MSP Math Circle Summer Camp Purple Haze and Purple Rain June 15, 2016 Purple Haze and Purple Rain MSP Math Circle Summer Camp
Purple Haze and Purple Rain Anne decides to repaint her house in purple. She can buy two different types of purple paint. Paint A is made up from blue and red paint in the ratio 1 : 2 . Paint B is made up from blue and red paint in the ratio 1 : 8 . She can mix the paints to produce different shades of purple. If Paint A and Paint B come in same size cans, what is the least number she would need of each type in order to produce purple paint containing blue and red in the ratio 1 : 3 . Source: NRICH enriching mathematics http://nrich.maths.org Purple Haze and Purple Rain MSP Math Circle Summer Camp
Things to do Outline of Activity and Solution: 1. Ask each group to present their solution to the problem on a poster. Sort the solutions (same answer, similar methods, type of 2. exposition etc.) and critique other groups solutions. 3. Compare different solutions and find the correct one. 4. See if we can find a solution if we use the ratios part to whole. Purple Haze and Purple Rain MSP Math Circle Summer Camp
Picture Paint A Paint B red red red red red red red red red red red red blue red blue red blue blue Purple Haze and Purple Rain MSP Math Circle Summer Camp
One Solution From the picture above we can see that one can of Paint A contains 3 parts of blue and 6 parts of red while one can of Paint B contains 1 part of blue and 8 parts of red. It might be a good idea to imagine that one can contains 9 ounces of paint. Note that combining one can of Paint A with one can of Paint B yields a mixture that contains 4 ounces of blue paint and 14 ounces of red paint, ratio of 4 : 14 . The table below lists the number of ounces of blue and the number of ounces of red paint for the corresponding mixtures of cans of A and B. Note that 5 cans of A and 3 cans of B yield 18 ounces of blue and 54 ounces of red, a ratio of 18:54, which equals the desired ratio of 1:3. Paint A 1 2 3 4 5 1 4:14 7:20 10:26 13:32 16:38 2 5:22 8:28 11:34 14:40 17:46 Paint B 3 6:30 9:36 12:42 15:48 18:54 4 7:38 10:44 13:50 16:56 19:62 5 8:46 11:52 14:58 17:64 20:70 Purple Haze and Purple Rain MSP Math Circle Summer Camp
Another Solution Let’s start with one can of Paint A. What we What we Too much blue or Correction want have too little blue? 1:3 3:6 too much blue add B Purple Haze and Purple Rain MSP Math Circle Summer Camp
Another Solution Let’s start with one can of Paint A. What we What we Too much blue or Correction want have too little blue? 1:3 3:6 too much blue add B 1:3 4:14 too little blue add A Purple Haze and Purple Rain MSP Math Circle Summer Camp
Another Solution Let’s start with one can of Paint A. What we What we Too much blue or Correction want have too little blue? 1:3 3:6 too much blue add B 1:3 4:14 too little blue add A 1:3 7:20 too much blue add B Purple Haze and Purple Rain MSP Math Circle Summer Camp
Another Solution Let’s start with one can of Paint A. What we What we Too much blue or Correction want have too little blue? 1:3 3:6 too much blue add B 1:3 4:14 too little blue add A 1:3 7:20 too much blue add B 1:3 8:28 too little blue add A Purple Haze and Purple Rain MSP Math Circle Summer Camp
Another Solution Let’s start with one can of Paint A. What we What we Too much blue or Correction want have too little blue? 1:3 3:6 too much blue add B 1:3 4:14 too little blue add A 1:3 7:20 too much blue add B 1:3 8:28 too little blue add A 1:3 11:34 too little blue add A Purple Haze and Purple Rain MSP Math Circle Summer Camp
Another Solution Let’s start with one can of Paint A. What we What we Too much blue or Correction want have too little blue? 1:3 3:6 too much blue add B 1:3 4:14 too little blue add A 1:3 7:20 too much blue add B 1:3 8:28 too little blue add A 1:3 11:34 too little blue add A 1:3 14:40 too much blue add B Purple Haze and Purple Rain MSP Math Circle Summer Camp
Another Solution Let’s start with one can of Paint A. What we What we Too much blue or Correction want have too little blue? 1:3 3:6 too much blue add B 1:3 4:14 too little blue add A 1:3 7:20 too much blue add B 1:3 8:28 too little blue add A 1:3 11:34 too little blue add A 1:3 14:40 too much blue add B 1:3 15:48 too little blue add A Purple Haze and Purple Rain MSP Math Circle Summer Camp
Another Solution Let’s start with one can of Paint A. What we What we Too much blue or Correction want have too little blue? 1:3 3:6 too much blue add B 1:3 4:14 too little blue add A 1:3 7:20 too much blue add B 1:3 8:28 too little blue add A 1:3 11:34 too little blue add A 1:3 14:40 too much blue add B 1:3 15:48 too little blue add A 1:3 18:54 just right done! Purple Haze and Purple Rain MSP Math Circle Summer Camp
Another Solution Let’s start with one can of Paint A. What we What we Too much blue or Correction want have too little blue? 1:3 3:6 too much blue add B 1:3 4:14 too little blue add A 1:3 7:20 too much blue add B 1:3 8:28 too little blue add A 1:3 11:34 too little blue add A 1:3 14:40 too much blue add B 1:3 15:48 too little blue add A 1:3 18:54 just right done! We started with one can of A and added 4 more cans of A and 3 cans of B. We used 5 cans of A and 3 cans of B. Does this algorithm always work? Purple Haze and Purple Rain MSP Math Circle Summer Camp
Yet Another Solution Note that this argument uses the ratio of part to whole. 1 / 3 of a can of Paint A is blue and 1 / 9 of one can of Paint B is blue. Purple Haze and Purple Rain MSP Math Circle Summer Camp
Yet Another Solution Note that this argument uses the ratio of part to whole. 1 / 3 of a can of Paint A is blue and 1 / 9 of one can of Paint B is blue. Both cans are of the same size. Purple Haze and Purple Rain MSP Math Circle Summer Camp
Yet Another Solution Note that this argument uses the ratio of part to whole. 1 / 3 of a can of Paint A is blue and 1 / 9 of one can of Paint B is blue. Both cans are of the same size. We want to mix x cans of Paint A and y cans of paint B in order to get the equivalent of x + y cans of a certain mixed paint, 1 / 4 of which is blue. Purple Haze and Purple Rain MSP Math Circle Summer Camp
Yet Another Solution Note that this argument uses the ratio of part to whole. 1 / 3 of a can of Paint A is blue and 1 / 9 of one can of Paint B is blue. Both cans are of the same size. We want to mix x cans of Paint A and y cans of paint B in order to get the equivalent of x + y cans of a certain mixed paint, 1 / 4 of which is blue. If we total the amount of red paint in x cans of A and y cans of B, we obtain x · 1 3 + y · 1 9 . Purple Haze and Purple Rain MSP Math Circle Summer Camp
Yet Another Solution Note that this argument uses the ratio of part to whole. 1 / 3 of a can of Paint A is blue and 1 / 9 of one can of Paint B is blue. Both cans are of the same size. We want to mix x cans of Paint A and y cans of paint B in order to get the equivalent of x + y cans of a certain mixed paint, 1 / 4 of which is blue. If we total the amount of red paint in x cans of A and y cans of B, we obtain x · 1 3 + y · 1 9 . On the other hand x + y cans containing 1 / 4 of red paint each would give the following total of red paint ( x + y ) · 1 4 . Purple Haze and Purple Rain MSP Math Circle Summer Camp
Yet Another Solution Setting the two sides equal to each other gives us x · 1 3 + y · 1 9 = ( x + y ) · 1 4 . Purple Haze and Purple Rain MSP Math Circle Summer Camp
Yet Another Solution Setting the two sides equal to each other gives us x · 1 3 + y · 1 9 = ( x + y ) · 1 4 . Multiplying both sides with the common denominator of 36 yields 12 x + 4 y = 9( x + y ) = 9 x + 9 y Purple Haze and Purple Rain MSP Math Circle Summer Camp
Yet Another Solution Setting the two sides equal to each other gives us x · 1 3 + y · 1 9 = ( x + y ) · 1 4 . Multiplying both sides with the common denominator of 36 yields 12 x + 4 y = 9( x + y ) = 9 x + 9 y Collecting terms results in 3 x = 5 y or y = 5 x 3 Purple Haze and Purple Rain MSP Math Circle Summer Camp
Yet Another Solution Setting the two sides equal to each other gives us x · 1 3 + y · 1 9 = ( x + y ) · 1 4 . Multiplying both sides with the common denominator of 36 yields 12 x + 4 y = 9( x + y ) = 9 x + 9 y Collecting terms results in 3 x = 5 y or y = 5 x 3 Hence the ratio of number of cans of Paint A to the number of cans of Paint B should be 5 : 3 in order to obtain the desired mixture. We need at least 5 cans of A and 3 cans of B. Purple Haze and Purple Rain MSP Math Circle Summer Camp
Poker Three players, Peggy, Jason and Susan, participated in South Alabama Poker Championship. At the beginning of the night the amount of money each had was in the ratios 7 : 6 : 5 . At the end of the night the ratio was 6 : 5 : 4 . One of the players won $1 , 200. How much money did the players have at the beginning of the evening? Purple Haze and Purple Rain MSP Math Circle Summer Camp
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