Difference between revisions of "1988 AJHSME Problems"

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== Problem 5 ==
 
== Problem 5 ==
 +
 +
If <math>\angle \text{CBD}</math> is a right angle, then this protractor indicates that the measure of <math>\angle \text{ABC}</math> is approximately
 +
 +
<asy>
 +
unitsize(36);
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pair A,B,C,D;
 +
A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20);
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draw((1.5,0)..(0,1.5)..(-1.5,0));
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draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle);
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draw(A--B); draw(C--B); draw(D--B);
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label("O",(-2.5,0),W);
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label("A",A,W);
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label("B",B,S);
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label("C",C,W);
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label("D",D,E);
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label("0",(-1.8,0),W);
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label("20",(-1.7,.5),NW);
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label("160",(1.6,.5),NE);
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label("180",(1.7,0),E);
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</asy>
 +
 +
<math>\text{(A)}\ 20^\circ \qquad \text{(B)}\ 40^\circ \qquad \text{(C)}\ 50^\circ \qquad \text{(D)}\ 70^\circ \qquad \text{(E)}\ 120^\circ</math>
  
 
[[1988 AJHSME Problems/Problem 5|Solution]]
 
[[1988 AJHSME Problems/Problem 5|Solution]]
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== Problem 16 ==
 
== Problem 16 ==
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 +
Placing no more than one <math>\text{X}</math> in each small square, what is the greatest number of <math>\text{X}</math>'s that can be put on the grid shown without getting three <math>\text{X}</math>'s in a row vertically, horizontally, or diagonally?
 +
 +
<math>\text{(A)}\ 2 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 5 \qquad \text{(E)}\ 6</math>
 +
 +
<asy>
 +
for(int a=0; a<4; ++a)
 +
{
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  draw((a,0)--(a,3));
 +
}
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for(int b=0; b<4; ++b)
 +
{
 +
  draw((0,b)--(3,b));
 +
}
 +
</asy>
  
 
[[1988 AJHSME Problems/Problem 16|Solution]]
 
[[1988 AJHSME Problems/Problem 16|Solution]]

Revision as of 19:36, 3 April 2009

Problem 1

The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of

[asy] draw((-3,0)..(0,3)..(3,0)); draw((-3.5,0)--(-2.5,0)); draw((0,2.5)--(0,3.5)); draw((2.5,0)--(3.5,0)); draw((1.8,1.8)--(2.5,2.5)); draw((-1.8,1.8)--(-2.5,2.5)); draw((0,0)--3*dir(120),EndArrow); label("$10$",(-2.6,0),E); label("$11$",(2.6,0),W); [/asy]

$\text{(A)}\ 10.05 \qquad \text{(B)}\ 10.15 \qquad \text{(C)}\ 10.25 \qquad \text{(D)}\ 10.3 \qquad \text{(E)}\ 10.6$

Solution

Problem 2

The product $8\times .25\times 2\times .125 =$

$\text{(A)}\ \frac18 \qquad \text{(B)}\ \frac14 \qquad \text{(C)}\ \frac12 \qquad \text{(D)}\ 1 \qquad \text{(E)}\ 2$

Solution

Problem 3

$\frac{1}{10}+\frac{2}{20}+\frac{3}{30} =$

$\text{(A)}\ .1 \qquad \text{(B)}\ .123 \qquad \text{(C)}\ .2 \qquad \text{(D)}\ .3 \qquad \text{(E)}\ .6$

Solution

Problem 4

Solution

Problem 5

If $\angle \text{CBD}$ is a right angle, then this protractor indicates that the measure of $\angle \text{ABC}$ is approximately

[asy] unitsize(36); pair A,B,C,D; A=3*dir(160); B=origin; C=3*dir(110); D=3*dir(20); draw((1.5,0)..(0,1.5)..(-1.5,0)); draw((2.5,0)..(0,2.5)..(-2.5,0)--cycle); draw(A--B); draw(C--B); draw(D--B); label("O",(-2.5,0),W); label("A",A,W); label("B",B,S); label("C",C,W); label("D",D,E); label("0",(-1.8,0),W); label("20",(-1.7,.5),NW); label("160",(1.6,.5),NE); label("180",(1.7,0),E); [/asy]

$\text{(A)}\ 20^\circ \qquad \text{(B)}\ 40^\circ \qquad \text{(C)}\ 50^\circ \qquad \text{(D)}\ 70^\circ \qquad \text{(E)}\ 120^\circ$

Solution

Problem 6

$\frac{(.2)^3}{(.02)^2} =$

$\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 20$

Solution

Problem 7

$2.46\times 8.163\times (5.17+4.829)$ is closest to

$\text{(A)}\ 100 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 300 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 500$

Solution

Problem 8

Betty used a calculator to find the product $0.075 \times 2.56$. She forgot to enter the decimal points. The calculator showed $19200$. If Betty had entered the decimal points correctly, the answer would have been

$\text{(A)}\ .0192 \qquad \text{(B)}\ .192 \qquad \text{(C)}\ 1.92 \qquad \text{(D)}\ 19.2 \qquad \text{(E)}\ 192$

Solution

Problem 9

Solution

Problem 10

Chris' birthday is on a Thursday this year. What day of the week will it be $60$ days after her birthday?

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

Solution

Problem 11

$\sqrt{164}$ is

$\text{(A)}\ 42 \qquad \text{(B)}\ \text{less than }10 \qquad \text{(C)}\ \text{between }10\text{ and }11 \qquad \text{(D)}\ \text{between }11\text{ and }12 \qquad \text{(E)}\ \text{between }12\text{ and }13$

Solution

Problem 12

Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is

$\text{(A)}\ 40\text{ dollars} \qquad \text{(B)}\ 50\text{ dollars} \qquad \text{(C)}\ 80\text{ dollars} \qquad \text{(D)}\ 100\text{ dollars} \qquad \text{(E)}\ 125\text{ dollars}$

Solution

Problem 13

If rose bushes are spaced about $1$ foot apart, approximately how many bushes are needed to surround a circular patio whose radius is $12$ feet?

$\text{(A)}\ 12 \qquad \text{(B)}\ 38 \qquad \text{(C)}\ 48 \qquad \text{(D)}\ 75 \qquad \text{(E)}\ 450$

Solution

Problem 14

$\diamondsuit$ and $\Delta$ are whole numbers and $\diamondsuit \times \Delta =36$. The largest possible value of $\diamondsuit + \Delta$ is

$\text{(A)}\ 12 \qquad \text{(B)}\ 13 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 20\ \qquad \text{(E)}\ 37$

Solution

Problem 15

The reciprocal of $\left( \frac{1}{2}+\frac{1}{3}\right)$ is

$\text{(A)}\ \frac{1}{6} \qquad \text{(B)}\ \frac{2}{5} \qquad \text{(C)}\ \frac{6}{5} \qquad \text{(D)}\ \frac{5}{2} \qquad \text{(E)}\ 5$

Solution

Problem 16

Placing no more than one $\text{X}$ in each small square, what is the greatest number of $\text{X}$'s that can be put on the grid shown without getting three $\text{X}$'s in a row vertically, horizontally, or diagonally?

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

[asy] for(int a=0; a<4; ++a)  {   draw((a,0)--(a,3));  } for(int b=0; b<4; ++b)  {   draw((0,b)--(3,b));  } [/asy]

Solution

Problem 17

Solution

Problem 18

The average weight of $6$ boys is $150$ pounds and the average weight of $4$ girls is $120$ pounds. The average weight of the $10$ children is

$\text{(A)}\ 135\text{ pounds} \qquad \text{(B)}\ 137\text{ pounds} \qquad \text{(C)}\ 138\text{ pounds} \qquad \text{(D)}\ 140\text{ pounds} \qquad \text{(E)}\ 141\text{ pounds}$

Solution

Problem 19

What is the $100\text{th}$ number in the arithmetic sequence: $1,5,9,13,17,21,25,...$?

$\text{(A)}\ 397 \qquad \text{(B)}\ 399 \qquad \text{(C)}\ 401 \qquad \text{(D)}\ 403 \qquad \text{(E)}\ 405$

Solution

Problem 20

Solution

Problem 21

A fifth number, $n$, is added to the set $\{ 3,6,9,10 \}$ to make the mean of the set of five numbers equal to its median. The number of possible values of $n$ is

$\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ \text{more than }4$

Solution

Problem 22

Tom's Hat Shoppe increased all original prices by $25\%$. Now the shoppe is having a sale where all prices are $20\%$ off these increased prices. Which statement best describes the sale price of an item?

$\text{(A)}\ \text{The sale price is }5\% \text{ higher than the original price.}$

$\text{(B)}\ \text{The sale price is higher than the original price, but by less than }5\% .$

$\text{(C)}\ \text{The sale price is higher than the original price, but by more than }5\% .$

$\text{(D)}\ \text{The sale price is lower than the original price.}$

$\text{(E)}\ \text{The same price is the same as the original price.}$

Solution

Problem 23

Maria buys computer disks at a price of $4$ for <dollar/>$5$ and sells them at a price of $3$ for <dollar/>$5$. How many computer disks must she sell in order to make a profit of <dollar/>$100$?

$\text{(A)}\ 100 \qquad \text{(B)}\ 120 \qquad \text{(C)}\ 200 \qquad \text{(D)}\ 240 \qquad \text{(E)}\ 1200$

Solution

Problem 24

Solution

Problem 25

A palindrome is a whole number that reads the same forwards and backwards. If one neglects the colon, certain times displayed on a digital watch are palindromes. Three examples are: $\boxed{1:01}$, $\boxed{4:44}$, and $\boxed{12:21}$. How many times during a $12$-hour period will be palindromes?

$\text{(A)}\ 57 \qquad \text{(B)}\ 60 \qquad \text{(C)}\ 63 \qquad \text{(D)}\ 90 \qquad \text{(E)}\ 93$

Solution

See also