Proportional Reasoning Meeting (Measurement and Proportional Reasoning)

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1 Proportional Reasoning Meeting (Measurement and Proportional Reasoning) Topic The topics of measurement and proportional reasoning are covered in this set of problems. Materials Needed Copies of the Proportional Reasoning problem set (Problems and answers can be viewed below. Complete solutions and a more student-friendly version of the problems with pictures and larger text are available for download from on the MCP Members Only page of the Club Program section.) Calculators Meeting Plan The problems for this meeting plan all focus on the idea of proportional reasoning. The problems are arranged such that they increase in difficulty. A general overview of proportional reasoning should be done with students before they begin work on the problems (or the first couple of problems can be done as a group to review key concepts and vocabulary). For the first 11 problems, a proportion can be set up to determine the answer, with x representing the missing value. These proportions have been included in our answers on the following page. However, as you may wish to discuss with your students, every proportion can be written in many ways. For instance, if a/b = c/d, then a/c = b/d, b/a = d/c, (a + b)/b = (c + d)/d and so on. Though students will need to use proportional reasoning for #12-19, these problems may require multiple steps, and therefore, asking for a simple proportion to represent each problem may not be as easy or appropriate. 1. A box of 100 personalized pencils costs $30. How many dollars does it cost to buy 2500 pencils? 2006 School Competition Countdown Round #3 2. Terry needs at least four square feet of floor space per bird in his chicken coop. Terry plans to build a 30-foot by 20-foot chicken coop. What is the maximum number of chickens he can house? 2006 School Competition Sprint Round #6 3. Kyle, Kate and Kendall split the cost of a pizza evenly between the three of them. If Kyle paid $3.24, what was the total cost of the pizza? 2007 School Competition Countdown Round #48 4. If the earth rotates at a constant rate and rotates 360 degrees in one day, how many degrees does it rotate in one hour? 2007 School Competition Countdown Round #14 5. A recipe uses three eggs, nine cups of flour and some other ingredients. The recipe makes enough batter for six servings. How many cups of flour are needed to make enough batter for exactly 10 servings? School Handbook Warm-Up 1-7 (modified) 6. One day, 45 frogs were captured from a pond, marked and then returned to the pond. The next day, 40 frogs were observed in the pond, of which 10 had been marked the previous day. Assuming the marked frogs are equally distributed among all the frogs in the pond, what is the best estimate of how many frogs live in the pond? 2006 School Competition Sprint Round #22 7. The shipping cost of a dozen lemons is directly proportional to how many miles they are shipped. A dozen lemons shipped the 200 miles to Taraville cost $5 to ship. How much will it cost to ship a dozen lemons 1000 miles? School Handbook Warm-Up Julia runs around the track at a steady rate, finishing 400 meters in exactly two minutes. At that rate, how many minutes will it take her to run 5000 meters? School Handbook Warm-Up MATHCOUNTS Club Resource Guide 43 LEMONS

2 9. Traveling at a constant rate, an astronaut takes 36 minutes to go 10,000 miles. At this rate, how far will she travel in the next 10 minutes? Express your answer to the nearest whole number School Handbook Workout A 75-pound bag of feed for livestock contains 30 pounds of bone meal, 30 pounds of limestone and 15 pounds of salt. If Mandelbrot Feed Co. wishes to sell this same mixture in 100-pound bags, how many pounds of salt should be in each bag? School Handbook Workout Purple paint is made with a 16:3:1 ratio of white paint:blue paint:red paint. How much white paint, in gallons, is needed in order to make one gallon of purple paint? Express your answer as a common fraction School Competition Countdown Round # The ratio of boys to girls in a class of 35 students is 3:4. How many more girls than boys are in the class? School Handbook Warm-Up Albert s salary is 34,000 Brunei dollars. Sharky s salary is 28,600 Canadian dollars. If one U.S. dollar is equivalent to 1.7 Brunei dollars or 1.3 Canadian dollars, by how much, in U.S. dollars, does one salary exceed the other? 2006 School Competition Countdown Round # When the temperature goes up 3 degrees on the Cantor scale, it goes up 8 degrees on the Frobenius scale. On both scales, 18 degrees is the same temperature. How many degrees Frobenius are equal to 30 degrees Cantor? 2006 School Competition Target Round #4 15. Equilateral triangle ABC has side length 400 cm and a perimeter equal to 100 times the perimeter of equilateral triangle DEF. How long is each side of triangle DEF, in centimeters? School Handbook Warm-Up A month ago the ratio of nurses to doctors on a hospital staff was 3:5. Since that time two additional nurses joined the staff, no nurses left and the number of doctors remained the same. The ratio of nurses to doctors on the hospital staff is now 4:5. How many nurses are now on the staff? School Handbook Warm-Up A carpenter draws a 1/10-size scale drawing of the top view of a coffee table s rectangular top. The scale drawing of the table s top has an area of 3 square inches. What is the actual area of the table s top going to be? School Handbook Warm-Up A picture measures 6 inches by 8 inches. If a copy machine increases each of the two dimensions by 30%, by what percent is the area of the original picture increased? School Handbook Workout It takes 4 people 3 days to eat 6 watermelons. At this rate, how many days will it take 2 people to eat 5 watermelons? Answers: 1) 100/30 = 2500/x, $750; 2) 1/4 = x/600, 150 chickens; 3) 3.24/1 = x/3, $9.72; 4) 360/24 = x/1, 15 degrees; 5) 9/6 = x/10, 15 cups; 6) 10/40 = 45/x, 180 frogs; 7) 200/5 = 1000/x, $25; 8) 400/2 = 5000/x, 25 minutes; 9) 36/10,000 = 10/x, 2778 miles; 10) 15/x = 75/100, 20 pounds; 11) 16/20 = x/1, 4/5 gallons; 12) 5 girls; 13) $2000 U.S.; 14) 50 degrees; 15) 4 centimeters; 16) 8 nurses; 17) 300 in 2 ; 18) 69%; 19) 5 days Possible Next Steps In the process of going over the answers, ask students to share the proportions they set up and solved. Students may see that though they came to the correct answer, there was a simpler or more efficient way of solving the problem than they used. Problems will have more interesting solutions and should be the focus of discussion, assuming your students were successful with the more basic problems at the beginning MATHCOUNTS Club Resource Guide

3 Proportional Reasoning Student Sheet 1. $ A box of 100 personalized pencils costs $30. How many dollars does it cost to buy 2500 pencils? 2. chickens Terry needs at least four square feet of floor space per bird in his chicken coop. Terry plans to build a 30-foot by 20-foot chicken coop. What is the maximum number of chickens he can house? 3. $ Kyle, Kate and Kendall split the cost of a pizza evenly between the three of them. If Kyle paid $3.24, what was the total cost of the pizza? 4. degrees If the earth rotates at a constant rate and rotates 360 degrees in one day, how many degrees does it rotate in one hour? 5. cups A recipe uses three eggs, nine cups of flour and some other ingredients. The recipe makes enough batter for six servings. How many cups of flour are needed to make enough batter for exactly 10 servings? 6. frogs One day, 45 frogs were captured from a pond, marked and then returned to the pond. The next day, 40 frogs were observed in the pond, of which 10 had been marked the previous day. Assuming the marked frogs are equally distributed among all the frogs in the pond, what is the best estimate of how many frogs live in the pond? 7. $ The shipping cost of a dozen lemons is directly proportional to how many miles they are shipped. A dozen lemons shipped the 200 miles to Taraville cost $5 to ship. How much will it cost to ship a dozen lemons 1000 miles? LEMONS 8. minutes Julia runs around the track at a steady rate, finishing 400 meters in exactly two minutes. At that rate, how many minutes will it take her to run 5000 meters? 9. miles Traveling at a constant rate, an astronaut takes 36 minutes to go 10,000 miles. At this rate, how far will she travel in the next 10 minutes? Express your answer to the nearest whole number. 10. pounds A 75-pound bag of feed for livestock contains 30 pounds of bone meal, 30 pounds of limestone and 15 pounds of salt. If Mandelbrot Feed Co. wishes to sell this same mixture in 100-pound bags, how many pounds of salt should be in each bag? 2010 MATHCOUNTS Foundation. All rights reserved. MATHCOUNTS Club Resource Guide Problem Set

4 11. gallons Purple paint is made with a 16:3:1 ratio of white paint:blue paint:red paint. How much white paint, in gallons, is needed in order to make one gallon of purple paint? Express your answer as a common fraction. 12. girls The ratio of boys to girls in a class of 35 students is 3:4. How many more girls than boys are in the class? 13. $ Albert s salary is 34,000 Brunei dollars. Sharky s salary is 28,600 Canadian dollars. If one U.S. dollar is equivalent to 1.7 Brunei dollars or 1.3 Canadian dollars, by how much, in U.S. dollars, does one salary exceed the other? 14. ºF When the temperature goes up 3 degrees on the Cantor scale, it goes up 8 degrees on the Frobenius scale. On both scales, 18 degrees is the same temperature. How many degrees Frobenius are equal to 30 degrees Cantor? 15. cm Equilateral triangle ABC has side length 400 cm and a perimeter equal to 100 times the perimeter of equilateral triangle DEF. How long is each side of triangle DEF, in centimeters? 16. nurses A month ago the ratio of nurses to doctors on a hospital staff was 3:5. Since that time two additional nurses joined the staff, no nurses left and the number of doctors remained the same. The ratio of nurses to doctors on the hospital staff is now 4:5. How many nurses are now on the staff? 17. sq in A carpenter draws a 1/10-size scale drawing of the top view of a coffee table s rectangular top. The scale drawing of the table s top has an area of 3 square inches. What is the actual area of the table s top going to be? 18. % A picture measures 6 inches by 8 inches. If a copy machine increases each of the two dimensions by 30%, by what percent is the area of the original picture increased? 19. days It takes 4 people 3 days to eat 6 watermelons. At this rate, how many days will it take 2 people to eat 5 watermelons? 2010 MATHCOUNTS Foundation. All rights reserved. MATHCOUNTS Club Resource Guide Problem Set

5 Proportional Reasoning Solutions 1. Since a box of 100 personalized pencils costs $30, we can let the cost of 2500 pencils be $x and set up the following proportion: 100/30 = 2500/x. The cross-product yields 100x = 2500(30) 100x = 75,000. Dividing both sides by 100 shows that x = $750. Notice that we are considering 25 times the amount of original pencils, so the cost will be 25 times the original cost (25)($30) = $ The chicken coop will be 30(20) = 600 square feet. Since each bird needs 4 square feet, we can let the greatest possible number of birds be x and set up the following proportion: 1/4 = x/600. The cross-product yields 4x = 600(1) 4x = 600. Dividing both sides by 4 shows that x = 150 chickens. 3. Since each of the three people ate the same amount of pizza, we can let x be the total cost of the pizza and set up the following proportion: 1/3 = ($3.24)/(x). The cross-product yields (x)(1) = 3($3.24) x = $ One day is equivalent to 24 hours. Letting x be the number of degrees the earth rotates in one hour, we can set up the following proportion: x/1 = 360/24. The cross-product yields 24x = 360(1) 24x = 360. Dividing both sides by 24 shows that x = 15. Notice that since the amount of time being considered was 1/24 of a full day, the rotation would be 1/24 of the full 360 degrees (1/24)(360) = 15 degrees. 5. Letting x be the number of cups of flour needed to make enough batter for exactly 10 servings, we can set up the following proportion: 9/6 = x/10. The cross-product yields 90 = 6x. Dividing both sides by 6 shows that x = 15 cups. 6. Letting x be the number of frogs in the pond, we can set up the following proportion: 45/x = 10/40. The cross-product yields 10x = 45(40) 10x = Dividing both sides by 10 shows that x = 180 frogs. 7. Both shipments involved a dozen lemons. The only factor manipulating the cost is the distance the lemons are shipped. Letting x be the cost to ship a dozen lemons 1000 miles, we can set up the following proportion: 200/$5 = 1000/x. The cross-product yields $5000 = 200x. Dividing both sides by 200 shows that x = $25. Notice that since the distance was five times longer, the cost was five times the original cost. 8. Letting x be the number of minutes it will take to run 5000 meters, we can set up the following proportion: 400/2 = 5000/x. The cross-product yields 10,000 = 400x. Dividing both sides by 400 shows that x = 25 minutes. 9. Letting x be the distance she will travel in the next 10 minutes, we can set up the following proportion: 36/10,000 = 10/x. The cross-product yields 100,000 = 36x. Dividing both sides by 36 shows us x = 2778 miles, to the nearest whole number. 10. Letting x be the number of pounds of salt in a 100-pound bag of feed, we can set up the following proportion: 15/75 = x/100. The cross-product yields 75x = Dividing both sides by 75 shows us x = 20 pounds MATHCOUNTS Foundation. All rights reserved. MATHCOUNTS Club Resource Guide Problem Set Solutions

6 11. Assuming the original ratio of 16:3:1 is in gallons, this would yield 20 gallons of the purple paint. Letting x be the number of gallons of white paint needed for 1 gallon of purple paint, we can set up the following proportion: 16/20 = x/1. The cross-product yields 20x = 16. Dividing both sides by 20 shows us that x = 16/20 = 4/5 gallons. 12. The ratio of boys to girls in the class being 3:4 tells us that there are = 7 parts in the class of 35 students. Each part then represents 35 7 = 5 students. Since there is one more part girls than boys, there are 5 more girls than boys in the class. 13. First we can put each of the original dollar amounts into U.S. dollars. Using $x US/ $34,000 BRU = $1 US/$1.7 BRU, we see the 34,000 Brunei dollars is equivalent to ($34,000) (1.7) = $20,000 US. Using $x US/$28,600 CAN = $1 US/$1.3 CAN, we see the 28,600 Canadian dollars is equivalent to ($28,600) (1.3) = $22,000 US. The difference is $22,000 $20,000 = $2000 US. 14. From 18ºC to 30ºC is an increase of 12ºC. Knowing that a rise of 3ºC is equivalent to a rise of 8ºF, we can let x be the increase in ºF and set up the following proportion: 12/x = 3/8 3x = 96 x = 32ºF. Therefore a reading of 30ºC on the thermometer is equal to 18º + 32º = 50ºF on the thermometer. 15. Any two equilateral triangles are similar, meaning that the ratio of their sides is equivalent to the ratio of their perimeters. Letting x be the side length of triangle DEF, we can set up the following proportion: 400/x = 100/1. The cross-product yields 100x = 400, and so x = The ratio of nurses to doctors being 3:5 tells us that there were 3x nurses and 5x doctors for some value of x. The number of nurses increased to 3x + 2, the number of doctors remained 5x and the ratio became 4:5. Therefore, (3x + 2)/(5x) = 4/5. The cross-product yields 5(3x + 2) = 4(5x) 15x + 10 = 20x. Subtracting 15x from both sides and then dividing both sides by 5 shows us x = 2. Therefore, there are 3(2) + 2 = 8 nurses on staff now. 17. The scale drawing and the actual table are similar figures. The ratio of the areas of the figures will be the square of the ratios of the side lengths of the figures. Since the ratio of the side lengths is 1/10, the ratio of the areas is 1/100. Letting x be the area of the actual table top, in square inches, we can set up the following proportion: 1/100 = 3/x. The cross-product yields x = 300 square inches. 18. We can use logic similar to that used for #17. A 30% increase in each side length means that the ratio of the sides of the original picture to the enlargement is 1/1.3. The ratio of the areas will then be (1/1.3) 2 = 1/1.69. This shows that the enlargement is an increase of 69%. 19. We are given that it takes 4 people 3 days to eat 6 watermelons. Notice that if we cut the number of people in half, we can either double the number of days or cut the number of watermelons in half to keep a true ratio: 2 people/6 days/6 watermelons or 2 people/3 days/3 watermelons. Notice that each of these leads to the realization that it takes 2 people 1 day to eat 1 watermelon. Therefore, it will take them 5 days to eat 5 watermelons MATHCOUNTS Foundation. All rights reserved. MATHCOUNTS Club Resource Guide Problem Set Solutions