{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 255 0 255 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 " Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 } } {SECT 0 {PARA 0 "" 0 "" {TEXT 265 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 264 21 " Mixture Word Problems" }}{PARA 0 "" 0 "" {TEXT -1 325 "\nMixture word \+ problems involve mixing of two quantities to obtain some quantity with a worth of some intermediate value. In this worksheet, we will explor e the concept of mixtures of coffee, nuts, coins, etc. We see a proper ly scaled graphic representations, and examine a box method of seting \+ up and solving thise problems. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 266 153 "[Directions : Execute the Code Resource section first. Although there will be no output immediately, these de finitions are used later in this worksheet.]" }}{SECT 0 {PARA 4 "" 0 " " {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "res tart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1056 "Mi xturePlot := proc( a, ar, b, br)\n local c,ct,cr,g,as,bs,cs, spacer, bo, co, pa, pb, pc1, pc2, tp, te, r, ch;\n\n c := a + b; \+ ct := a*ar + b*br; cr := ct/c;\n g := 1.6;\n as := sqrt( a/g); bs := sqrt( b/g); cs := sqrt( c/g);\n spacer := (as+bs+c s)/10; \n \n pa := polygonplot( [ [0,0],[as,0],[as,as*g],[0,a s*g] ], \n color = gold, axes = none):\n bo := as + spacer; \n \n pb := polygonplot( [ [bo, 0],[bo+bs, 0] ,[bo+bs,bs*g],[bo,bs*g] ], \n color = green, ax es = none):\n co := bo + spacer*1.5 + bs; \n r := a/(a+b); \+ ch := r*cs*g;\n pc1 := polygonplot( [ [co,0],[co+cs,0],[co+cs,ch], [co,ch] ], \n color = gold, axes = none):\n \+ pc2 := polygonplot( [ [co,ch],[co+cs,ch],[co+cs,cs*g],[co,cs*g] ], \n color = green, axes = none):\n\n tp := tex tplot( [ as + spacer/2, max(as,bs)*g/2, `+`]);\n te := textplot( [ bo + bs + spacer/2, max(as,bs)*g/2, `=`]);\n\n display( pa, pb, p c1, pc2, tp, te)\n end proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 537 "MixtureTableTes t := proc( a, ar, b, br )\n # a = quanity of a, ar = a's rate, a nd so forth for b\n local rt, M ;\n rt := (a*ar + b*br)/(a+ b);\n if( not( (whattype(rt) = integer) or\n (what type(rt) = float) or\n (whattype(rt) = fraction) )) t hen rt := `?` fi;\n array( [ [``, `Quantity *`,` Rate`,` = Resu lt`], \n [ `A`, a, ar, a*ar], \n \+ [ `B`, b, br, b*br], \n [ `A+B `, a+b, rt, a*ar + b*br] ]);\n end proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 692 "MixtureTable1 := proc( a, ar, b, br, cr )\n \+ # a = quanity of a, ar = a's rate, and so forth for b\n local \+ rt, M, equation;\n rt := (a*ar + b*br)/(a+b);\n if( not( (w hattype(rt) = integer) or\n (whattype(rt) = float) o r\n (whattype(rt) = fraction) )) then rt := `?` fi;\n \+ M := array( [ [``, `Quantity *`,` Rate`,` = Result`], \n \+ [ `A`, a, ar, a*ar], \n [ `B`, \+ b, br, b*br], \n [ `A+B `, a+b, cr, (a+ b)*cr] ]);\n print(M);\n print(` `):\n equation := M[2,4] + M[3,4] = M[4,4];\n print( equation );\n solve( equation, x); print(x = %);\n end proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 624 "Mixtu reTableEq := proc( ar, br, cr, c) #||a rate|b rate|c rate|c amo unt||\n \n local a, b, rt, M, equation;\n rt := cr ; #(a* ar + b*br)/(a+b);\n a := x; b := c-a;\n \n M := arra y( [ [``, `Quantity *`, ` Rate`,` = Result`], \n [ `A`, a, ar, a*ar ], \n [ `B`, b, br, (c-a)*br ], \n [ `A+B `, c, cr, c* cr ] ]);\n print(M);\n print(` `):\n equation := M [2,4] + M[3,4] = M[4,4];\n print( equation );\n solve( equati on, x); print(` `); print(A = %, B = c-%); print(` `);\n \n end pr oc:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 457 "MixtureTableSol n := proc( ar, br, cr, c) #||a rate|b rate|c rate|c amount|| \n local a, rt, M, equation;\n rt := cr ; #(a*ar + b*br)/(a+b );\n a := solve( x*ar + (c-x)*br = c*cr, x);\n \n arra y( [ [``, `Quantity *`, ` Rate`,` = Result`], \n [ `A`, a, ar, a*ar ], \n [ `B`, c- a, br, (c-a)*br ], \n [ `A+B `, c, cr, \+ c*cr ] ]);\n \n end proc:" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "1. General Idea" }}{PARA 0 "" 0 "" {TEXT -1 240 "There are many variations of mixture problems such as mxitures of coffee, nuts, and \+ coins. First, lets just get a rough idea of what is going on.\n\n_____ ______________________________________________________________________ ________________\n\n" }{TEXT 256 13 "Problem 1.1 :" }{TEXT -1 72 " You mix 5 pounds of nuts worth $2/lb and 3 pounds of nuts worth $4/lb.\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "MixturePlot( 5, 2, 3, 4) ; #(a amount) | (a rate) | (b amount) | (b rate)" }}}{PARA 0 "" 0 " " {TEXT -1 188 "\nThere are three items we can compute :\n 1. H ow many pounds are in the mixture?\n 2. How much does the entir e mixture cost?\n 3. How much is the mixture worth per pound?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "`Total number of pounds` : = 5 + 3;\n`Total cost` := 5*2 + 3*4;\n`Cost per pound` := evalf( %/%%, 3) ;" }}}{PARA 0 "" 0 "" {TEXT -1 90 "_______________________________ ________________________________________________________\n\n\n" } {TEXT 257 13 "Problem 1.2 :" }{TEXT -1 88 " You mix 12 pounds of coffe e A worth $5.50/lb and 19 pounds of coffee B worth $6.75/lb.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "MixturePlot( 12, 5.50, 19, 6 .75 ); #(a amount) | (a rate) | (b amount) | (b rate)\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 188 "\nThere are th ree items we can compute :\n 1. How many pounds are in the mixt ure?\n 2. How much does the entire mixture cost?\n 3. Ho w much is the mixture worth per pound?" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "`Total number of pounds ` := 12 + 19;\n`Total cost` := 12*5.50 + 19*6.75;\n`Cost per pound` := evalf( %/%%, 3) ;" }}}{PARA 0 "" 0 "" {TEXT -1 92 "\n\n______________ ______________________________________________________________________ ___\n\n\n" }{TEXT 267 13 "Problem 1.2 :" }{TEXT -1 172 " You mix 12 po unds of coffee A worth $5.50/lb and 19 pounds of coffee B worth $6.75/ lb.\n\nNotice the difference between a 50-50 mixture, a 99-1 mixture, \+ and a 1-99 mixture.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "Mix turePlot( 50, 1, 50, 1);\nMixturePlot( 99, 1, 1, 1 );\nMixturePlot( 1, 1, 99, 1 );" }}}{PARA 0 "" 0 "" {TEXT -1 91 "\n______________________ _________________________________________________________________\n\n \n" }{TEXT 268 13 "Problem 1.2 :" }{TEXT -1 145 " You mix 12 pounds of coffee A worth $5.50/lb and 19 pounds of coffee B worth $6.75/lb.\n\n \nThere are also other types of mixture problems .....\n\n\n" }{TEXT 259 13 "Problem 1.3 :" }{TEXT -1 97 " You combine 17 nickels and 23 qu arters. How many coins to you have? And what is the total value?" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "`number of coins` = 17 + 23; \n`total value` = 17*.05 + 23*.25;" }}}{PARA 0 "" 0 "" {TEXT -1 101 " \n____________________________________________________________________ ______________________________\n\n" }{TEXT 258 13 "Problem 1.4 :" } {TEXT -1 90 " You combine 12 gallons of a 15% acid solution with 34 ga llons of a 22% acid solution. \n " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{SECT 0 {PARA 4 "" 0 "" {TEXT -1 29 "2. Mixture Problems " } {TEXT 269 30 " (only one known quantity)" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 262 13 "Problem 2.1 :" }{TEXT -1 359 " How many pound s of A nuts worth $2/lb should be combined with 3 pounds of B nuts wor th $4/lb to get a mixture worth $3.50 per pound?\n\nJust for fun, we c an guess a few answers to get a feel for what is going on. We know all the entries except the quantity of the A nuts. Lets guess 3, 5 and 12 pounds of A to see what price per pound comes out for the result." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 148 "MixtureTableTest( 3, 2, 3, \+ 4); # (a quantity) | (a rate) | (b quantity) | (b rate)\nMixtureTab leTest( 5, 2, 3, 4);\nMixtureTableTest( 12, 2, 3, 4);" }}}{PARA 0 "" 0 "" {TEXT -1 265 "Look at the A+B rate in each case ... $3/lb, $11/ 4 = $2.75 /lb, and $12/5 = $2.40/lb. \n\n\n\n\nLets find the solution. We form the table, form an equation by adding the first two entries i n the last column to be equal to the last entry, then solve to find th e answer." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "MixtureTable1( \+ x, 2, 3, 4, 3.5); #||a amount|a rate|b amount|b rate|c rate||" }}} {PARA 0 "" 0 "" {TEXT -1 99 "_________________________________________ _________________________________________________________\n" }{TEXT 260 15 "\n\nProblem 2.2 :" }{TEXT -1 129 " How many kilograms of candy worth $6.50kg should be combined with 12 kg of candy worth $11/kg to \+ get a mixture worth $8 per kg?\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "MixtureTable1( x, 6.50, 12, 11, 8);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 13 "Problem 2.3 :" }{TEXT -1 137 " 100 grams of chemicals worth $30/g should be mixed with how many gra ms of a chemical worth $44/gram to obtain a mixture worth $38/gram." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "MixtureTable1( 100, 30, x, \+ 44, 38);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 33 "3. Mixture Problems \020 " }{TEXT 270 25 "(bot h quantities unknown)" }}{PARA 0 "" 0 "" {TEXT -1 4 "\n " }{TEXT 273 23 "Coffee & Rice Problems\n" }{TEXT -1 1 "\n" }{TEXT 261 13 "Prob lem 3.1 :" }{TEXT -1 378 " How many pounds of coffee A worth $8/lb and coffee B worth $7.25/lb should be mixed to obtain 50 pounds of a mixt ure worth $7.70 per pound?\n\nWe create a table. Since both quantities (of A and B) are unknown, we need to make one x, and the other 50-x. \+ Then we multiply across, add down, to form an equation. In this case : (8x) + (7.25 - 362.50) = 385. Then we solve and enjoy.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "MixtureTableEq( 8, 7.25, 7.70, 50); #||a rate|b rate|c rate|c amount|| " }}}{PARA 0 "" 0 "" {TEXT -1 39 "\nHere is a view of the entire solution." }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 74 "MixtureTableSoln( 8, 7.25, 7.70, 50 ); #||a r ate|b rate|c rate|c amount||" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" } {TEXT 275 13 "Problem 3.2 :" }{TEXT -1 333 " How many pounds of basmat i rice worth $2.70 cents per kg, and wild rice worth $4.57 per kg sho uld be mixed to obtain 22 kg of a mixture worth $3.55 per kg? Again, w e create a table. Since both quantities of rice are unknown, we need t o make one x, and the other 22-x. Then we \"multiply across\" and \"ad d down\" to form an equation. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "MixtureTableEq( 2.70, 4.57, 3.55, 22); #||a rate|b rate|c rate|c amount|| " }}}{PARA 0 "" 0 "" {TEXT -1 231 "\nWe solve the equ ation : -1.87*x + 100.54 = 78.10, to get x, which is the quantity of \+ A (Basmati rice). We find that A is 12 kg. To find B (Wild rice), we s ubtract this from 22 to get 10 kg.\n\nHere is a view of the entire sol ution." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "MixtureTableSoln( \+ 2.60, 4.20, 3.15, 18 ); #||a rate|b rate|c rate|c amount||" }}}{PARA 0 "" 0 "" {TEXT -1 7 "\n\n\n " }{TEXT 274 14 "Coin Problems\n" } {TEXT -1 319 "\nCoin problems are similar to coffee, candy, and other \+ \"price per pound\" quantites. In the case of coins, we don't give the m a value per pound, we give them a value per coin. So instead of \"nu mber of pounds\" and \"price per pound\", we use \"number of coins\" a nd \"value per coin\". The rest is then the same, essentially.\n\n" } {TEXT 271 13 "Problem 3.3 :" }{TEXT -1 310 " How many pennies and quar ters are needed to have 64 coins worth $4.72?\n\nNote that both the nu mber of pennies and quarters are both unknown. (Note - not for doing t he math, but for using this maple procedure. The \"average value\" of \+ a coin is the total value of all coins divided by the total number of \+ coins.)\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "MixtureTableEq( .01,.25, 4.72/64 , 64); #||a rate|b rate|c rate|c amount|| " }}} {PARA 0 "" 0 "" {TEXT -1 1 "\n" }{TEXT 272 13 "Problem 3.4 :" }{TEXT -1 68 " How many dimes and nickels are needed to have 80 coins worth $ 4.35?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "MixtureTableEq( .10 ,.05, 4.35/80 , 80); " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 36 "\n \+ \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {MARK "0 0" 22 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }