Individual Problems and Solutions
For problems and detailed solutions to each of the 1953 AHSME problems, please refer below:
Problem 1: A boy buys oranges at 3 for 10 cents. He will sell them for 5 for 20 cents. In order to make a profit of , he must sell
Answer Choices:
A. oranges
B. oranges
C. oranges
D. an infinite number of oranges
E. none of these
Solution:
Problem 2: A refrigerator is offered for sale at $$250.00$ less successive discounts of and . The sale price of the refrigerator is
Answer Choices:
A. less than
B. of
C. of
D. of
E. none of these
Solution:
Problem 3: The factors of the expression are
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 4: The roots of are
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 5: If , the value of is
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 6: Charles has quarters and Richard has quarters. The difference in their money in dimes is
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 7: The fraction reduces to
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 8: The value of at the intersection of and is
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 9: The number of ounces of water needed to reduce 9 ounces of shaving lotion containing 50% alcohol to a lotion containing 30% alcohol is
Answer Choices:
A.
B.
C.
D.
E. 7
Solution:
Problem 10: The number of revolutions of a wheel, with fixed center and with an outside diameter of 6 feet, required to cause a point on the rim to go one mile is
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 11: A running track is the ring formed by two concentric circles. It is 10 feet wide. The circumferences of the two circles differ by about
Answer Choices:
A. feet
B. feet
C. feet
D. feet
E. none of these
Solution:
Problem 12: The diameters of two circles are 8 inches and 12 inches respectively. The ratio of the area of the smaller to the area of the larger circle is
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 13: A triangle and a trapezoid are equal in area. They also have the same altitude. If the base of the triangle is 18 inches, the median of the trapezoid is
Answer Choices:
A. inches
B. inches
C. inches
D. not enough information
E. none of these answers
Solution:
Problem 14: Given the larger of two circles with center and radius and the smaller with center and radius . Draw . The following statement is false.
Answer Choices:
A. can be equal to
B. can be equal to
C. can be less than
D. can be less than
E. none of these
Solution:
Problem 15: A circular piece of metal of maximum size is cut out of a square piece and then a square piece of maximum size is cut out of the circular piece. The total amount of metal wasted is
Answer Choices:
A. the area of the original square
B. the area of the original square
C. the area of the circular piece
D. the area of the circular piece
E. none of these
Solution:
Problem 16: Adams plans a profit of on the selling price of an article and his expenses are of sales. The rate of mark-up on an article that sells for $$5.00$ is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 17: A man has part of invested at and the rest at . If his annual return on each investment is the same, the average rate of interest which he realizes on the is
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 18: One of the factors of is
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 19: In the expression , the values of and are each decreased , the value of the expression is
Answer Choices:
A. decreased
B. decreased
C. decreased of its value
D. decreased of its value
E. none of these
Solution:
Problem 20: If , then becomes
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 21: If , the value of is
Answer Choices:
A. or
B. or
C. or
D. or
E. none of these
Solution:
Problem 22: The logarithm of to the base is
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 23: The equation has
Answer Choices:
A. an extraneous root between and
B. an extraneous root between and
C. a real root between and
D. two real roots
E. two extraneous roots
Solution:
Problem 24: If and are positive integers less than 10 , then equals if
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 25: In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. Then the common ratio is
Answer Choices:
A.
B. about
C.
D.
E.
Solution:
Problem 26: The base of a triangle is 15 inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three equal areas. The length of the parallel close to the base is
Answer Choices:
A. inches
B. inches
C. inches
D. inches
E. none of these
Solution:
Problem 27: The radius of the first circle is 1 inch, that of the second inch, that of the third inch and so on indefinitely. The sum of the areas of the circles is
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 28: In triangle , sides and are opposite angles and respectively. bisects angle and meets at . Then if and the correct proportion is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 29: The number of significant digits in the measurement of the side of a square whose computed area is 1.1025 square inches to the nearest ten-thousandth of a square inch is
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 30: A house worth is sold by Mr. to Mr. at a loss. Mr. sells it to Mr. at a profit. Mr. then sells it to Mr. at a loss. Mr. sells it back to Mr. at a profit. The price paid by Mr. for the house is
Answer Choices:
A. Mr. breaks even
B. Mr. gains
C. Mr. loses
D. Mr. loses
E. Mr. gains
Solution:
Problem 31: The rails on a railroad are feet long. As the train passes over the point where the rails are joined, there is an audible click. The speed of the train in miles per hour is approximately the number of clicks heard in:
Answer Choices:
A. seconds
B. minutes
C. minutes
D. minutes
E. none of these
Solution:
Problem 32: Each angle of a rectangle is trisected. The intersections of the pairs of trisectors adjacent to the same side always form:
Answer Choices:
A. a square
B. a rectangle
C. a parallelogram with unequal sides
D. a rhombus
E. a trapezium
Solution:
Problem 33: The perimeter of an isosceles right triangle is . Its area is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 34: If one side of a triangle is 12 inches and the opposite angle is 30 degrees, then the diameter of the circumscribed circle is:
Answer Choices:
A. inches
B. inches
C. inches
D. inches
E. none of these
Solution:
Problem 35: If , then equals:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 36: Determine so that is divisible by . The obtained value, , is an exact divisor of:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 37: The base of an isosceles triangle is 6 inches and one of the equal sides is 12 inches. The radius of the circle through the vertices of the triangle is:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 38: If and , then is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 39: The product, is equal to:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 40: The contradictory of the statement, "all men are honest," is:
Answer Choices:
A. no men are honest
B. all men are dishonest
C. some men are dishonest
D. no men are dishonest
E. some men are honest
Solution:
Problem 41: A girls' camp is located 300 rods from a straight road. On this road, a boys' camp is located 500 rods from the girls' camp. It is desired to build a canteen on the road which shall be exactly the same distance from each camp. The distance of the canteen from each of the camps is:
Answer Choices:
A. rods
B. rods
C. rods
D. rods
E. none of these answers
Solution:
Problem 42: The centers of two circles are 41 inches apart. The smaller circle has a radius of 4 inches and the larger one has a radius of 5 inches. The length of the common internal tangent is:
Answer Choices:
A. inches
B. inches
C. inches
D. inches
E. inches
Solution:
Problem 43: If the price of an article is increased by per cent , then the decrease in per cent of sales must not exceed in order to yield the same income. The value of is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 44: In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equation and obtains 8 and 2 for the roots. Another student makes a mistake only in the coefficient of the first degree term and finds -9 and -1 for the roots. The correct equation was:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 45: The lengths of two line segments are units and units respectively. Then the correct relation between them is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 46: Instead of walking along two adjacent sides of a rectangular field, a boy took a short-cut along the diagonal of the field and saved a distance equal to the longer side. The ratio of the shorter side of the rectangle to the longer side was:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 47: If is greater than zero, then the correct relationship is:
Answer Choices:
A.
B.
C.
D.
E. none of these
Solution:
Problem 48: If the larger base of an isosceles trapezoid equals a diagonal and the smaller base equals the altitude, then the ratio of the smaller base to the larger base is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 49: The coordinates of and are and respectively. The value of that makes as small as possible is:
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Problem 50: One of the sides of a triangle is divided into segments of 6 and 8 units by the point of tangency of the inscribed circle. If the radius of the circle is 4 , then the length of the shortest side of the triangle is:
Answer Choices:
A. units
B. units
C. units
D. units
E. units
Solution:
The problems on this page are the property of the MAA's American Mathematics Competitions