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# Recent problems solved by 'ptaylor'

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 Travel_Word_Problems/106198: On a river a boat travels 240 miles downstream in the same amount of time that it takes to travel 160 miles upstream. The speed of the boat without current is 25 mph. Find the speed of the river. Be sure to define your variable, write an equation, and show how you solved it.1 solutions Answer 77326 by ptaylor(2048)   on 2007-10-25 14:06:20 (Show Source): You can put this solution on YOUR website! distance(d) equals rate(r) times time(t) or d=rt; r=d/t and t=d/r Let r=speed (rate) of the river rate of the boat downstream=25+r rate of the boat upstream=25-r Time to travel downstream=(distance downstream)/(rate downstream)=240/(25+r) Time to travel upstream=(distance upstream)/(rate upstream)=160/(25-r) Now we are told that these two times are equal. So: 240/(25+r)=160/(25-r) multiply both sides by (25+r)(25-r) 240(25-r)=160(25+r) get rid of parens 6000-240r=4000+160r subtract 6000 and also 160r from both sides 6000-6000-240r-160r=4000-6000+160r-160r collect like terms -400r=-2000 divide both sides by -400 r=5 mph-------------------rate (speed) of the river CK 240/30=160/20 8=8 Hope this helps---ptaylor
 Travel_Word_Problems/106148: This question is from textbook beginning algebra Martina leaves home at 9 A.M., bicycling at a rate of 24 mi/h. Two hours later, John leaves, driving at the rate of 48 mi/h. At what time will John catch up with Martina?1 solutions Answer 77292 by ptaylor(2048)   on 2007-10-25 05:45:37 (Show Source): You can put this solution on YOUR website! distance(d)=rate(r) times time(t) or d=rt;t=d/r and r=d/t Let t=time that elapses after John leaves until he catches Martina We know that when John leaves, Martina has already travelled 24*2 or 48 miles (d=rt) Distance Martina has travelled when John catches her=48+24t Distance John has travelled when he catches Martina=48t Now we know that John will have caught up with Martina when they have both travelled the same distance. So our equation to solve is: 48+24t=48t subtract 24t from both sides 48+24t-24t=48t-24t collect like terms 48=24t divide both sides by 24 t=2 hrs---------------------time that elapses after John leaves until he catches Martina We are also told that John leaves at 11:00AM. So 11:00AM plus 2 hours=1:00PM ---------time that John catches Martina CK 48+24*2=48*2 48+48=96 96=96 Hope this helps---ptaylor
 Age_Word_Problems/106246: john is x years old. his father is 3 times as old as he is. his mother is 3 years younger than his father. find the sum of their ages in terms of x in 10 years time.1 solutions Answer 77291 by ptaylor(2048)   on 2007-10-25 05:27:28 (Show Source): You can put this solution on YOUR website!NOW: x=John's age 3x=father's age 3x-3=mother's age TEN YEARS FROM NOW: x+10=John's age 3x+10=father's age (3x-3)+10=mother's age SO THE SUM OF THEIR AGES IN TEN YEARS TIME IS: x+10+3x+10+(3x-3)+10 get rid of parens x+10+3x+10+3x-3+10 collect like terms 7x+27---------------ans Hope this helps---ptaylor
 Expressions-with-variables/106136: This question is from textbook Algebra:Structure and Method Problem solving using chart: Question: The Eiffel tower is 497 ft. taller than the Washington monument. If each of the monuments were 58ft. shorter, the eiffel tower would be twice as tall as the Washington monument. How tall is each? 1 solutions Answer 77290 by ptaylor(2048)   on 2007-10-25 05:18:29 (Show Source): You can put this solution on YOUR website! Let x=height of Washington Monument: (x-58)----58 ft shorter Then x+497=height of Eiffel Tower: ((x+497)-58)---------58 ft shorter Now we are told that: ((x+497)-58)=2(x-58) get rid of parens x+497-58=2x-116 simplify x+439=2x-116 add 116 to and subtract x from both sides x-x+439+116=2x-x-116-116 collect like terms 555=x or x=555 ft--------------------------height of Washington Monument x+497=555+497=1052 ft---------- height of Eiffel Tower CK 1052-58=2*(555-58) 994=2*497 994=994 Hope this helps-----ptaylor
 Percentage-and-ratio-word-problems/106135: This question is from textbook Algebra: Structure and Method Problem solving using chart:Question: Brian O'Reilly earns twice as much each week as a tutor than he does pumping gas. His total weekly wages are \$150 more than that of his younger sister. She earns one quarter as much as Brian does as a tutor. How much does Brian earn as a tutor?1 solutions Answer 77289 by ptaylor(2048)   on 2007-10-25 05:05:27 (Show Source): You can put this solution on YOUR website!Let x=amount that Brian earns as a tutor---weekly wages Then (x/2)=amount that Brian earns pumping gas So x+(x/2)= Brian's total weekly wages And (x/4)=amount that Brian's sister earns Now we are told that: x+(x/2)=(x/4)+150 multiply both sides by 4 4x+2x=x+600 subtract x from both sides 4x+2x-x=x-x+600 collect like terms 5x=600 divide both sides by 5 x=\$120------Brian's weekly wages as a tutor (x/2)=(120/2)=\$60-------------------Brian's weekly wages pumping gas (x/4)=(200/4)=\$50 ----------- Brian's sister's weekly wages CK \$120+\$60=\$30+\$150 \$180=\$180 Hope this helps---ptaylor
 Mixture_Word_Problems/106238: A 5 gallon radiator containing a mixture of water and antifreeze was supposed to contain a 50% antifreeze solution. When tested, it was found to have only 40% antifreeze. How much must be drained out and replaced with pure antifreeze so that the radiator will contain the desired 50% antifreeze. thank u for sparing time for my problem...thanks....1 solutions Answer 77288 by ptaylor(2048)   on 2007-10-25 04:47:22 (Show Source): You can put this solution on YOUR website!Let x=amount that needs to be drained and replaced with pure antifreeze Now we know that the amount of pure antifreeze left after we drain x amount out (0.40(5-x))plus the amount of pure antifreeze added (x) has to equal the amount of pure antifreeze in the final mixture (0.50*5). So our equation to solve is: 0.40(5-x)+x=0.50*5 get rid of parens and simplify 2-0.40x+x=2.5 subtract 2 from both sides 2-2-0.40x+x=2.5-2 collect like terms 0.60x=0.5 divide both sides by 0.60 x=0.833333-- gal----------amount that needs to be drained and replaced with pure antifreeze CK 0.40(5-0.83333333333)+0.8333333333333=2.5 1.666666666666+0.8333333333333333333=2.5 ~2.5=2.5 Hope this helps----ptaylor
 Average/105863: 5 times a number, increased by 1, is 36. find the number? I am not good with these can you help1 solutions Answer 77039 by ptaylor(2048)   on 2007-10-23 19:03:37 (Show Source): You can put this solution on YOUR website! Let x=the number Then 5x= five times the number And 5x+1=five times the number increased by 1 and we are told that this equals 36 So: 5x+1=36 subtract 1 from both sides 5x+1-1=36-1 collect like terms 5x=35 divide both sides by 5 x=7 Ck 5*7+1=36 36=36 Hope this helps--ptaylor
 Percentage-and-ratio-word-problems/105531: In a family there are two cars. In a given week, the first car gets an average of 15 miles per gallon, and the second car gets 40 miles per gallon. The two cars combined drive a total of 65 miles in that week, for a total gas consumption of 1600 gallons. How many gallons were consumed by each of the two cars that week? 1 solutions Answer 76865 by ptaylor(2048)   on 2007-10-22 15:43:12 (Show Source): You can put this solution on YOUR website!Let x=miles driven by the 15 mpg car Then 15x=amount of gas used by the 15 mpg car And 65-x=miles driven by the 40 mpg car Then 40(65-x)=amount of gas used by the 40 mpg car Now we are told that together the two cars use 1600 gallons. So our equation to solve is: 15x+40(65-x)=1600 get rid of parens 15x+2600-40x=1600 subtract 2600 from both sides 15x+2600-2600-40x=1600-2600 collect like terms -25x=-1000 divide both sides by -25 x=40 miles--------------------miles driven by the 15 mpg car 65-x=65-40=25 miles -----------miles driven by the 40 mpg car CK 40*15+25*40=1600 600+1000=1600 1600=1600 Hope this helps----ptaylor
 Equations/105517: Which equation is an identity? a) 3w+6-w=4w-2(w-3) b) 8-(5v+4)=-5v-4 c) 5y+7=5y-7 d) 4m-6=5m+9-m1 solutions Answer 76811 by ptaylor(2048)   on 2007-10-22 04:03:00 (Show Source): You can put this solution on YOUR website!The answer is a). a) 3w+6-w=4w-2(w-3) get rid of parens 3w+6-w=4w-2w+6 collect like terms on each side of the equation 2w+6=2w+6---------------this is an identity---true for all values of w b) 8-(5v+4)=-5v-4 get rid of parens 8-5v-4=-5v-4 collect like terms on each side of the equation -5v+4=-5v-4 This problem does not have a solution. There are no values for v that will satisfy this equation. c) 5y+7=5y-7 This problem does not have a solution. There are no values for y that will satisfy this equation. d) 4m-6=5m+9-m collect like terms on each side of the equation 4m-6=4m+9 This problem does not have a solution. There are no values for m that will satisfy this equation. Hope this helps---ptaylor
 Miscellaneous_Word_Problems/105228: I'm not even sure if you would consider this a word problem. I do not even know where to begin. The area of a square is 82 ft^2. Find the length of the side to the nearest hundredth. Your help is appreciated.1 solutions Answer 76605 by ptaylor(2048)   on 2007-10-20 11:30:41 (Show Source): You can put this solution on YOUR website! Well, lets begin by letting S=one of the sides of the square. Now we know that the area of a square is S*S or S^2 and we are told that this equals 82 ft^2. So: S^2=82 ft^2 take the square root of both sides S=+or-SQRT(82) forget the negative value because lengths are positive S=9.055 now to the nearest hundreth, S would be either 9.06 or 9.05 depending on our convention for rounding. Hope this helps----ptaylor
 Linear_Equations_And_Systems_Word_Problems/105258: This question is from textbook University of Phoenix Elementary and Intermediate Algebra Ok my favorite tutors, I only have 2 more questions, at least until monday:) You guys rock!!! Ziggy's famous yogurt blends regular yogurt that is 3% fat with its no-fat yogurt to obtain low-fat yogurt that is 1% fat. How many pounds of regular yogurt and how many pounds of no-fat yogurt should be mixed to obtain 60 pounds of low-fat yogurt? Can you please help me set this up and solve it???1 solutions Answer 76601 by ptaylor(2048)   on 2007-10-20 11:20:07 (Show Source): You can put this solution on YOUR website!Let x=required amount of yogurt that is 3% fat Then 60-x=required amount of no-fat yogurt Now we know that the amount of pure fat in the 3% yogurt (0.03x) plus the amount of pure fat in the no-fat yogurt (0) has to equal the amount of pure fat in the final 1% yogurt mixture(0.01*60) so or equation to solve is: 0.03x=0.01*60 simplify 0.03x=0.6 divide both sides by 0.03 x=20 lbs-----------------------------amount of yogurt that is 3% fat 60-x=60-20=40 lb-------------------------amount of no-fat yogurt needed CK 0.03*20=0.01*60 0.6=0.6 Hope this helps---ptaylor
 Mixture_Word_Problems/104828: hector is performing a chemistry experiment that requires 140 milliliters of a 30% copper sulfate solution. He has a 25% copper sulfate solution and a 60% copper sulfate solution. How many milliliters of each solution should he mix to obtain the needed solution?1 solutions Answer 76408 by ptaylor(2048)   on 2007-10-18 15:58:21 (Show Source): You can put this solution on YOUR website! Let x=amount of 25% copper sulfate solution needed Then 140-x=amount of 60% copper sulfate solution needed Now we know that the amount of pure copper sulfate in the 25% solution (0.25x) plus the amount of pure copper sulfate in the 60% solution (0.60(140-x)) has to equal the amount of pure copper sulfate in the final mixture(0.30*140). So our equation to solve is: 0.25x+0.60(140-x)=).30*140 get rid of parens 0.25x+84-0.60x=42 subtract 84 from both sides 0.25x+84-84-0.60x=42-84 collect like terms -0.35x=--42 divide both sides by -0.35 x=120 ml ---------------------------------- amount of 25% solution 140-x=140-120=20 ml------------------------------amount of 60% solution ck 0.25*120+0.60(140-120)=0.30*140 simplify 30+12=42 42=42 Hope this helps---ptaylor
 Linear-systems/104818: This question is from textbook Pretice hall mathematics. Algebra 1 The math club and science club had fundraisers to buy supplies for a hospice. the math club spent \$135 buying six cases of juice and one case of bottled water. The science club spent \$110 buying four cases of juice and two cases of bottled water. how much did a case of juice cost? how much did a case of bottled water cost? K this one also needs to be written in a linear equation form and solved..i'm really bad at this stuff! please help me1 solutions Answer 76405 by ptaylor(2048)   on 2007-10-18 15:45:43 (Show Source): You can put this solution on YOUR website!Let x=cost of a case of juice And y=cost of a case of bottled water Now we are told that: 6x+y=\$135---------------eq1----cases and cost for the math club 4x+2y=\$110---------------eq2-----cases and cost for the science club Multiply eq1 by 2 and we get: 12x+2y=\$270-------------------eq1a 4x+2y=\$110-----------------------eq2 subtract eq2 from eq1a and we get: 8x=\$160 divide both sides by 8 x=\$20----------------cost of a case of juice substitute x=\$20 into eq2 and we have: 4*\$20+2y=\$110 or \$80+2y=\$110 subtract \$80 from both sides \$80-\$80+2y=\$110-\$80 collect like terms 2y=\$30 divide both sides by 2 y=\$15----------------------------cost of a case of bottled water CK 6*20+1*15=135 120+15=135 135=135 also 4*20+2*15=110 80+30=110 110=110 Hope this helps----ptaylor
 decimal-numbers/103628: How do u convert fractions into decimals?1 solutions Answer 75366 by ptaylor(2048)   on 2007-10-11 16:12:27 (Show Source): You can put this solution on YOUR website!How do u convert fractions into decimals? You divide the denominator of the fraction into the numerator Ex1-------1/4= 1 divided by 4 =0.25 Ex2--------1/3=1 divided by 3=0.333333333333333333--- Ex3-------2/3=2 divided by 3=0.6666666666666666666----- Ex4----------2/4=2 divided by 4=0.50 Generally, when we are dealing with decimalss, we round them off depending on the level of accuracy that we need. In example 2, if we are content with accuracy to the nearest tenth, then the answer is 0.3; If we want an accuracy to the nearest hundreth, then the answer is 0.33; If we want an accuracy to the nearest millionth, then the answer is 0.333333 and so forth. In example 3, if we want an accuracy to the nearest thousandth, then the answer would be 0.667. If someone tells you they walked 4 miles, then they could have walked anywhere between 3.5 miles and 4.5 miles (to the nearest mile) but if they tell you they walked 4.0 miles, they must have walked somewhere between 3.95 miles and 4.05 miles (to the nearest 1/10 of a mile) Hope this helps---ptaylor
 Linear-systems/103564: Solve the system by substitution. Help me please. x + y = 12 y = 2x 1 solutions Answer 75361 by ptaylor(2048)   on 2007-10-11 14:47:31 (Show Source): You can put this solution on YOUR website! x + y = 12---------------eq1 y = 2x ----------------------eq2 Lets substitute y=2x from eq2 into eq1 and we get: x+2x=12 collect like terms 3x=12 divide both sides by 3 x=4 Now we can substitute x=4 into eq2 and we get: y=2*4=8 So our answer is: x=4 and y=8 CK From eq1: x+y=12 4+8=12 12=12 From eq2: y=2x 8=2*4 8=8 Hope this helps---ptaylor
 Travel_Word_Problems/103532: two people in a canoe travel 6 mi upstream against a 2mi/h current. They then returned to the same point downstream. If the entire trip took 4 hours, how fast can they paddle in still water?1 solutions Answer 75360 by ptaylor(2048)   on 2007-10-11 14:39:24 (Show Source): You can put this solution on YOUR website!distance(d)=rate(r) times time(t) or d=rt; r=d/t and t=d/r Let r=rate (speed) in still water. distance upstream = distance downstream=6 mi time to travel upstream + time to travel downstream=4 hrs (Note: When travelling against the current, you must subtract the speed of the current from your speed in still water. Likewise, when travelling with the current, you must add the speed of the current to your speed in still water) time to travel upstream=6/(r-2) time to travel downstream=6/(r+2) So, our equation to solve is: 6/(r-2)+6/(r+2)=4 multiply both sides by (r-2)(r+2) or (r^2-4) 6(r+2)+6(r-2)=4(r^2-4) get rid of parens 6r+12+6r-12=4r^2-16 subtract 4r^2 from and add 16 to both sides 6r+6r+12-12-4r^2+16=0 collect like terms -4^r2+12r+16=0 multiply each term by -1 4r^2-12r-16=0 quadratic in standard form and it can be factored (2r+2)(2r-8)=0 2r+2=0 r=-1---------------------disregard negative speed in this problem and 2r-8=0 r=4 mph----------------------speed in still water CK time upstream=6/(4-2)=3 hrs time downstream=6(4+2)=1 hr 3 hrs+1 hr=4 hr 4 hr=4 hr Hope this helps----ptaylor
 Mixture_Word_Problems/103233: How many liters of a solution containing 22% sugar and how many liters of another solution containing 31% sugar must be mixed together to make 680 liters of solution containing 27% sugar?1 solutions Answer 75319 by ptaylor(2048)   on 2007-10-11 07:15:40 (Show Source): You can put this solution on YOUR website!Let x=number of liters of the 22% sugar solution needed Then 680-x=number of liters of the 31% solution needed Now we know that the pure sugar in the 22% solution (0.22x) plus the pure sugar in the 31% solution (0.31(680-x)) has to equal the pure sugar in the final mixture(0.27*680). So our equation to solve is: 0.22x+0.31(680-x)=0.27*680 get rid of parens 0.22x+210.8-0.31x=183.6 subtract 210.8 from both sides 0.22x+210.8-210.8-0.31x=183.6-210.8 collect like terms -0.09x=-27.2 divide both sides by -0.09 x=302.222222222---- liters of 22% solution 680-x=680-302.222222222222222222---=377.77777777777777----liters of 31% solution CK 302.2222*0.0.22+377.7777*0.31=680*0.27 66.4888888888888+117.11111111111111=183.6 183.599999999----~183.6 Hope this helps----ptaylor
 Miscellaneous_Word_Problems/102896: A small pump takes 14 hrs to empty the storage tank. A medium pump would take 8 hrs. A large pump would take 6 hours. How long would it take the small and medium together? 1 solutions Answer 74846 by ptaylor(2048)   on 2007-10-08 04:43:02 (Show Source): You can put this solution on YOUR website! Let x=length of time it takes the small and medium pump working together The small pump empties the tank at the rate of 1/14 tank per hour The medium pump empties the tank at the rate of 1/8 tank per hour Together they empty the tank at the rate of (1/14+1/8 or 4/56+7/56=11/56) tank per hour. So, the question is, if both tanks working together can empty 11/56 tank per hour, how long will it take them to empty 56/56 or all of the tank? (11/56)x=56/56 multiply each side by 56 11x=56 divide both sides by 11 x=5.090909--- hours Hope this helps----ptaylor
 Miscellaneous_Word_Problems/102385: how much pure acid should be mixed with 7 gallons of a 40% acid solution in order to get a 70% acid solution?1 solutions Answer 74533 by ptaylor(2048)   on 2007-10-05 19:40:36 (Show Source): You can put this solution on YOUR website!Let x=amount of pure acid needed Now we know that the amount of pure acid in the acid that's added (x) plus the amount of pure acid in the 40% solution ((7)*(0.40)) equals the amount of pure acid in the final mixture ((0.70)(7+x)). So our equation to solve is: x+7*0.40=0.70(7+x) get rid of parens x+2.8=4.9+0.70x subtract 0.70x and also 2.8 from both sides x+2.8-2.8-0.70x=4.9-2.8+0.70x-0.70x collect like terms 0.30x= 2.1 divide both sides by 0.30 x=7 gallons---------------------------amount of pure acid needed CK 7+7*0.40=0.70(14) 7+2.8=9.8 9.8=9.8 Hope this helps---ptaylor
 Human-and-algebraic-language/102176: This question is from textbook Algebra The sum of two consecutive even integers is -298. What are the integers?1 solutions Answer 74324 by ptaylor(2048)   on 2007-10-04 05:01:20 (Show Source): You can put this solution on YOUR website!Let x= one integer Then x+2=the other Now we are told that x+x+2=-298 subtract 2 from both sides x+x+2-2=-298-2 collect like terms 2x=-300 divide both sides by 2 x=-150--------------one integer x+2=-150+2=-148-----the other CK (-150)+(-148)=-298 -298=-298 Hope this helps---ptaylor
 Human-and-algebraic-language/102172: This question is from textbook Algebra The sum of two consecutive even integers is 118.1 solutions Answer 74323 by ptaylor(2048)   on 2007-10-04 04:56:00 (Show Source): You can put this solution on YOUR website!Let x= one integer Then x+2= the other Now we are told that x+x+2=118 subtract 2 from both sides x+x+2-2=118-2 collect like terms 2x=116 divide both sides by 2 x=58------------------------one integer x+2=58+2=60------------------------the other CK 58+60=118 118=118 Hope this helps---ptaylor
 Percentage-and-ratio-word-problems/102266: two investments were made totaling \$15,000. For a certain year these investments yielded \$1432 in simple intrest. part of the \$15,000 was invested at 9% and part at 10%. find the amount invested at each rate1 solutions Answer 74322 by ptaylor(2048)   on 2007-10-04 04:51:07 (Show Source): You can put this solution on YOUR website! Interest(I)=Principal(P) times Rate(R) times Time(T) or I=PRT Let P-amount invested at 9% (1) Interest accrued at 9%=P*0.09*1 or 0.09P Then 15000-P=amount invested at 10% (2) Interest accrued at 10%=(15000-P)*0.10*1 or 0.10(15000-P) Now we are told that (1)+(2)=1432 or 0.09P+0.10(15000-P)=1432 get rid of parens 0.09P+1500-0.10P=1432 subtract 1500 from both sides 0.09P+1500-1500-0.10P=1432-1500 collect like terms -0.01P=-68 divide both sides by -0.01 P=\$6800---------------------amount invested at 9% 15000-P=15000-6800=\$8200----------------amount invested at 10% CK 6800*0.09+8200*0.10=1432 \$612+\$820=\$1432 \$1432=\$1432 Hope this helps----ptaylor
 percentage/102271: A broker has invested \$17,500 in two mutual funds, one earning 10% annual interest, and the other earning 12%. After 1 year, his combined interest is \$1,941. how much was invested at each rate? Fill up the given table, set up and equation and solve. Mutual fund 1 Mutual fund 2 1 solutions Answer 74321 by ptaylor(2048)   on 2007-10-04 04:38:01 (Show Source): You can put this solution on YOUR website!Interest(I)=Principal(P) times Rate(R) times Time(T) or I=PRT Let P=amount invested at 12% (1) Interest earned at 12%=P*0.12*1=0.12P Then (17,500-P)=amount invested at 10% (2) Interest earned at 10%=(\$17,500-P)*0.10*1=0.10(\$17,500-P) Now we are told that (1)+(2)=\$1,941, so our equation to solve is: 0.12P+0.10(\$17,500-P)=\$1,941 get rid of parens 0.12P+\$1,750-0.10P=\$1,941 subtract \$1,750 from both sides 0.12P+\$1,750-\$1,750-0.10P=\$1,941-\$1,750 collect like terms 0.02P=\$191 divide both sides by 0.02 P=\$9,550-------------------------------amount invested at 12% (\$17,500-P)=\$17,500-\$9,550)=\$7,950----------amount invested at 10% CK \$9550*0.12+\$7950*0.10=\$1941 \$1146+\$795=\$1941 \$1941=\$1941 Hope this helps----ptaylor
 Travel_Word_Problems/102118: A driver sets out on a journey. For the first half of the distance, she travels at 30 miles per hour. For the second half of the distance, she travels at 60 miles per hour. What is her average speed on this trip? I know that speed=distance/time so, 30=distance/t1 and 60=distance/t2 but i'm not sure where to go from there. 1 solutions Answer 74320 by ptaylor(2048)   on 2007-10-04 04:18:41 (Show Source): You can put this solution on YOUR website!YOU ARE ON THE RIGHT TRACK!!!! SOLVE YOUR EQUATIONS FOR t1 AND t2; ADD THEM TO GET TOTAL TIME AND OUR CALCULATIONS BEGIN TO AGREE. distance(d) equals rate(r) times time(t) or d=rt;r=d/t and t=d/r Let S=average speed on this trip S=(Total distance travelled)/(Total time required) (Note: Here's another problem that sort of underscores what average speed is all about: "You are going on a 60-mile trip. The first 30 miles, you travel at the rate of 30 mph. How fast must you travel the remaining 30 miles in order to average 60 mph???" ANSWER: Impossible!!! Why??? In order to average 60 mph, you must travel the entire 60 miles in an hour. Unfortunately, you have already used up that hour in the first 30 miles.) Now back to your problem: Let d=total distance travelled Let d1=distance travelled for the first half of the trip And let d2=distance travelled for the second half of the trip *****Now we know that d1=d2. So total distance (d)=d1+d2=2d1 (or 2d2) Let t=total time required Let t1=time to travel the first half and this equals d1/30 Let t2=time to travel the second half and this equals d1/60 *****So Total time (t)=t1+t2=(d1/30+d1/60)=(2d1/60+d1/60)=(3d1/60)=d1/20 So S=2d1/(d1/20)---Now we multiply numerator and denominator by (20/d1) to get rid of the complex fraction and we get: S=(2d1(20/d1))/(d1/20*20/d1)=(40d1/d1)/1 or 40 mph (the d1's cancel) S=40 mph-------------------------------ans Hope this helps---ptaylor
 Percentage-and-ratio-word-problems/101946: apples are collected in a basket for six people. One-third, one-forth, one-eighth and one-fifth are given to four people, respectively. The fifth person gets ten apples with one apple remaining for the sixth person. Find the original number of apples in the basket.1 solutions Answer 74170 by ptaylor(2048)   on 2007-10-03 08:56:50 (Show Source): You can put this solution on YOUR website!Let x=original number of apples in the basket 1st person gets (1/3)x 2nd person gets(1/4)x 3rd person gets(1/8)x 4th person gets (1/5)x 5th person gets ---10 apples 6th person gets-------1 apple After the first four persons get their apples, we'll find out what fraction of the apples are left for the 5th and 6th persons and we are told that whatever that fraction is, will equal 11: (1/3)x+(1/4)x+(1/8)x+(1/5)x= Multiply each term by 120/120, the LCM: (40/120)x+(30/120)x+(15/120)x+(24/120)x=(109/120)x So, the apples that are left for the 5th & 6th persons equals: (120/120)x-(109/120)x=(11/120)x ---and this equals 11. (11/120)x=11 multiply both sides by 120 11x=1320 divide both sides by 11 x=120--------------------------number of apples CK (40/120)*120+(30/120)*120+(15/120)*120+(24/120)*120+11=120 or all the apples 40+30+15+24+11=120 109+11=120 120=120 Another way to work this problem is as follows: (1/3)x+(1/4)x+(1/8)x+(1/5)x+11=x (109/120)x-x=-11 -(11/120)x=-11 -11x=-1320 x=120 Hope this helps---ptaylor
 Expressions-with-variables/101829: For what integral values of k can 4x^2+kx+3 be factored over the integers?1 solutions Answer 74032 by ptaylor(2048)   on 2007-10-02 14:49:54 (Show Source): You can put this solution on YOUR website!The possible factors of this quadratic are as follows: (2x+1)(2x+3) (4x+1)(x+3) (4x+3)(x+1) We will use the FOIL crutch (First, Inner, Outer, Last) to expand the above: (2x+1)(2x+3)----4x^2+2x+6x+3=4x^2+8x+3;k=8 (4x+1)(x+3)-----4x^2+x+12x+3=4x^2+13x+3; k=13 (4x+3)(x+1)-----4x^2+3x+4x+3=4x^2+7x+3; k=7 So the answer is: k=7, 8 or 13 There may be a more scientific approach to solving this problem, but if there is I'm not aware of it. Generally, when the A coefficient of a quadratic is >1, some level of trial and error is needed to get at the factors. There are some tricks, however, that simplifies the trial and error process. In this quadratic, for example, since all signs are positive then we know there cannot be any negative signs in the factors. Also, if the C term is positive and the B term is negative,then we know, right off, that both signs in each of the two factors must be negative. If both the B term and the C term are negative, then we know that one factor is positive and one is negative. Information such as this can frequently help to reduce the number of possibilities. Hope this helps----ptaylor
 Mixture_Word_Problems/101750: This question is from textbook algebra 1 the nut house sells walnuts for \$4.00 a pound and cashews for \$7.00 a pound. how many pounds of CASHEWS should be mixed with 10 pounds of walnuts to obtain a mixture that sells for \$5.50 a pound? how do i figure out how many pounds of cashews should be mixed with the 10 pounds of walnuts?1 solutions Answer 73990 by ptaylor(2048)   on 2007-10-02 05:33:59 (Show Source): You can put this solution on YOUR website!Let x=number of pounds of cashews needed Value of the cashews=\$7x Value of the walnuts=\$4*10 Value of the final mixture \$5.50*(10+x) Now we know that the value of the nuts before the mixture is equal to the value of the nuts after the mixture. So we have: (\$ are understood) 7x+4*10=5.50*(10+x) get rid of parens 7x+40=55+5.5x subtract 40 and also 5.5x from both sides 7x+40-40-5.5x=55-40+5.5x-5.5x collect like terms 1.5x=15 x=10---------------------------------number of pounds of cashews needed CK 7*10+4*10=5.5*20 70+40=110 110=110 Hope this helps----ptaylor
 Travel_Word_Problems/101608: One car travels 50 miles per hour and another one travels 55 miles per hour. If they start from the same place at the same time and travel in the same direction, after how many hours will the faster car be 35 miles ahed of the slower car? 1 solutions Answer 73948 by ptaylor(2048)   on 2007-10-01 19:41:07 (Show Source): You can put this solution on YOUR website!distance(d)=rate(r) times time(t) or d=rt; r=d/t and t=d/r Let t=number of hours that it takes the faster car to be 35 mi ahead of the slower car Distance faster car travels=(rate of faster car)*t=55t Distance slower car travels=(rate of slower car)*t=50t Now the difference between these distances needs to equal 35 mi, so: 55t-50t=35 5t=35 divide both sides by 5 t=7 hours-------------ans CK in 7 hrs faster car travels 7*55 mi=385 mi in 7 hrs slower car travels 7*50=350 mi 385-350=35 35=35 Hope this helps--ptaylor
 Numbers_Word_Problems/101637: This question is from textbook Pre-Algebra workbook Please help us solve this equation: 4. One more than the difference between 18 and 7 times a number is -9 This is what we have tried: 7n + 18 +1 = -9 7n + 19 = -9 7n = -28 n= -4 Is this correct? 1 solutions Answer 73945 by ptaylor(2048)   on 2007-10-01 19:29:09 (Show Source): You can put this solution on YOUR website!4. One more than the difference between 18 and 7 times a number is -9 This is what we have tried: 7n + 18 +1 = -9 7n + 19 = -9 7n = -28 n= -4 Is this correct? YOU ARE VERY CLOSE!!!!!! WORD PROBLEMS ARE GENERALLY FAR EASIER IF YOU LAY OUT THE PROBLEM, STEP BY STEP, AS I HAVE DONE BELOW. Let n=the number 7 times the number is 7n Difference between 18 and 7 times a number: 18-7n One more than the difference between 18 and 7 times a number: (18-7n)+1 --------and we are told that this equals -9. So: 18-7n+1=-9 simplify 19-7n=-9 subtract 19 from both sides 19-19-7n=-9-19 collect like terms -7x=-28 divide both sides by -7 n=4------------the number CK 18-(7*4)+1=-9 18-28+1=-9 -10+1=-9 -9=-9 Hope this helps---ptaylor