Points A(0, 0), B(6, 0), C(6, 10), and D(0, 10) are vertices of rectangle ABCD; and Eis on segment CDat (2, 10). What is the ratio of the area of triangle ADE to the area of quadrilateral ABCE?Express your answer as a common fraction.
A palindromeis a number that reads the same forward as backward. For example, 343 and 1221 are palindromes. What is the least natural number that can be added to 40305 to create a palindrome?
The arithmetic mean of nine numbers is 54. If two numbers, uand v,are added to the list, the mean of the eleven-number list becomes 66. What is the mean of uand v?
Suppose that
and that
What is the value of
Express your answer as a common fraction.
z ? x+y
.
= y z
14 3
= x y
4 7
Each of the four digits 2, 4, 6, and 9 is placed in one of the boxes to form a frac-tion. The numerator and the denominator
are both two-digit whole numbers. What is the small-est value of all the common fractions that can be formed? Express your answer as a com-mon fraction.
The numbers on a standard six-faced die are arranged such that numbers on oppo-site faces always add to 7. The product of the numbers appearing on the four lateral faces of a rolled die is calculated (ignoring the numbers on the top and bottom). What is the maximum possible value of this product?
The gasoline gauge on a van initially read 1/8 full. When 15 gallons of gasoline were added to the tank, the gauge read 3/4 full. How many more gallons are needed to fill the tank?
When Tomas enters a classroom at exactly 9:00 AM, the twelve-hour analog clock on the wall is behaving strangely. The clock reads 4:20, and the second hand is racing. It makes one complete circle every four seconds. The minute hand and hour hand behave as if every full rotation of the second hand indi-cates that a minute has passed. When Tomas leaves the class at 9:50 AM, what
time does the clock on the wall read?
At 10:00 AM, Boon Tee is the 225th person in line to ride the Rocker Roller Coaster. Each roller coaster train holds 36 people. A full train leaves every four minutes. If the first 36 people in line leave on the 10:01 train, what time will Boon Tee’s train leave?
Angle PQRis a right angle. The three quadrilaterals shown are squares. The sum
of the areas of the three squares is 338 square centimeters. What is the number of square centimeters in the area of the largest square?
Three fair, standard six-faced dice of dif-ferent colors are rolled. In how many ways can the dice be rolled such that the sum of the numbers rolled is 10?
If
then what is the value of + x
x ?
1
3 3 + = x
x ,
1 4
The triangle with vertices A(6, 1), B(4, 1), and C(4, 4) is rotated 90 degrees counterclockwise about B. What are the coordinates of the image of C(the point where Cis located after the rotation)? Express your answer as an ordered pair.
Container I holds eight red balls and four green balls; containers II and III each hold two red balls and four green balls. A container is selected at random, and a ball is randomly selected from that con-tainer. What is the probability that the ball selected is green? Express your answer as a common fraction.
In the counting game bing-bong, Arlene starts counting at 1 but skips all multiples of 3 and all numbers that contain the digit 3. For example, Arlene counts as follows: 1, 2, 4, 5, 7, 8, 10, 11, 14, 16, . . . . What is the fortieth number in this sequence? A cube is sliced by a plane that goes through
two opposite corners and the midpoints of two edges, as shown. If the cube has an edge length of one unit, how many square units
are in the area of the rhombus formed by the intersection of the plane and the cube? Express your answer as a common fraction in simplest radical form.
The positive integers are written consecu-tively in the pattern below. What integer will be the eighth entry in row A?
Rich invested $100 seven years ago. Since then, his investment has doubled in value to $200. If Rich’s money continues to dou-ble every seven years, in how many years will his $200 grow to $1600?
A stack of 100 nickels has a height of 6.25 inches. What is the value, in dollars, of an 8-foot-high stack of nickels? Express your answer to the nearest hundredth.
Of the 6.25 billion people in the world, 310 million live in North America. What percent of the world’s population lives in North America? Express your answer to the nearest whole number percent.
How many full seven-day weeks are in seven consecutive years? Assume that the first day of the first year is the first day of the week.
What is the slope of a line parallel to 2x+4y=–17? Express your answer as a common fraction.
On the grid below, Beatrice draws all the lines with integral y-intercepts and slope 1 or –1. The lines
form many inter-section points. How many of these intersection points lie in the interior of the shaded region?
Paco uses a spinner to select a number from 1 through 5, each of which has equal probability of being selected. Manu uses a different spinner to select a number from 1 through 10, each of which has equal probability. What is the probability that the product of Manu’s number and Paco’s number is less than 30? Express your answer as a common fraction.
In the arithmetic sequence 17, a, b, c,41, what is the value of b?
In a three-digit number, the hundreds digit is greater than 5, the tens digit is greater than 4 but less than 8, and the units digit is the smallest prime number. How many three-digit numbers satisfy all these conditions?
The sum of nine consecutive integers is 9. What is the least of these nine integers?
What is the smallest positive integer n such that the value 7 +30 ×nis not a prime number?
Two complementary angles, Aand B,have measures in the ratio of 7 to 23, respec-tively. What is the ratio of the measure of the complement of angle Ato the measure of the complement of angle B?Express your answer as a common fraction.
The trip from Carville to Nikpath requires 4 1/2 hours when traveling at an average speed of 70 miles per hour. How many hours does the trip require when traveling at an average speed of 60 miles per hour? Express your answer as a decimal to the nearest hundredth.
NOVEMBER
© National Council of Teachers of Mathematics, 1906 Association Drive, Reston, VA 20191-1502
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Row A 4 10 Row B 3 5 9 11 Row C 2 6 8 12 14 Row D 1 7 13
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x 5
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1. $76.80. Set up a ratio comparing the number of nickels and the height of the nickels; let nrepresent the number of nickels in an 8-foot-high stack. Rewrite 8 feet as 96 inches, and the ratio is
which can be rewritten as 9600 =6.25x. Dividing both sides by 6.25 produces x=
1536; thus, 1536 nickels will be in a stack that is 8 feet tall. Since each nickel is worth $0.05, we multiply 0.05 by 1536 to get 76.8, so the total value of the nickels is $76.80.
2. 8. Draw the lines described and count the number of intersection points that are in the shaded area.
3. –3. If we let xrepresent the first of the integers, each successive integer will be 1 more than the previous one, so the sum of all the integers is
x+(x+1) +(x+2) +(x+3) +(x +4) +(x+5) +(x+6) +(x+7) +(x+8)
=9x+36. The sum should be 9, so we have the equation 9x+36 =9, which is equiva-lent to 9x =–27, which is equivalent to
x=–3.
4. square units.
First, we must find the length of the side of the rhombus. The side of the rhombus, along with the edge of the cube and half of the edge of the cube, as illustrated in bold in the drawing, form a right triangle. Since the lengths of the two legs are 1/2 and 1, the remaining side length can be found:
Thus,
To find the area of the rhombus, we note that the rhombus is made up of two isosceles triangles whose side lengths are the same as the length of the side of the rhombus. The remaining side of the tri-angle is the segment connecting the two midpoints of the sides. That segment has the same length as the diagonal of one of the faces. Since the length of the edges of each face is 1, the diagonal has length 12. To find the area of the isosceles triangle, we note that it has two sides of length
15/2 and a base of 12. When drawing the altitude, we will have a right triangle with hypotenuse 15/2 and a leg that is half of the base, or 12/2. Thus, to find the height h,we need to solve
=
s 5. 2 = + ⎛⎝⎜ ⎞⎠⎟ = + =
s 1 1 2 1 1
4 5 4
2
2
6 2/
y
x
5 4 3 2 1
1 2 3 4 = x
. ,
100 6 25 96
Edited by JUDITH COVINGTON,jcovingt@ pilot.lsus.edu, Louisiana State University Shreveport, Shreveport, LA 71115
Problems 1–7 are from MATHCOUNTS 2003 State Target Round. Problems 8–30 are from MATHCOUNTS State Spring Round.
The Editorial Panel of the Mathematics
Teacheris considering sets of problems sub-mitted by individuals, classes of prospective teachers, and mathematics clubs for publica-tion in the monthly “Calendar” during the 2005–2006 academic year. Please write to the Mathematics Teacher editor, 1906 Association Drive, Reston, VA 20191-1502, for guidelines. Calendar problems can also be sent to [email protected].
Three other sources of problems in calendar
form are available from NCTM: Calendar
Problems from the Mathematics Teacher (a book featuring more than 400 problems, organized by topic, order number 12059, $23.95), “Calendars for the Calculating,” vol. 2 (a set of nine monthly calendars that originally appeared from September 1987 to May 1988, order number 496, $13.50), and “A Year of Mathematics” (one annual calendar that origi-nally appeared in September 1982, order num-ber 311, $4.00; set of five, order numnum-ber 312, $8.00). Individual members receive a 20 per-cent discount off these prices. Write to NCTM to request the catalog of educational materials, which includes a listing for the publication
Exploratory Problems in Mathematics (order number 495, $23.95). An online version of the
catalog is available at www.nctm.org. —Ed.
SOLUTIONS
which is equivalent to
which yields h2=
3/4 and h=13/2. The area of one of the triangles is
The two triangles will thus have an area twice that, or 16/2.
5. 5 percent. To find the percent, we need to divide the total population by the population that lives in North Amer-ica. First, however, we need to be sure that the units of both are the same; this unit will be millions. We write 6.25 bil-lion as 6250 milbil-lion, and divide 310 by 6250 to get 0.0496, which is approxi-mately 5 percent.
6. 41/50. Since Paco chooses between five numbers, the probability that he chooses any one of them is 1/5. Since Manu chooses between ten numbers, the probability that he chooses any one of them is 1/10. Examine the probability based on Paco’s choice. If Paco chooses 1 or 2, any number that Manu chooses re-sults in a product that is less than 30. The probability of choosing 1 is 1/5, and the probability of choosing 2 is 1/5. If Paco chooses 3, nine of the numbers that Manu can choose will result in a product that is less than 30, so the probability is
If Paco chooses 4, Manu can choose seven numbers that will result in a product that is less than 30. This probability is
Finally, if Paco chooses 5, five numbers will result in a product that is less than 30, and the probability will be
Writing all the probabilities with a de-nominator of 50 and adding produces
7. 6. We can easily see that if n =7, then 7 +30(7) is divisible by 7 and is not a prime number. We must then examine whether this outcome will occur with any values that are less than 7. We re-place nwith 6, which results in the value of 7 +30(6), or 187; since 187 equals 11 ×17, we know that 187 is not prime. We next examine the values that will result in replacing nwith 5, 4, 3, 2, and 1, resulting in values of 157, 127, 97, 67, and 37. All these values are prime, so 6 is the smallest integer that results in 7 +30nnot being prime. 8. 46. Each number in row A is 6 more than the previous number. We can write the nth number in row A as 4 +6(n– 1). If n=8, the value is
4 +6(7) =4 +42 =46.
9. 365. A year consists of 365 days, with some years having an extra day. During a seven-year period, one or two leap years can occur. At least 7(365), or 2555, days plus 1 or 2 extra occur in seven years. In all three cases, the total number of days divided by 7 days per week produces a whole number of 365. 10. 29. In an arithmetic sequence, the same value is added to each term to ob-tain the next term. Represent this value by k.To get from 17 to 41, this value has been added four times. Then 17 +4k=
41, so 4k=24 and k=6. Thus,
b=17 +2k
=17 +12 =29.
11. 23/7. Represent one angle measure by 7x;then the other angle measure must be 23xto have a ratio of 7 to 23. The an-gles are complementary, so 90 – 7x=23x
and 90 – 23x =7x. The complement of the 7xangle is 23x, and the complement of the 23xangle is 7x. The ratio of the complementary angles is 23x/7x,or 23/7. 12. Twenty-one years. To make $200 become $1600, we must double the
money three times: $200 will double to $400, then $400 will double to $800, and $800 will double to $1600. Each doubling period takes seven years, for a total of twenty-one years.
13. –1/2. If a line is in the form Ax +By
=C,the slope will be –A/B. Thus, the slope of the given line is –2/4, or –1/2. Since parallel lines have the same slope, the parallel line also has a slope of –1/2. 14. 12. Three numbers are chosen for a three-digit number. The first digit must be greater than 5, so four choices are possible for the first digit. Three choices (5, 6, or 7) are possible for the second digit. The units digit is the smallest prime number, which is 2, so only one choice is possible for the last digit. A total of 4 ×3 ×1, or twelve, choices are possible for the digits, so twelve three-digit numbers satisfy the given condition.
15. 5.25 hours. The distance can be cal-culated by multiplying the speed by time, so the distance is 4.5 times 70, which is 315 miles. To find the time when the speed is 60 MPH, divide 315 by 60, which is 5.25 hours.
16. 1/5. Graphing the four points indi-cates that the lower-left corner of the rectangle is at the origin. The lengths of the sides of the rectangle are 6 and 10. The point (2, 10) lies on the top side of the rectangle. Connecting point
Ato point Eproduces right triangle
ADE. Triangle ADEhas AD=10 and
DE=2, so the area of triangle ADEis 0.5(10)(2), or 10. The area of the rec-tangle is 6(10) =60, so the area of quadrilateral ABCEis 60 – 10 =50. Thus, the ratio is 10/50, which is 1/5.
D E C
B A
y
x 0 2 4 6
50 10 10
50 50
10 50
9 50
7 50
5 50
41 50
+ + + + = .
× = .
1 5
5 10
5 50
× = .
1 5
7 10
7 50
× = .
1 5
9 10
9 50
× × = .
1
2 2
3 2
6 4 = +h , 5
4 2 4
2
⎛ ⎝⎜
⎞ ⎠⎟ =
⎛ ⎝⎜
⎞ ⎠⎟ +h , 5
2
2 2
2 2 2
17. 1/4. To make a fraction as small as possible, make the numerator as small as possible and the denominator as large as possible. With the given digits, the small-est two-digit number that can be formed is 24 and the largest two-digit number that can be formed is 96. The fraction is 24/96, which is equivalent to 1/4. 18. 10:25 AM. To figure out how many groups of 36 are in 225, divide 225 by 36, which gives an answer of 6.25, so there are 6 full groups of 36 and one partial group. Since Boon Tee is the 225th person, he is in the 7th group and will have to wait for six trains before he gets on. Waiting for six trains will take 6 ×4 minutes, or 24 minutes. Since the first train leaves at 10:01 AM, the seventh train will leave at 10:25 AM.
19. (1, 1). The side BAis parallel to the
x-axis, from 4 to 6. The side BCis per-pendicular to BA,so rotating the point (4, 4) counterclockwise places it on the line through BA,which is the line y=1. Since Cis 3 units above B,after rotating, it will be 3 units to the left of B. The x -coordinate of the image of Cis 4 – 3, or 1; and the y-coordinate of y is 1; thus, C
will be rotated to (1, 1).
20. 99. To make 40305 a palindrome, the first and last digits must match. Ei-ther changing the 4 to a 5 or the 5 to a 4 will do so. To make the 4 become a 5, 10,000 must be added. To change the 5 to a 4, a minimum of 9 must be added. However, adding 9 to 40305 produces 40314. The first and last digits then match, but the second and fourth digits no longer match. To make the 1 become a 0, we must add 90, which produces 40404. The total value that we have added is 99. This is the smallest value that can be added, since we worked on making the digits with the smallest place value have the properties needed to achieve a palindrome.
21. 120. The numbers will be paired on opposite faces as 1 and 6, 2 and 5, and 3 and 4. The product of all six numbers is 720. If we calculate the product of four of the numbers, leaving out one pair, then either 1 times 6, 2 times 5, or 3 times 4 is left out of the total product. The products are 120, 72, and 60. Thus, the maximum possible value is 120. 22. 169 square centimeters. Triangle PQR
is a right triangle, so PR2= PQ2+
QR2
. To find the area of each rectangle, square the length of each side; the total area of the squares is PQ2=
QR2+ PR2
. Using the re-sult of the Pythagorean theorem gives an area of 2PR2
. Then 2PR2=
338, so PR2=
169. Since PRis the hypotenuse of the tri-angle, it is the longest side, so it will pro-duce the square with the largest area. 23. 5/9. The probability of choosing each container is 1/3. Next, the probability of selecting a green ball must be examined for each container. In the first container, the probability of selecting a green ball is 4/12, or 1/3. Thus, the probability in this container is (1/3)(1/3), or 1/9. In the sec-ond container, the probability of selecting a green ball is 4/6, or 2/3. Thus, the prob-ability in this container is (1/3)(2/3), or 2/9. That probability is the same as for the third container. Adding the probabilities gives 1/9 +2/9 +2/9, or 5/9.
24. 120. If there are nine numbers and the mean is 54, then the sum of the nine numbers is 9 ×54, or 486. After adding two numbers, the eleven numbers will have a mean of 66, so the sum of all eleven numbers is 11 ×66, or 726. The difference between 726 and 486 is 240, which is the sum of the two numbers added. To find the mean, divide 240 by 2, giving 120. 25. Six gallons. The tank starts off 1/8 full and then is 3/4 full. The difference between 3/4 and 1/8 is the amount of the tank that was filled: 3/4 – 1/8 =5/8. The fifteen gallons that were added filled 5/8 of the tank. Think of 5/8 as five groups out of eight total. Thus, fifteen gallons is five groups, and each group consists of three gallons. To have a full tank, all eight groups are needed. Cur-rently, the tank is 3/4 full, which is the same as 6/8 full, so two more groups are
needed. Each group consists of three gal-lons, so six gallons must be added. 26. 27. When rolling two standard dice, the possible sums are 2 through 12; and the only possibilities that lead to a sum of 10 with three dice are 4, 5, 6, 7, 8, and 9. Each of these sums requires a specific value on the third dice of 6, 5, 4, 3, 2, and 1, respectively. A sum of 4 can be found in three ways, a sum of 5 can be found in four ways, a sum of 6 can be found in five ways, a sum of 7 can be found in six ways, a sum of 8 can be found in five ways, and a sum of 9 can be found in four ways. Thus, the number of possibilities is 3 +4 +5 +6 +5 +4, or twenty-seven. 27. 76. If no numbers were skipped, the fortieth number would be 40. However, before 40, Arlene will not count twenty numbers. These numbers are 3, 6, 9, 13, 15, 18, 21, 23, 24, 27, 30, 31, 32, 33, 34, 35, 36, 37, 38, and 39. Thus, we must move twenty numbers beyond 40 to 60. However, between 40 and 60, ten num-bers, including 60, must be excluded; these numbers are 42, 43, 45, 48, 51, 53, 54, 56, 57, and 60. Thus, we need to go ten numbers beyond 60. However, three of them must be excluded; they are 63, 66, and 69. We need to add 3 to 70, pro-ducing 73, but 73 must be excluded, which leads to 74; however, since 71 was excluded, one more number is needed. After 74 comes 75, which must be ex-cluded, which finally leads to 76. 28. 22/3. First note that if y/z=14/3, then z/y=3/14. Also, 7x=4y,so x=
(4/7)y.Then
Thus,
8 3
14 3 22
3 + = +
= +
= .
x y z
x z
y z
4 7 4 7
14 3 8 3 = = =
• • .
x z
y z
y = 1
y
x C'
C
B A
29. 4:50. Fifty minutes of actual time has passed, so the amount of time on the clock needs to be calculated. For every 4 seconds of actual time, the clock indicates that 60 seconds have passed. Thus, 1 sec-ond of actual time is 15 secsec-onds on the clock. So the clock shows time passing at 15 times its actual rate. Thus, in 50 utes, the clock will indicate that 750 min-utes have passed. Translating this num-ber to hours requires that 750 be divided by 60, which produces 12.5 hours, which is 12 hours 30 minutes. Twelve hours brings the clock back to where it started at 4:20; thirty minutes later gives 4:50. 30. 52. Expanding
gives
∞
+ + + =
+ + ⎛⎝⎜ + ⎞⎠⎟ = + +
( )
=+ = −
=
x x x x x
x x x x
x x
x
.
3 3 1
3
64
1 1
64 1
3 4 64 1
64 12 52
3
3 3
3 3
3 3
3
+ ⎛
⎝⎜x x⎞⎠⎟ = 1
64
3
Share with readers and the Editorial Panel your opinions
about any of the articles or departments appearing in this
issue by writing to “Reader Reflections,” NCTM, 1906 Association Drive, Reston, VA
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