 factors $\Big(1 - \frac12\Big)\Big(1 - \frac13\Big)\Big(1 - \frac14\Big)\cdots\Big(1 - \frac {1}{10}\Big) =$

$\textbf{(A)}\ \frac {1}{10} \qquad \textbf{(B)}\ \frac {1}{9} \qquad \textbf{(C)}\ \frac {1}{2} \qquad \textbf{(D)}\ \frac {10}{11} \qquad \textbf{(E)}\ \frac {11}{2}$

Solution

Problem 10

The fraction halfway between $\frac{1}{5}$ and $\frac{1}{3}$ (on the number line) is

[asy] unitsize(12); draw((-1,0)--(20,0),EndArrow); draw((0,-.75)--(0,.75)); draw((10,-.75)--(10,.75)); draw((17,-.75)--(17,.75)); label("$0$",(0,-.5),S); label("$\frac{1}{5}$",(10,-.5),S); label("$\frac{1}{3}$",(17,-.5),S); [/asy]

$\textbf{(A)}\ \frac{1}{4} \qquad \textbf{(B)}\ \frac{2}{15} \qquad \textbf{(C)}\ \frac{4}{15} \qquad \textbf{(D)}\ \frac{53}{200} \qquad \textbf{(E)}\ \frac{8}{15}$

Solution

Problem 11

A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $\text{X}$ is

[asy] draw((0,0)--(0,1)--(2,1)--(2,2)--(3,2)--(3,0)--(2,0)--(2,-2)--(1,-2)--(1,0)--cycle); draw((1,0)--(1,1)); draw((2,0)--(2,1)); draw((1,0)--(2,0)); draw((1,-1)--(2,-1)); draw((2,1)--(3,1)); label("U",(.5,.3),N); label("V",(1.5,.3),N); label("W",(2.5,.3),N); label("X",(1.5,-.7),N); label("Y",(2.5,1.3),N); label("Z",(1.5,-1.7),N); [/asy]

$\textbf{(A)}\ \text{Z} \qquad \textbf{(B)}\ \text{U} \qquad \textbf{(C)}\ \text{V} \qquad \textbf{(D)}\ \ \text{W} \qquad \textbf{(E)}\ \text{Y}$

Solution

Problem 12

A square and a triangle have equal perimeters. The lengths of the three sides of the triangle are $6.2 \text{ cm}$, $8.3 \text{ cm}$ and $9.5 \text{ cm}$. The area of the square is

$\textbf{(A)}\ 24\text{ cm}^2 \qquad \textbf{(B)}\ 36\text{ cm}^2 \qquad \textbf{(C)}\ 48\text{ cm}^2 \qquad \textbf{(D)}\ 64\text{ cm}^2 \qquad \textbf{(E)}\ 144\text{ cm}^2$

Solution

Problem 13

If you walk for $45$ minutes at a rate of $4 \text{ mph}$ and then run for $30$ minutes at a rate of $10\text{ mph,}$ how many miles will you have gone at the end of one hour and $15$ minutes?

$\textbf{(A)}\ 3.5\text{ miles} \qquad \textbf{(B)}\ 8\text{ miles} \qquad \textbf{(C)}\ 9\text{ miles} \qquad \textbf{(D)}\ 25\frac{1}{3}\text{ miles} \qquad \textbf{(E)}\ 480\text{ miles}$

Solution

Problem 14

The difference between a $6.5\%$ sales tax and a $6\%$ sales tax on an item priced at $$20$ before tax is

$\textbf{(A)}$ $$.01$

$\textbf{(B)}$ $$.10$

$\textbf{(C)}$ $$ .50$

$\textbf{(D)}$ $$ 1$

$\textbf{(E)}$ $$10$

Solution

Problem 15

How many whole numbers between $100$ and $400$ contain the digit $2$?

$\textbf{(A)}\ 100 \qquad \textbf{(B)}\ 120 \qquad \textbf{(C)}\ 138 \qquad \textbf{(D)}\ 140 \qquad \textbf{(E)}\ 148$

Solution


Problem 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?

$\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 2$

Solution

Problem 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

$\textbf{(A)}\ 86 \qquad \textbf{(B)}\ 88 \qquad \textbf{(C)}\ 90 \qquad \textbf{(D)}\ 91 \qquad \textbf{(E)}\ 92$

Solution

Problem 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?

$\textbf{(A)}$ $$1.07$

$\textbf{(B)}$ $$1.08$

$\textbf{(C)}$ $$1.09$

$\textbf{(D)}$ $$1.10$

$\textbf{(E)}$ $$1.11$

Solution

Problem 19

If the length and width of a rectangle are each increased by $10\%$, then the perimeter of the rectangle is increased by

$\textbf{(A)}\ 1\% \qquad \textbf{(B)}\ 10\% \qquad \textbf{(C)}\ 20\% \qquad \textbf{(D)}\ 21\% \qquad \textbf{(E)}\ 40\%$

Solution

Problem 20

In a certain year, January had exactly four Tuesdays and four Saturdays. On what day did January $1$ fall that year?

$\textbf{(A)}\ \text{Monday} \qquad \textbf{(B)}\ \text{Tuesday} \qquad \textbf{(C)}\ \text{Wednesday} \qquad \textbf{(D)}\ \text{Friday} \qquad \textbf{(E)}\ \text{Saturday}$

Solution

Problem 21

Mr. Green receives a $10\%$ raise every year. His salary after four such raises has gone up by what percent?

$\textbf{(A)}\ \text{less than }40\% \qquad \textbf{(B)}\ 40\% \qquad \textbf{(C)}\ 44\% \qquad \textbf{(D)}\ 45\% \qquad \textbf{(E)}\ \text{more than }45\%$

Solution

Problem 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 $0$?

$\textbf{(A)}\ \frac{1}{63} \qquad \textbf{(B)}\ \frac{1}{80} \qquad \textbf{(C)}\ \frac{1}{81} \qquad \textbf{(D)}\ \frac{1}{90} \qquad \textbf{(E)}\ \frac{1}{100}$

Note: All telephone numbers are 7-digit whole numbers.

Solution


Problem 23

King Middle School has $1200$ students. Each pupil 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?

$\textbf{(A)}\ 30 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 45 \qquad \textbf{(E)}\ 50$

Solution


Problem 24

In a magic triangle, each of the six whole numbers $10\text{--}15$ 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

[asy] draw(circle((0,0),1)); draw(dir(60)--6*dir(60)); draw(circle(7*dir(60),1)); draw(8*dir(60)--13*dir(60)); draw(circle(14*dir(60),1)); draw((1,0)--(6,0)); draw(circle((7,0),1)); draw((8,0)--(13,0)); draw(circle((14,0),1)); draw(circle((10.5,6.0621778264910705273460621952706),1)); draw((13.5,0.86602540378443864676372317075294)--(11,5.1961524227066318805823390245176)); draw((10,6.9282032302755091741097853660235)--(7.5,11.258330249197702407928401219788)); [/asy]

$\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 37 \qquad \textbf{(C)}\ 38 \qquad \textbf{(D)}\ 39 \qquad \textbf{(E)}\ 40$

Solution


Problem 25

Five cards are lying on a table as shown.

\[\begin{matrix} & \qquad & \boxed{\tt{P}} & \qquad & \boxed{\tt{Q}} \\ \\ \boxed{\tt{3}} & \qquad & \boxed{\tt{4}} & \qquad & \boxed{\tt{6}} \end{matrix}\]

Each card has a letter on one side and a whole number on the other side. 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?

$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ \text{P} \qquad \textbf{(E)}\ \text{Q}$

Solution

See Also

1985 AJHSME (Problems • Answer Key • Resources)
Preceded by
First
AJHSME 		Followed by
1986 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

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