How many of these intersection points lie in the interior of the shaded region? If 1. then what is the value of

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1 NOVEMBER A stack of 00 nickels has a height of 6 inches What is the value, in dollars, of an 8-foot-high stack of nickels? Epress our answer to the nearest hundredth A cube is sliced b 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 man square units are in the area of the rhombus formed b the intersection of the plane and the cube? Epress our answer as a common fraction in simplest radical form The positive integers are written consecutivel in the pattern below What integer will be the eighth entr in row A? Row A 0 Row B 9 Row C 6 8 Row D Rich invested $00 seven ears ago Since then, his investment has doubled in value to $00 If Rich s mone continues to double ever seven ears, in how man ears will his $00 grow to $600? Points A(0, 0), B(6, 0), C(6, 0), and D(0, 0) are vertices of rectangle ABCD; and E is on segment CD at (, 0) What is the ratio of the area of triangle ADE to the area of quadrilateral ABCE? Epress our answer as a common fraction A palindrome is a number that reads the same forward as backward For eample, and are palindromes What is the least natural number that can be added to 00 to create a palindrome? Of the 6 billion people in the world, 0 million live in North America What percent of the world s population lives in North America? Epress our answer to the nearest whole number percent How man full seven-da weeks are in seven consecutive ears? Assume that the first da of the first ear is the first da of the week What is the slope of a line parallel to +? Epress our answer as a common fraction Each of the four digits,, 6, and 9 is placed in one of the boes to form a fraction The numerator and the denominator are both two-digit whole numbers What is the smallest value of all the common fractions that can be formed? Epress our answer as a common fraction The numbers on a standard si-faced die are arranged such that numbers on opposite faces alwas add to 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 maimum possible value of this product? On the grid below, Beatrice draws all the lines with integral -intercepts and slope or The lines form man intersection points How man of these intersection points lie in the interior of the shaded region? Paco uses a spinner to select a number from through, each of which has equal probabilit of being selected Manu uses a different spinner to select a number from through 0, each of which has equal probabilit What is the probabilit that the product of Manu s number and Paco s number is less than 0? Epress our answer as a common fraction In the arithmetic sequence, a, b, c,, what is the value of b? In a three-digit number, the hundreds digit is greater than, the tens digit is greater than but less than 8, and the units digit is the smallest prime number How man threedigit numbers satisf all these conditions? At 0:00 AM, Boon Tee is the th person in line to ride the Rocker Roller Coaster Each roller coaster train holds 6 people A full train leaves ever four minutes If the first 6 people in line leave on the 0:0 train, what time will Boon Tee s train leave? Angle PQR is a right angle The three quadrilaterals shown are squares The sum of the areas of the three squares is 8 square centimeters P What is the number of Q R square centimeters in the area of the largest square? 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 + 0 n is not a prime number? 6 Two complementar angles, A and B, have measures in the ratio of to, respectivel What is the ratio of the measure of the complement of angle A to the measure of the complement of angle B? Epress our answer as a common fraction The trip from Carville to Nikpath requires / hours when traveling at an average speed of 0 miles per hour How man hours does the trip require when traveling at an average speed of 60 miles per hour? Epress our answer as a decimal to the nearest hundredth The triangle with vertices A(6, ), B(, ), and C(, ) is rotated 90 degrees counterclockwise about B What are the coordinates of the image of C (the point where C is located after the rotation)? Epress our 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 randoml selected from that container What is the probabilit that the ball selected is green? Epress our answer as a common fraction 0 The arithmetic mean of nine numbers is If two numbers, u and v, are added to the list, the mean of the eleven-number list becomes 66 What is the mean of u and v? The gasoline gauge on a van initiall read /8 full When gallons of gasoline were added to the tank, the gauge read / full How man more gallons are needed to fill the tank? Three fair, standard si-faced dice of different colors are rolled In how man was can the dice be rolled such that the sum of the numbers rolled is 0? In the counting game bing-bong, Arlene starts counting at but skips all multiples of and all numbers that contain the digit For eample, Arlene counts as follows:,,,,, 8, 0,,, 6, What is the fortieth number in this sequence? 6 Suppose that and that What is the value of +? Epress our answer as a common fraction When Tomas enters a classroom at eactl 9:00 AM, the twelve-hour analog clock on the wall is behaving strangel The clock reads :0, and the second hand is racing It makes one complete circle ever four seconds The minute hand and hour hand behave as if ever full rotation of the second hand indicates that a minute has passed When Tomas leaves the class at 9:0 AM, what time does the clock on the wall read? If +, then what is the value of +? National Council of Teachers of Mathematics, 906 Association Drive, Reston, VA 09-0 Copright 00 The National Council of Teachers of Mathematics, Inc wwwnctmorg All rights reserved This material ma not be copied or distributed electronicall or in an other format without written permission from NCTM

2 SOLUTIONS to calendar Edited b JUDITH COVINGTON, jcovingt@ pilotlsusedu, Louisiana State Universit Shreveport, Shreveport, LA Problems are from MATHCOUNTS 00 State Target Round Problems 8 0 are from MATHCOUNTS State Spring Round The Editorial Panel of the Mathematics Teacher is considering sets of problems submitted b individuals, classes of prospective teachers, and mathematics clubs for publication in the monthl Calendar during the academic ear Please write to the Mathematics Teacher editor, 906 Association Drive, Reston, VA 09-0, for guidelines Calendar problems can also be sent to mt@nctmorg Three other sources of problems in calendar form are available from NCTM: Calendar Problems from the Mathematics Teacher (a book featuring more than 00 problems, organied b topic, order number 09, $9), Calendars for the Calculating, vol (a set of nine monthl calendars that originall appeared from September 98 to Ma 988, order number 96, $0), and A Year of Mathematics (one annual calendar that originall appeared in September 98, order number, $00; set of five, order number, $800) Individual members receive a 0 percent discount off these prices Write to NCTM to request the catalog of educational materials, which includes a listing for the publication Eplorator Problems in Mathematics (order number 9, $9) An online version of the catalog is available at wwwnctmorg Ed $680 Set up a ratio comparing the number of nickels and the height of the nickels; let n represent the number of nickels in an 8-foot-high stack Rewrite 8 feet as 96 inches, and the ratio is 00, 6 96 which can be rewritten as Dividing both sides b 6 produces 6; thus, 6 nickels will be in a stack that is 8 feet tall Since each nickel is worth $00, we multipl 00 b 6 to get 68, so the total value of the nickels is $680 8 Draw the lines described and count the number of intersection points that are in the shaded area If we let represent the first of the integers, each successive integer will be more than the previous one, so the sum of all the integers is + ( + ) + ( + ) + ( + ) + ( + ) + ( + ) + ( + 6) + ( + ) + ( + 8) The sum should be 9, so we have the equation , which is equivalent to 9, which is equivalent to 6/ 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 / and, the remaining side length can be found: + s Thus, + s 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 triangle 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, the diagonal has length To find the area of the isosceles triangle, we note that it has two sides of length / and a base of When drawing the altitude, we will have a right triangle with hpotenuse / and a leg that is half of the base, or / Thus, to find the height h, we need to solve 0 MATHEMATICS TEACHER Vol 98, No November 00

3 + h, which is equivalent to + h, which ields h / and h / The area of one of the triangles is 6 The two triangles will thus have an area twice that, or 6/ percent To find the percent, we need to divide the total population b the population that lives in North America First, however, we need to be sure that the units of both are the same; this unit will be millions We write 6 billion as 60 million, and divide 0 b 60 to get 0096, which is approimatel percent 6 /0 Since Paco chooses between five numbers, the probabilit that he chooses an one of them is / Since Manu chooses between ten numbers, the probabilit that he chooses an one of them is /0 Eamine the probabilit based on Paco s choice If Paco chooses or, an number that Manu chooses results in a product that is less than 0 The probabilit of choosing is /, and the probabilit of choosing is / If Paco chooses, nine of the numbers that Manu can choose will result in a product that is less than 0, so the probabilit is If Paco chooses, Manu can choose seven numbers that will result in a product that is less than 0 This probabilit is 0 0 Finall, if Paco chooses, five numbers will result in a product that is less than 0, and the probabilit will be 0 0 Writing all the probabilities with a denominator of 0 and adding produces We can easil see that if n, then + 0() is divisible b and is not a prime number We must then eamine whether this outcome will occur with an values that are less than We replace n with 6, which results in the value of + 0(6), or 8; since 8 equals, we know that 8 is not prime We net eamine the values that will result in replacing n with,,,, and, resulting in values of,, 9, 6, and All these values are prime, so 6 is the smallest integer that results in + 0n not being prime 8 6 Each number in row A is 6 more than the previous number We can write the nth number in row A as + 6(n ) If n 8, the value is + 6() A ear consists of 6 das, with some ears having an etra da During a seven-ear period, one or two leap ears can occur At least (6), or, das plus or etra occur in seven ears In all three cases, the total number of das divided b das per week produces a whole number of In an arithmetic sequence, the same value is added to each term to obtain the net term Represent this value b k To get from to, this value has been added four times Then + k, so k and k 6 Thus, b + k + 9 / Represent one angle measure b ; then the other angle measure must be to have a ratio of to The angles are complementar, so 90 and 90 The complement of the angle is, and the complement of the angle is The ratio of the complementar angles is /, or / Twent-one ears To make $00 become $600, we must double the mone three times: $00 will double to $00, then $00 will double to $800, and $800 will double to $600 Each doubling period takes seven ears, for a total of twent-one ears / If a line is in the form A + B C, the slope will be A/B Thus, the slope of the given line is /, or / Since parallel lines have the same slope, the parallel line also has a slope of / Three numbers are chosen for a three-digit number The first digit must be greater than, so four choices are possible for the first digit Three choices (, 6, or ) are possible for the second digit The units digit is the smallest prime number, which is, so onl one choice is possible for the last digit A total of, or twelve, choices are possible for the digits, so twelve three-digit numbers satisf the given condition hours The distance can be calculated b multipling the speed b time, so the distance is times 0, which is miles To find the time when the speed is 60 MPH, divide b 60, which is hours 6 / Graphing the four points indicates that the lower-left corner of the rectangle is at the origin The lengths of the sides of the rectangle are 6 and 0 The point (, 0) lies on the top side of the rectangle Connecting point A to point E produces right triangle ADE Triangle ADE has AD 0 and DE, so the area of triangle ADE is 0(0)(), or 0 The area of the rectangle is 6(0) 60, so the area of quadrilateral ABCE is Thus, the ratio is 0/0, which is / D 0 E 0 C B A 0 6 Vol 98, No November 00 MATHEMATICS TEACHER

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 smallest two-digit number that can be formed is and the largest two-digit number that can be formed is 96 The fraction is /96, which is equivalent to / 8 0: AM To figure out how man groups of 6 are in, divide b 6, which gives an answer of 6, so there are 6 full groups of 6 and one partial group Since Boon Tee is the th person, he is in the th group and will have to wait for si trains before he gets on Waiting for si trains will take 6 minutes, or minutes Since the first train leaves at 0:0 AM, the seventh train will leave at 0: AM 9 (, ) The side BA is parallel to the -ais, from to 6 The side BC is perpendicular to BA, so rotating the point (, ) counterclockwise places it on the line through BA, which is the line Since C is units above B, after rotating, it will be units to the left of B The - coordinate of the image of C is, or ; and the -coordinate of is ; thus, C will be rotated to (, ) 0 99 To make 00 a palindrome, the first and last digits must match Either changing the to a or the to a will do so To make the become a, 0,000 must be added To change the to a, a minimum of 9 must be added However, adding 9 to 00 produces 0 The first and last digits then match, but the second and fourth digits no longer match To make the become a 0, we must add 90, which produces 00 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 C' A' B C A 0 The numbers will be paired on opposite faces as and 6, and, and and The product of all si numbers is 0 If we calculate the product of four of the numbers, leaving out one pair, then either times 6, times, or times is left out of the total product The products are 0,, and 60 Thus, the maimum possible value is 0 69 square centimeters Triangle PQR is a right triangle, so PR PQ + QR To find the area of each rectangle, square the length of each side; the total area of the squares is PQ QR + PR Using the result of the Pthagorean theorem gives an area of PR Then PR 8, so PR 69 Since PR is the hpotenuse of the triangle, it is the longest side, so it will produce the square with the largest area /9 The probabilit of choosing each container is / Net, the probabilit of selecting a green ball must be eamined for each container In the first container, the probabilit of selecting a green ball is /, or / Thus, the probabilit in this container is (/)(/), or /9 In the second container, the probabilit of selecting a green ball is /6, or / Thus, the probabilit in this container is (/)(/), or /9 That probabilit is the same as for the third container Adding the probabilities gives /9 + /9 + /9, or /9 0 If there are nine numbers and the mean is, then the sum of the nine numbers is 9, or 86 After adding two numbers, the eleven numbers will have a mean of 66, so the sum of all eleven numbers is 66, or 6 The difference between 6 and 86 is 0, which is the sum of the two numbers added To find the mean, divide 0 b, giving 0 Si gallons The tank starts off /8 full and then is / full The difference between / and /8 is the amount of the tank that was filled: / /8 /8 The fifteen gallons that were added filled /8 of the tank Think of /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 Currentl, the tank is / full, which is the same as 6/8 full, so two more groups are needed Each group consists of three gallons, so si gallons must be added 6 When rolling two standard dice, the possible sums are through ; and the onl possibilities that lead to a sum of 0 with three dice are,, 6,, 8, and 9 Each of these sums requires a specific value on the third dice of 6,,,,, and, respectivel A sum of can be found in three was, a sum of can be found in four was, a sum of 6 can be found in five was, a sum of can be found in si was, a sum of 8 can be found in five was, and a sum of 9 can be found in four was Thus, the number of possibilities is , or twent-seven 6 If no numbers were skipped, the fortieth number would be 0 However, before 0, Arlene will not count twent numbers These numbers are, 6, 9,,, 8,,,,, 0,,,,,, 6,, 8, and 9 Thus, we must move twent numbers beond 0 to 60 However, between 0 and 60, ten numbers, including 60, must be ecluded; these numbers are,,, 8,,,, 6,, and 60 Thus, we need to go ten numbers beond 60 However, three of them must be ecluded; the are 6, 66, and 69 We need to add to 0, producing, but must be ecluded, which leads to ; however, since was ecluded, one more number is needed After comes, which must be ecluded, which finall leads to 6 8 / First note that if / /, then / / Also,, so (/) Then Thus, MATHEMATICS TEACHER Vol 98, No November 00

5 9 :0 Fift minutes of actual time has passed, so the amount of time on the clock needs to be calculated For ever seconds of actual time, the clock indicates that 60 seconds have passed Thus, second of actual time is seconds on the clock So the clock shows time passing at times its actual rate Thus, in 0 minutes, the clock will indicate that 0 minutes have passed Translating this number to hours requires that 0 be divided b 60, which produces hours, which is hours 0 minutes Twelve hours brings the clock back to where it started at :0; thirt minutes later gives :0 0 Epanding + 6 gives ( ) Do You Have Something to Add? Share with readers and the Editorial Panel our opinions about an of the articles or departments appearing in this issue b writing to Reader Reflections, NCTM, 906 Association Drive, Reston, VA 09-0, or b sending to mt@nctmorg Vol 98, No November 00 MATHEMATICS TEACHER