MATHCOUNTS
https://www.mathcounts.org
enMiniature Golf
https://www.mathcounts.org/resources/problem-of-the-week/miniature-golf
<div class="field field-name-field-date-of-problem field-type-datetime field-label-hidden field--date-of-problem"><div class="field-items"><div class="field-item even"><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2020-09-21T11:00:00-07:00">September 21, 2020</span></div></div></div><div class="field field-name-field-problem-category field-type-taxonomy-term-reference field-label-inline clearfix field--problem-category"><div class="field-label">Topics/Content Areas: </div><div class="field-items"><div class="field-item even"><div class="textformatter-list"><a href="/problem-topic-content-area/algebraic-expressions-equations">Algebraic Expressions & Equations</a>, <a href="/problem-topic-content-area/plane-geometry">Plane Geometry</a></div></div></div></div><div class="l-constrain l-constrain--prose">
<p><span style="font-size:11pt"><span style="background:white"><span style="line-height:normal"><span style="vertical-align:top"><span style="font-family:Calibri,sans-serif"><span style="color:black">Gerald is going mini golfing at Golf-O-Rama with his family. For his family of 5 to play it costs $34. They were charged for three student tickets and two adult tickets. If a student ticket costs 3/4 as much as an adult ticket costs, how much does one student ticket cost?</span></span></span></span></span></span></p>
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<p><span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><span style="background:white"><span style="color:black">This is a diagram of the first hole.</span></span></span></span></span></p>
<p><span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><span style="background:white"><span style="color:black"><img alt="" src="/sites/default/files/u70427/Golf%20photo%201.jpg" style="width: 240px; height: 297px;" /></span></span></span></span></span></p>
<p><span style="font-size:11pt"><span style="background:white"><span style="line-height:normal"><span style="vertical-align:top"><span style="font-family:Calibri,sans-serif"><span style="color:black">Gerald knows that when his ball hits the wall it will bounce off at the same angle at which it hit the wall. That means, if his ball hits the wall with an angle of 50 degrees relative to the wall, for example, it will bounce off with an angle of 50 degrees relative to the wall. Keeping this in mind, Gerald lines up his shot. The dotted line represents the path of his ball.</span></span></span></span></span></span></p>
<p><img alt="" src="/sites/default/files/u70427/Golf%20photo%202.jpg" style="width: 240px; height: 296px;" /></p>
<p><span style="font-size:11pt"><span style="background:white"><span style="line-height:normal"><span style="vertical-align:top"><span style="font-family:Calibri,sans-serif"><span style="color:black">Gerald’s ball traveled 6 yards before it hit the wall and then traveled another 6 yards to the hole. Using a straight segment from his ball’s original position to the hole, how far was the ball’s original position from the hole? Express your answer in simplest radical form.</span></span></span></span></span></span></p>
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<p><span style="font-size:11.0pt"><span style="background:white"><span style="line-height:107%"><span style="font-family:"Calibri",sans-serif"><span style="color:black">The next hole has a windmill as shown below. Gerald decides that he is never going to be able to time his shot right, so he decides to just randomly swing and hope that the ball makes it past the blades. If the end of each blade is 6 inches and the distance from the center of the fan to the end of a blade is 3 feet, what is the probability of Gerald’s shot making it past the windmill? Express your answer as a percent to the nearest tenth.</span></span></span></span></span></p>
<p><img alt="" src="/sites/default/files/u70427/Golf%20photo%203.jpg" style="width: 306px; height: 150px;" /></p>
<p> </p>
<p style="text-align: center;"><strong>CHECK THE <a href="https://www.mathcounts.org/resources/problem-archive"> PROBLEM OF THE WEEK ARCHIVE </a><br /><br />
FOR SOLUTIONS TO PREVIOUS PROBLEMS </strong></p>
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Fri, 18 Sep 2020 16:10:01 +0000michellelefrancois308303 at https://www.mathcounts.orghttps://www.mathcounts.org/resources/problem-of-the-week/miniature-golf#commentsSchool Makeover
https://www.mathcounts.org/resources/problem-of-the-week/school-makeover-2
<div class="field field-name-field-date-of-problem field-type-datetime field-label-hidden field--date-of-problem"><div class="field-items"><div class="field-item even"><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2020-09-14T11:00:00-07:00">September 14, 2020</span></div></div></div><div class="field field-name-field-problem-category field-type-taxonomy-term-reference field-label-inline clearfix field--problem-category"><div class="field-label">Topics/Content Areas: </div><div class="field-items"><div class="field-item even"><div class="textformatter-list"><a href="/problem-topic-content-area/proportional-reasoning">Proportional Reasoning</a>, <a href="/problem-topic-content-area/statistics-data">Statistics & Data</a></div></div></div></div><div class="l-constrain l-constrain--prose">
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif">Lou has been contracted to tile classroom floors as part of the renovation of a local school. During the first five days on the job, Lou completed tiling 4, 5, 6, 4 and 8 classrooms, respectively. What is the average number of classrooms Lou tiled each day during that five-day period? Express your answer as a decimal to the nearest tenth.</span></span></span></p>
<p style="margin-bottom: 11px; margin-left: 40px;"><em><span style="color:#19a2bf;">To determine the average number of classrooms, we need to calculate the total number of classrooms Lou tiled over the five days and divide by five. Doing so, we find that the average number of classrooms that Lou can tile in a day is (4 + 5 + 6 + 4 + 8) ÷ 5 = 27 ÷ 5 = </span><span style="color:#272561;"><strong>5.4</strong></span><span style="color:#19a2bf;"> classrooms. </span></em></p>
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif">Electricians Chris and Dave have been contracted to wire every classroom in the same school. Each of the classrooms requires exactly the same electrical work. Working alone, Chris can complete the wiring for one classroom in 20 hours, and it takes Dave 15 hours to do the same job by himself. After 120 hours of work, they had completed wiring 1/2 of the classrooms in the school. How many classrooms are in the school?</span></span></span></p>
<p style="margin-left: 40px;"><em><span style="color:#19a2bf;">A common measure would make Chris' and Dave's rates easier to compare. Using the least common multiple of 20 and 15, which is 60, will work. At a rate of 1 classroom every 20 hours, Chris can wire 3 classrooms in 60 hours. At a rate of 1 classroom every 15 hours, Dave can wire 4 classrooms in 60 hours. So, every 60 hours, they complete wiring a total of 7 classrooms. That means in 120 hours, they will complete wiring (120/60) × 7 = 2 × 7 = 14 classrooms. If this is 1/2 of the total number of classrooms, the school must have a total of 2 × 14 = </span><span style="color:#272561;"><strong>28</strong></span><span style="color:#19a2bf;"> classrooms. </span></em></p>
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif">In the school’s main office there are four smaller offices that each require the exact same wiring. Chris and Dave can each wire an office in exactly half the time it took them each to wire a classroom. Working together, how long will it take Chris and Dave to complete wiring the four smaller offices? Express your answer as a decimal to the nearest hundredth.</span></span></span></p>
<p style="margin-left: 40px;"><em><span style="color:#19a2bf;">If it takes Chris half the time to wire the office as the classroom, he can wire an office in 20 ÷ 2 = 10 hours. Similarly, it takes Dave 15 ÷ 2 = 7.5 hours to wire an office. It follows that Chris can complete 1/10 of the required wiring for an office in 1 hour. Likewise, Dave can complete 1/7.5 = 2/15 of the wiring for an office in 1 hour. Together, they will be able to complete 1/x of the wiring in 1 hour. That means 1/10 + 2/15 = 1/x → 3/30 + 4/30 = 1/x → 7/30 = 1/x → x = 30/7. Since it takes 30/7 hours for Chris and Dave to wire one office working together, four offices will take 4(30/7) = 17.14285714 ≈ </span><span style="color:#272561;"><strong>17.14</strong></span><span style="color:#19a2bf;"> hours. </span></em></p>
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<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"> <strong><span style="font-size:11.0pt"><span style="font-family:"Segoe UI Symbol",sans-serif"><span style="color:#3b3838">♦</span></span></span></strong><strong> </strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 1</span></span></span></strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""> <span style="color:#3b3838">of the linked PDF contains PROBLEMS & SOLUTIONS.</span></span></span></strong></span></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 2</span><strong> </strong><span style="color:#3b3838">contains ONLY PROBLEMS. </span></span></span><span style="font-size:11.0pt"><span style="color:#3b3838">♦</span></span></strong></span></span></p>
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Fri, 18 Sep 2020 15:47:58 +0000michellelefrancois308301 at https://www.mathcounts.orghttps://www.mathcounts.org/resources/problem-of-the-week/school-makeover-2#commentsBack to School, Part 2
https://www.mathcounts.org/resources/problem-of-the-week/back-school-part-2-0
<div class="field field-name-field-date-of-problem field-type-datetime field-label-hidden field--date-of-problem"><div class="field-items"><div class="field-item even"><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2020-09-07T11:00:00-07:00">September 7, 2020</span></div></div></div><div class="field field-name-field-problem-category field-type-taxonomy-term-reference field-label-inline clearfix field--problem-category"><div class="field-label">Topics/Content Areas: </div><div class="field-items"><div class="field-item even"><div class="textformatter-list"><a href="/problem-topic-content-area/general-math">General Math</a>, <a href="/problem-topic-content-area/number-theory">Number Theory</a></div></div></div></div><div class="l-constrain l-constrain--prose">
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif">Friends Megan and Heather go to different schools. Megan has math class during first period on each of the 180 days she goes to school. She will be in math class a total of 8640 minutes this year. Heather’s school year also has 180 days, but her math class only meets every other day, so her class periods are longer. (One day she’ll have math class, and then the next day she won’t.) If Megan and Heather end up with the same number of minutes of math class each year, how long are the class periods at Heather’s school?</span></span></span></p>
<p style="margin-bottom: 11px; margin-left: 40px;"><span style="color:#19a2bf;"><em>We could solve this a couple of ways. We could see that Heather will only use 90 days to reach her 8640 minutes of math, so she will spend 8640 ÷ 90 = </em></span><span style="color:#272561;"><em><strong>96</strong></em></span><span style="color:#19a2bf;"><em> minutes in math class each time it meets. We also could have seen that Megan will spend 8640 ÷ 180 = 48 minutes in math class each day, and if we double this number, we'll have the length of Heather's math classes: 48 × 2 = </em></span><span style="color:#272561;"><em><strong>96</strong></em></span><span style="color:#19a2bf;"><em> minutes. </em></span></p>
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif">Carter received his locker combination from his first period teacher. He noticed that each of the three integers of the combination is a prime, the sum of the three integers is 50, and the three integers are in decreasing order. What is the value of the last integer of Carter’s locker combination?</span></span></span></p>
<p style="margin-bottom: 11px; margin-left: 40px;"><span style="color:#19a2bf;"><em>The key to doing the least amount of work for this problem is to realize that only one prime is even, and all of the others are odd. If we add together three odd numbers, the sum will be an odd number. Since the sum is even (50), our combination must consist of the even prime and two of the odd primes. The even prime is 2, and no matter what the other numbers are in the combination, </em></span><span style="color:#272561;"><em><strong>2</strong></em></span><span style="color:#19a2bf;"><em> is the smallest prime, so it is the last integer of the combination. </em></span></p>
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif">Soon Yee was putting together her notebook after the first day of school. She bought a pack of 200 sheets of loose-leaf notebook paper and some dividers. She wanted to put the same amount of paper in each of the four sections for her core subjects: math, history, English and science. She figured that her two elective courses (art and music) each only needed half of the amount of paper that a core subject needed. How many sheets of paper did she put into the art section of her notebook?</span></span></span></p>
<p style="margin-left: 40px;"><span style="color:#19a2bf;"><em>Since her art and music sections each only had half the number of sheets of paper as a core subject, together the two sections had the same amount of paper as a core subject. Therefore, it is almost like her notebook had five core subjects, rather than four core subjects and two electives. If she divided the 200 sheets equally among the five core subjects, there would be 200 ÷ 5= 40 sheets in each section. Now we can see that art would actually have half of this amount, or </em></span><span style="color:#272561;"><em><strong>20</strong></em></span><span style="color:#19a2bf;"><em> sheets of paper. </em></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:11.0pt"><span style="font-family:"Segoe UI Symbol",sans-serif"><span style="color:#3b3838">♦</span></span></span></strong><strong> </strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 1</span></span></span></strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""> <span style="color:#3b3838">of the linked PDF contains PROBLEMS & SOLUTIONS.</span></span></span></strong></span></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 2</span><strong> </strong><span style="color:#3b3838">contains ONLY PROBLEMS. </span></span></span><span style="font-size:11.0pt"><span style="color:#3b3838">♦</span></span></strong></span></span></p>
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Mon, 14 Sep 2020 14:33:59 +0000michellelefrancois307983 at https://www.mathcounts.orghttps://www.mathcounts.org/resources/problem-of-the-week/back-school-part-2-0#commentsFlight To Nowhere
https://www.mathcounts.org/flight-nowhere
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<div class="media-object__content"><div class="field field-name-field-media-body field-type-text-long field-label-hidden field--media-body"><div class="field-items"><div class="field-item even"><p><strong>Original Handbook Problem: </strong>It is the policy of Sky Airlines to cancel any flight that has fewer than 80% of its seats filled. A Sky Airlines plane has 24 rows with 6 seats per row. What is the maximum number of empty seats on a flight that is not canceled? (2019-20 Problem #181)</p>
<p> </p>
<p><strong>Why the Number Ninjas Team got high scores in creativity: </strong></p>
<p>The use of Legos was a clever idea and gave the video a unique look.</p>
<p>The team used different shots to help give more depth to their scene and establish which characters were talking.</p>
<p> </p>
<p><strong>What else judges liked: </strong></p>
<p>The dialog was clear and easy to understand.</p>
<p> </p>
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Tue, 08 Sep 2020 18:37:01 +0000david.dumas307644 at https://www.mathcounts.orghttps://www.mathcounts.org/flight-nowhere#commentsCookies, Cars, and Calculations
https://www.mathcounts.org/cookies-cars-and-calculations
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<iframe class="" width="455px" height="260px" src="//www.youtube.com/embed/yIAQvqSnAbo?width%3D455px%26amp%3Bheight%3D260px%26amp%3Btheme%3Ddark%26amp%3Bautoplay%3D0%26amp%3Brel%3D0%26amp%3Bshowinfo%3D1%26amp%3Bmodestbranding%3D1%26amp%3Biv_load_policy%3D1%26amp%3Bautohide%3D2%26amp%3Bvq%3Dhd720%26amp%3Bwmode%3Dopaque" frameborder="0" allowfullscreen></iframe> </div>
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<div class="media-object__content"><div class="field field-name-field-media-body field-type-text-long field-label-hidden field--media-body"><div class="field-items"><div class="field-item even"><p><strong>Original Handbook Problem: </strong>Bob and Sam each are going to drive from Lincoln to Denver, a driving distance of 488.5 miles. Bob leaves at 6:00 a.m., traveling at an average speed of 75 mi/h. Sam leaves at 8:30 a.m., traveling at an average speed of 60 mi/h. How many minutes after Bob arrives in Denver will Sam arrive? Express your answer as a decimal to the nearest tenth. (2019-20 Problem #177)</p>
<p> </p>
<p><strong>Why the Number Ninjas Team got high scores in creativity: </strong></p>
<p>Their use of stopmotion animation was simple and effective.</p>
<p>Their animation of background objects and use of sounds effects helped their video come alive.</p>
<p> </p>
<p><strong>What else judges liked: </strong></p>
<p>The problem and solution were displayed in an easy to understand way.</p>
<p>The dialog was clear and easy to follow.</p>
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Tue, 08 Sep 2020 17:48:15 +0000david.dumas307642 at https://www.mathcounts.orghttps://www.mathcounts.org/cookies-cars-and-calculations#commentsBack to School
https://www.mathcounts.org/resources/problem-of-the-week/back-school-0
<div class="field field-name-field-date-of-problem field-type-datetime field-label-hidden field--date-of-problem"><div class="field-items"><div class="field-item even"><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2020-08-31T11:00:00-07:00">August 31, 2020</span></div></div></div><div class="field field-name-field-problem-category field-type-taxonomy-term-reference field-label-inline clearfix field--problem-category"><div class="field-label">Topics/Content Areas: </div><div class="field-items"><div class="field-item even"><div class="textformatter-list"><a href="/problem-topic-content-area/probability-counting-combinatorics">Probability, Counting & Combinatorics</a>, <a href="/problem-topic-content-area/proportional-reasoning">Proportional Reasoning</a></div></div></div></div><div class="l-constrain l-constrain--prose">
<p style="text-align:justify; margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif">It’s a staple of American school lunches: peanut butter sandwiches (usually with jelly, too). The average American child will eat 1500 peanut butter sandwiches by the time she graduates from high school! (On average, that’s one sandwich every four days.) It takes 850 peanuts to make an 18-oz jar of peanut butter, and an average sandwich has two ounces of peanut butter on it. How many peanuts will be needed to make all of the sandwiches a child will eat by graduation?</span></span></span></p>
<p style="margin-bottom: 16px; margin-left: 40px;"><span style="color:#19a2bf;"><em>The total amount of peanut butter on 1500 sandwiches is 1500 × 2 = 3000 ounces. At 18 ounces a jar, that's equal to 3000 ÷ 18 = 166.666667 jars. Since it takes 850 peanuts to make a jar of peanut butter, 850 × 166.666667 = 141,666.666667 ≈ </em></span><span style="color:#272561;"><em><strong>141,667</strong></em></span><span style="color:#19a2bf;"><em> peanuts are needed to make all of the sandwiches for one child. </em></span></p>
<p style="margin-bottom:16px"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif">A staple of art classes, Crayola Crayons have been around since 1903. The original boxes of Crayola contained eight different-colored crayons. The crayons were arranged in two rows of four crayons each. In how many different ways could the crayons have been arranged in the box?</span></span></span></p>
<p style="margin-bottom: 16px; margin-left: 40px;"><span style="color:#19a2bf;"><em>Any of the crayons could be in the first spot on the top row, any of the remaining seven could be in the next spot, and so on. There are, therefore, 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 8! = </em></span><span style="color:#272561;"><em><strong>40,320</strong></em></span><span style="color:#19a2bf;"><em> ways to arrange the crayons. </em></span></p>
<p style="margin-bottom:16px"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif">There are 15 students in the 8th grade. The students are randomly placed into three different algebra classes of 5 students each. Trevor, Terry and Evan are best friends. What is the probability that all three of them will be in the same algebra class?</span></span></span></p>
<p style="margin-left: 40px;"><span style="color:#19a2bf;"><em>First, place Trevor into one of the classrooms. Next, let's place Evan. Since there are 14 possible spots for him and 4 of them are in the same class as Trevor, there is a 4/14 chance of Evan being placed in the same class as Trevor. Finally, let's place Terry. Of the remaining 13 spots, 3 of them will be in the same class as Trevor and Evan. Hence, there is a 4/14 × 3/13 = </em></span><span style="color:#272561;"><em><strong>6/91</strong></em></span><span style="color:#19a2bf;"><em> probability that Trevor, Evan and Terry will all be in the same algebra class. </em></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:11.0pt"><span style="font-family:"Segoe UI Symbol",sans-serif"><span style="color:#3b3838">♦</span></span></span></strong><strong> </strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 1</span></span></span></strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""> <span style="color:#3b3838">of the linked PDF contains PROBLEMS & SOLUTIONS.</span></span></span></strong></span></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 2</span><strong> </strong><span style="color:#3b3838">contains ONLY PROBLEMS. </span></span></span><span style="font-size:11.0pt"><span style="color:#3b3838">♦</span></span></strong></span></span></p>
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Tue, 08 Sep 2020 14:09:32 +0000michellelefrancois307627 at https://www.mathcounts.orghttps://www.mathcounts.org/resources/problem-of-the-week/back-school-0#commentsMini #100 - Careful Counting: A Measurement Lesson
https://www.mathcounts.org/resources/video-library/mathcounts-minis/mini-100-careful-counting-measurement-lesson
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<h3>September 2020</h3>
<p><em>There's no activity sheet, but we know you will enjoy this video. </em></p>
<p><strong><span style="color:#19a2bf;"><em>It's the <span style="color:#ff8a3b;">100<sup>th</sup></span> MATHCOUNTS MINI Lesson! </em></span></strong></p>
<p><em>We're taking a break from uploading new Mini Lessons for a while, which makes this the perfect time to dive into our collection of 100 Mini Lessons to revisit some of your favorites. </em></p>
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<p><em>Video by Art of Problem Solving's Richard Rusczyk, a MATHCOUNTS alum. Visit <a href="http://www.artofproblemsolving.com">Art of Problem Solving</a> for many more educational resources.</em></p>
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<div class="field field-name-field-video-category field-type-taxonomy-term-reference field-label-inline clearfix field--video-category"><div class="field-label">Video Category: </div><div class="field-items"><div class="field-item even"><a href="/resources/video-library/mathcounts-minis">MATHCOUNTS Minis</a></div></div></div>Tue, 01 Sep 2020 21:53:16 +0000kera.johnson307152 at https://www.mathcounts.orghttps://www.mathcounts.org/resources/video-library/mathcounts-minis/mini-100-careful-counting-measurement-lesson#commentsNational Dog Day!
https://www.mathcounts.org/resources/problem-of-the-week/national-dog-day-0
<div class="field field-name-field-date-of-problem field-type-datetime field-label-hidden field--date-of-problem"><div class="field-items"><div class="field-item even"><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2020-08-24T11:00:00-07:00">August 24, 2020</span></div></div></div><div class="field field-name-field-problem-category field-type-taxonomy-term-reference field-label-inline clearfix field--problem-category"><div class="field-label">Topics/Content Areas: </div><div class="field-items"><div class="field-item even"><div class="textformatter-list"><a href="/problem-topic-content-area/general-math">General Math</a>, <a href="/problem-topic-content-area/percents-fractions">Percents & Fractions</a>, <a href="/problem-topic-content-area/plane-geometry">Plane Geometry</a></div></div></div></div><div class="l-constrain l-constrain--prose">
<p><span style="font-size:11pt"><span style="background:white"><span style="line-height:normal"><span style="vertical-align:top"><span style="font-family:Calibri,sans-serif"><span style="color:black">Olivia is planning to adopt a puppy on National Dog Day (August 26). The adoption fee at the local shelter is $150 plus a 4% tax. On her way to pick up the puppy, she stops at a store to buy a food dish, dog food, a water dish and a bed for her new pup that cost (before tax) $5, $10, $5 and $12, respectively. If these purchases have a 5% sales tax added to them at check out, how much does Olivia spend in total (after tax) on the adoption fees and supplies combined?</span></span></span></span></span></span></p>
<p style="margin-left: 40px;"><span style="color:#19a2bf;"><em>The adoption fee at the shelter, including tax, would be $150(1.04) = $156. The supplies Olivia bought at the store, including tax, would be ($5 + $10 + $5 + $12)(1.05) = $33.60. Thus, in total, Olivia spends $156 + $33.60 = $</em></span><span style="color:#272561;"><em><strong>189.60</strong></em></span><span style="color:#19a2bf;"><em>. </em></span></p>
<p><span style="font-size:11pt"><span style="line-height:normal"><span style="font-family:Calibri,sans-serif"><span style="background:white"><span style="color:black">Gavin’s dog had a litter of puppies. All of the puppies in this litter were either yellow or black. If there was one less black puppy, the ratio of yellow puppies to black puppies would be 2/3. If there was one more black puppy, the ratio of yellow puppies to black puppies would be 1/2. How many puppies were in the litter?</span></span></span></span></span></p>
<p style="margin-bottom: 11px; margin-left: 40px;"><span style="color:#19a2bf;"><em>We can express "with one less black puppy, the ratio of yellow puppies to black puppies would be 2/3" as y/(b - 1) = 2/3 <strong>→</strong> 3y = 2b - 2. We can express "with one more black puppy, the ratio of yellow puppies to black puppies would be 1/2" as y/(b + 1) = 1/2 <strong>→</strong> 2y = b + 1 <strong>→</strong> b = 2y - 1. Using substitution, we get that 3y = 2(2y - 1) - 2. Now, we can solve for y: 3y = 4y - 2 - 2 <strong>→</strong> -1y = -4 <strong>→</strong> y = 4. Now, we can plug in the value of y to find the value of b: 3(4) = 2b - 2 <strong>→</strong> 12 = 2b - 2 <strong>→</strong> 14 = 2b<strong> →</strong> b = 7. Thus, the total number of puppies in the litter was 4 + 7 = </em></span><span style="color:#272561;"><em><strong>11</strong></em></span><span style="color:#19a2bf;"><em>. </em></span></p>
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif"><span style="color:black">Jermaine’s back yard measures 60 feet by 90 feet and is fenced in, as shown in the diagram, so that his dog can run around and play. His house is rectangular. How many square feet does Jermaine’s dog have in which to play?</span></span></span></span></p>
<p style="margin-bottom:11px"> </p>
<p style="margin-bottom:11px"><img alt="" src="/sites/default/files/u70427/National%20Dog%20Day%20PoTW%20image.jpg" /></p>
<p style="margin-left: 40px;"><span style="color:#19a2bf;"><em>If the fence were an enclosed rectangle, we could find its area by taking 90 × 60 to get 5400 square feet. However, we must take into account the portion of the house that overlaps with the yard, which has an area of 55 × 15 = 825 square feet. So, Jermaine's dog has 5400 - 825 = </em></span><span style="color:#272561;"><em><strong>4575</strong></em></span><span style="color:#19a2bf;"><em> square feet in which to play. </em></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:11.0pt"><span style="font-family:"Segoe UI Symbol",sans-serif"><span style="color:#3b3838">♦</span></span></span></strong><strong> </strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 1</span></span></span></strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""> <span style="color:#3b3838">of the linked PDF contains PROBLEMS & SOLUTIONS.</span></span></span></strong></span></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 2</span><strong> </strong><span style="color:#3b3838">contains ONLY PROBLEMS. </span></span></span><span style="font-size:11.0pt"><span style="color:#3b3838">♦</span></span></strong></span></span></p>
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Mon, 31 Aug 2020 13:35:49 +0000michellelefrancois307087 at https://www.mathcounts.orghttps://www.mathcounts.org/resources/problem-of-the-week/national-dog-day-0#commentsMath in the Middle
https://www.mathcounts.org/resources/problem-of-the-week/math-middle-2
<div class="field field-name-field-date-of-problem field-type-datetime field-label-hidden field--date-of-problem"><div class="field-items"><div class="field-item even"><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2020-08-17T11:00:00-07:00">August 17, 2020</span></div></div></div><div class="field field-name-field-problem-category field-type-taxonomy-term-reference field-label-inline clearfix field--problem-category"><div class="field-label">Topics/Content Areas: </div><div class="field-items"><div class="field-item even"><div class="textformatter-list"><a href="/problem-topic-content-area/percents-fractions">Percents & Fractions</a></div></div></div></div><div class="l-constrain l-constrain--prose">
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif">At Tri-City Middle School, half of the 472 enrolled students are taking algebra. One-fifth of the students not taking algebra are taking geometry, and each of the remaining students is taking pre-algebra. How many students enrolled at Tri-City Middle School are taking pre-algebra? Express your answer to the nearest whole number.</span></span></span></p>
<p style="margin-left: 40px;"><span style="color:#19a2bf;"><em>Since the total number of enrolled students at Tri-City Middle School is 472, that means 472 ÷ 2 = 236 students are taking algebra, and 236 students are taking either pre-algebra or geometry. We are told that 1/5 of these 236 students are taking geometry, which means that 1 - (1/5) = 4/5 of them are taking pre-algebra. Therefore, (4/5) × 236 = 188.8 ≈ </em></span><span style="color:#272561;"><em><strong>189 </strong></em></span><span style="color:#19a2bf;"><em>students at Tri-City Middle School are taking pre-algebra.</em></span></p>
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif">What percent of the students enrolled at Tri-City Middle School are currently taking geometry?</span></span></span></p>
<p style="margin-bottom: 11px; margin-left: 40px;"><span style="color:#19a2bf;"><em>From the previous problem, we know that 1/5 of 1/2 of the enrolled students at Tri-City Middle School are taking geometry. Therefore, (1/5) × (1/2) = 1/10 = </em></span><strong><span style="color:#272561;"><em>10</em></span></strong><span style="color:#19a2bf;"><em>% of the students at Tri-City Middle School are taking geometry. </em></span></p>
<p style="margin-bottom:11px"><span style="font-size:11pt"><span style="line-height:107%"><span style="font-family:Calibri,sans-serif">Ms. Sanchez, a math teacher at Tri-City Middle School, teaches two algebra classes, with 27 students in each class. What fraction of the students currently taking algebra at Tri-City Middle School are in an algebra class taught by Ms. Sanchez? Express your answer as a common fraction.</span></span></span></p>
<p style="margin-left: 40px;"><span style="color:#19a2bf;"><em>Ms. Sanchez has 27 students in each of her two algebra classes, which means that 54 students at Tri-City Middle School are in an algebra class taught by Ms. Sanchez. From the first problem, we know that 236 Tri-City Middle School students are taking algebra. Therefore, 54/236 = </em></span><span style="color:#272561;"><em><strong>27/118 </strong></em></span><span style="color:#19a2bf;"><em>of the students currently taking algebra are in an algebra class taught by Ms. Sanchez.</em></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:11.0pt"><span style="font-family:"Segoe UI Symbol",sans-serif"><span style="color:#3b3838">♦</span></span></span></strong><strong> </strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 1</span></span></span></strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""> <span style="color:#3b3838">of the linked PDF contains PROBLEMS & SOLUTIONS.</span></span></span></strong></span></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 2</span><strong> </strong><span style="color:#3b3838">contains ONLY PROBLEMS. </span></span></span><span style="font-size:11.0pt"><span style="color:#3b3838">♦</span></span></strong></span></span></p>
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Mon, 24 Aug 2020 15:11:37 +0000michellelefrancois306108 at https://www.mathcounts.orghttps://www.mathcounts.org/resources/problem-of-the-week/math-middle-2#commentsTent Geometry
https://www.mathcounts.org/resources/problem-of-the-week/tent-geometry-0
<div class="field field-name-field-date-of-problem field-type-datetime field-label-hidden field--date-of-problem"><div class="field-items"><div class="field-item even"><span class="date-display-single" property="dc:date" datatype="xsd:dateTime" content="2020-08-10T11:00:00-07:00">August 10, 2020</span></div></div></div><div class="field field-name-field-problem-category field-type-taxonomy-term-reference field-label-inline clearfix field--problem-category"><div class="field-label">Topics/Content Areas: </div><div class="field-items"><div class="field-item even"><div class="textformatter-list"><a href="/problem-topic-content-area/solid-geometry">Solid Geometry</a></div></div></div></div><div class="l-constrain l-constrain--prose">
<p>Veronica bought a tent for her camping trip this weekend. The shape of the tent is a right triangular prism lying on its side, as shown. (The front and back sides of the tent, which are the prism's bases, are isosceles triangles.) If the height of the tent is 6 feet at the tallest point, the front base of the tent is 5 feet across and the tent is 7 feet long, what is the volume of Veronica's new tent, in cubic feet?</p>
<p><img alt="" src="/sites/default/files/u70427/Tent%20for%20Tent%20Problem%20of%20the%20Week.jpg" style="width: 328px; height: 310px;" /></p>
<p style="margin-left: 40px;"><span style="color:#19a2bf;"><em>The volume of a right triangular prism can be found using the formula V = B × h, where B = the area of the base and h = the height of the prism. So, given the measurements above, we can calculate the volume to be V = B × h = (1/2)(5)(6) × 7 = </em></span><span style="color:#272561;"><em><strong>105 </strong></em></span><span style="color:#19a2bf;"><em>cubic feet. </em></span></p>
<p>Veronica's tent came with a fly to put over the tent to provide additional protection in the rain. If the fly covers all of the sides of the tent (except the bottom) perfectly, how many square feet is the fly? </p>
<p style="margin-left: 40px;"><span style="color:#19a2bf;"><em>The front and back of the tent each have an area of (1/2)(5)(6) = 15 square feet. Now, to figure out the area of the two slanted sides of the tent, we must determine the slant height of the tent. By drawing in the height of the front of the tent, we get two right triangles, so we can use the Pythagorean Theorem to determine the slant height: <span style="font-size:11.0pt"><span style="line-height:107%"><span style="font-family:"Calibri",sans-serif">√</span></span></span>(6<sup>2</sup> + (5/2)<sup>2</sup>) = <span style="font-size:11.0pt"><span style="line-height:107%"><span style="font-family:"Calibri",sans-serif">√</span></span></span>(36 + (25/4)) = 13/2 feet. This tells us that each slanted side of the tent has an area of 7 × (13/2) = 91/2 square feet. Therefore, since the fly covers the sides of the tent perfectly, the fly must be (15)(2) + (91/2)(2) = 30 + 91 = </em></span><span style="color:#272561;"><em><strong>121 </strong></em></span><span style="color:#19a2bf;"><em>square feet. </em></span></p>
<p>Veronica plans to put an air mattress under her sleeping bag for extra cushion. When the mattress is full of air, it is 4 inches thick, contains 6 cubic feet of air and forms a right rectangular prism. When lying flat on the ground parallel to the position of the tent floor, the mattress fits completely inside of the tent. If the length and width are each an integer number of feet, what are the dimensions of the base of the mattress? </p>
<p style="margin-left: 40px;"><span style="color:#19a2bf;"><em>Since the volume of a rectangular prism is length × width × height, we can say that 6 = l × w × (1/3), since 4 inches is the same as 1/3 of a foot. This tells us that l × w = 18 square inches. If the length and width are both integers and it must fit within the 5 feet by 7 feet base of the tent, the dimensions of the base of the mattress must be </em></span><span style="color:#272561;"><em><strong>6 feet by 3 feet</strong></em></span><span style="color:#19a2bf;"><em>.</em></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:11.0pt"><span style="font-family:"Segoe UI Symbol",sans-serif"><span style="color:#3b3838">♦</span></span></span></strong><strong> </strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 1</span></span></span></strong><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""> <span style="color:#3b3838">of the linked PDF contains PROBLEMS & SOLUTIONS.</span></span></span></strong></span></span></p>
<p style="margin-left: 40px; text-align: center;"><span style="font-size:12pt"><span style="font-family:"Times New Roman",serif"><strong><span style="font-size:9.0pt"><span style="font-family:"Akzidenz Grotesk BE Regular""><span style="color:#ff8a3b">Page 2</span><strong> </strong><span style="color:#3b3838">contains ONLY PROBLEMS. </span></span></span><span style="font-size:11.0pt"><span style="color:#3b3838">♦</span></span></strong></span></span></p>
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Mon, 17 Aug 2020 13:10:28 +0000michellelefrancois305847 at https://www.mathcounts.orghttps://www.mathcounts.org/resources/problem-of-the-week/tent-geometry-0#comments