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# Recent problems solved by 'Theo'

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 Numbers_Word_Problems/490234: please help me solve: ten times the sum of 30 and 15 is the same as the sum of a number and 12, what is the number? Have tried: 10 x 30 = 300 + 15 = 315, 315 - 12 = 303 the number is 303 have tried: 10 x (30 + 15)= 450, 450 - 12 = 438 the number is 4381 solutions Answer 333948 by Theo(3464)   on 2011-09-02 10:09:16 (Show Source): You can put this solution on YOUR website!the problem is: ten times the sum of 30 and 15 is the same as the sum of a number and 12, what is the number? let x be equal to the number. 10 times the sum of (30 + 15) is equal to 10 * 45 which is equal to 450. this is the same as the sum of a number and 12. sum of a number and 12 is equal to x + 12 because we just assigned x to be equal to the number. your equation becomes 450 = x + 12 subtract 12 from both sides of this equation to get: 450 - 12 = x which becomes: 438 = x your number should be 438 if we did this right. to confirm, assume your number is 438 and go back to the original problem and and see if it holds true. the original problem statement is: ten times the sum of 30 and 15 is the same as the sum of a number and 12, what is the number? ten times the sum of 30 and 15 is still equal to 450. the sum of 438 and 12 is equal to 450. they are both equal to 450 so the problem statement holds true when the number is assumed to be 438. looks like you got it right on the second try. ten times the sum of 30 and 15 means you take the sum of 30 and 15 first and then multiply that by ten. the key is the statement "the sum of" that implies the sum has to be done before the multiplication.
 Equations/490212: How fast is 360,000 cm/hr in m/s? Use conversion factors to cancel out one unit and introduce a new unit.1 solutions Answer 333947 by Theo(3464)   on 2011-09-02 10:03:41 (Show Source): You can put this solution on YOUR website!i presume that cm equals centimeters and m equals meters. you need to know the conversion factors first of all. 1 meter is equal to 10 centimeters this means that 1 centimeters is equal to 1/10 = .1 meter. 1 hour is equal to 60 minutes 1 minute is equal to 60 seconds 1 hour is therefore equal to 60 * 60 = 3600 seconds. this means that 1 second is equal to 1/3600 = .0002777778 of an hour. you start with 360,000 centimeters per hour. since 1 meters is equal to 10 centimeters, then you need to divide centimeters by 10 in order to get equivalent meters. 360,000 centimeters / 10 = 36,000 meters. your 360,000 centimeters is equivalent to 36,000 meters. since 1 hour is equal to 3600 seconds, you need to multiply 1 hour by 3600 to get 3600 seconds. your 1 hour has been 3600 seconds. 360,000 centimeters per hour becomes 36,000 meters per 3600 seconds. to find the number of meters per second you need to divide 36,000 by 3600 to get 10 meters per second. you 360,000 centimeters per hour becomes 36,000 meters per 3600 seconds which becomes 10 meters per second. that should be your answer. 360,000 centimeters per hour is equivalent to 10 meters per second. to see if this is good, try to get back to centimeters per hour from this equation. since 1 meter is equal to 10 centimeters, multiply 10 by 10 to get 100 centimeters per second. 10 meters per second becomes equivalent to 100 centimeters per second. since there are 3600 seconds in an hour, multiply 100 centimeters per second by 3600 to get 360,000 centimeters per hour. the conversion have been done successfully. 360,000 centimeters per hour is equivalent to 10 meters per second.
 Geometry_proofs/489628: given ac is = to df bc is = to ef prove ab is = to de1 solutions Answer 333711 by Theo(3464)   on 2011-09-01 08:37:36 (Show Source): You can put this solution on YOUR website!presumably these are all part of the same line. you have the line of af equal to ab + bc + cd + de + ef the line would look like this: ``` --------------------------------------------------- a b c d e f ``` you are given that the line segment ac is equal to df. you are also given that the line segment bc is equal to ef that would look like this: ``` --------------------------------------------------------- a b c d e f --------------------- --------------------- a c equals d f ----------- ----------- b c equals e f ``` you are given that ac = df you know that ac = ab + bc you know that df = de + ef this means that ab + bc = de + ef by substitution of equals. you are given that bc = ef this means that you can substitute bc for ef or you can substitute ef for bc in any equation. the equation of interest is: ab + bc = de + ef substitute bc for ef to get: ab + bc = de + bc subtract bc from both sides of this equation to get: ab = de you can do this because of the basic algebraic property that says: if a + c = b + c then a = b this follows from the fact that if you subtract the same quantity from both sides of an equality, then the equality is preserved. you are left with ab = de which is what you wanted to prove so you are done.
 Proportions/489031: I need help on solving a proportion problem. Could you help me by showing me step by step. 3r-2 all over r = 7 over 81 solutions Answer 333521 by Theo(3464)   on 2011-08-31 13:47:31 (Show Source): You can put this solution on YOUR website!you have: (3r-2) / r = 7/8 multiply both sides of this equation by r to get: (3r-2) = r * (7/8) multiply both sides of this equation by 8 to get: 8 * (3r - 2) = r * 7 simplify this equation to get: 24r - 16 = 7r add 16 to both sides of this equation and subtract 7r from both sides of this equation to get: 24r - 7r = 16 simplify to get: 17r = 16 divide both sides of this equation by 17 to get: r = 16/17 substitute 16/17 for r in your original equation to see if the equation holds true. your original equation is: (3r-2) / r = 7/8 substitute for r to get: (3 * (16/17) - 2) / (16/17) = 7/8 multiply both sides of this equation by (16/17) to get: (3 * (16/17) - 2) = (7/8) * (16/17) simplify this equation to get: (48/17 - 2) = (112/136) simplify further to get: (48/17 - 34/17) = (112/136) simplify further to get: (14/17) = (112/136) 112/136 is equivalent to 14/17 so you get: (14/17) = (14/17) which is true, confirming the value of (16/17) for r is good.
 Linear-equations/489034: 9x + 3 = 9x + b For what value(s) of b does the equation have no solution? the answer i thought was 3 but than i seen that if i put 3 it would be infienetly solution than i thought it was two but that was incorrect also 1 solutions Answer 333516 by Theo(3464)   on 2011-08-31 13:34:00 (Show Source): You can put this solution on YOUR website!your equation is: 9x + 3 = 9x + b subtract 9x from both sides of this equation to get: b = 3 in order for this equation to have a solution, the value of b has to be equal to 3. any other value of b will result in no solution. if you set b equal to 3, the equation reduces to 3 = 3 which is true, confirming the value of 3 for b is a solution. if you set b to any other value, like 2, the equation reduces to 3 = 2 which is false, confirming the value of 2 for b is not a solution. try any value for b other than 3 and you will see that the equation does not have a solution. the fact that you have 9x + 3 = 9x + b should be your clue that b has to be equal to 3. the 9x terms will always be equal regardless of the value of x. that means the other term in each expression has to also be equal. that can only happen when b = 3.
 Polynomials-and-rational-expressions/489023: Help me please? I don't get this at all.... Essay: Show all work. A school designer wants to create a whiteboard with the optimal dimensions to enhance learning. It is determined that if one side of the whiteboard is x + y inches, the other side should be x^2-3xy+7y^2 inches. Write an algebraic expression for the area of such a whiteboard, simplify it, and include correct units with your solution.1 solutions Answer 333512 by Theo(3464)   on 2011-08-31 13:28:43 (Show Source): You can put this solution on YOUR website!one side of the white board is x + y inches. the other side of the white board is x^2 - 3xy + 7y^2 inches. the area of the whiteboard is equal to length * width. let W equal the width and set W equal to x + y inches. let L equal the length and set L equal to x^2 - 3xy + 7y^2 inches. let A equal the area which is equal to L * W. you get: A = L * W = (x + y) * (x^2 - 3xy + 7y^2) square inches. you need to multiply these factors together. (x + y) * (x^2 - 3xy + 7y^2) will be equal to: (x * (x^2 - 3xy + 7y^2)) + (y * (x^2 - 3xy + 7y^2)) which becomes equal to: (x^3 - 3x^2y + 7xy^2) + (yx^2 - 3xy^2 + 7y^3) remove the parentheses and reorder the variables in each term so that the x variable is first and the y variable is second so you can see the like terms easier. you get: x^3 - 3x^2y + 7xy^2 + x^2y - 3xy^2 + 7y^3 now reorder the terms so that the like terms are next to each other. you get: x^3 [- 3x^2y + x^2y] [+ 7xy^2 - 3xy^2] + 7y^3 i enclosed the like terms in brackets so you can see them better. you would not ordinarily do that. i was careful to include the operation signs so you can see whether they needed to be added together or subtracted from each other. now combine like terms to get: x^3 -2x^2y + 4xy^2 + 7y^3 that's your area. it is expressed in square inches. you would say: A = x^3 - 2x^2y + 4xy^2 + 7y^3 square inches. if you want to confirm that you did the calculations correctly, then the easiest way is to provide some random values for x and y and use them to solve the original equation and the final equation after simplification. the original equation is: A = (x + y) * (x^2 - 3xy + 7y^2) square inches. the final equation is: A = x^3 - 2x^2y + 4xy^2 + 7y^3 square inches. i did the check using x = 2 and y = 3. i got an answer of 245 both times so i'm reasonably confident in did the calculations and the simplifications correctly. the multiplication uses the distributive property of mathematics. (a + b) * (c + d) would be equal to: a * (c + d) + b * (c + d) a + b) * (c + d + e) would be equal to: a * (c + d + e) + b * (c + d + e)
 Probability-and-statistics/488895: Can you please show steps how to calculate the profile score: Student received 9/20 in maths, 8/20 in english, 16/30 in General Ability Test. The school assessment marks were given as 80/100 - English, 87/100 - Maths. The calculated profile score used a mean of 60 and standard deviation of 12. The result was 199.99 as the profile score for the student. Please show steps how this was calculated to get 199.99? Thanks.1 solutions Answer 333474 by Theo(3464)   on 2011-08-31 05:57:43 (Show Source): You can put this solution on YOUR website!i was not able to come up with an example on how the score is actually calculated, but i was able to find a link to a place that might be able to help you. http://www.schools.nsw.edu.au/learning/k-6assessments/shsplacement/ss_marksexplain.php this assessment includes scores for: reading writing math general ability i think you only had 3 mentions (english, math, general ability). never the less, i think i'm in the right place. i sent them an email requesting an example of how the calculation would be performed. you should probably contact them as well for information if you can't find it any other way. send me an email through algebra.com so if i do get some information from them, i can relay it back to you.
 Money_Word_Problems/488896: what is the interest if the Principal is \$720, the Rate is 5 1/4% and the Time is 3 years?1 solutions Answer 333472 by Theo(3464)   on 2011-08-31 04:18:12 (Show Source): You can put this solution on YOUR website!if the interest is compounded monthly (normal bank practice), then \$720.00 would be equal to \$847.2093772in 3 years. if the interest is compounded annually, then \$720.00 would be equal to \$844.1213401 in 3 years. if the interest was compounded daily, then \$720.00 would be equal to \$847.4908701 in 3 years. if the interest was compounded continuously, then \$720.00 would be equal to \$847.50 continuous compounding is given by the formula: F = P * e^(i*y) annual, monthly, daily compounding is given by the formula: F = P * (1+(i/c)^(y*c) P is the present amount F is the future amount i is the annual interest rate y is the number of year c is the compounding period. e is the scientific constant of 2.718281828 example: for monthly compounding, the variables become: P = \$725.00 F = F i = .0525 y = 3 c = 12 the formula of F = P * (1+(i/c))^(y*c) becomes: F = \$724.00 * (1 + (.0525/12))^(*3*12) which becomes: F = \$724.00 * (1.004375)^36 which becomes: F = \$847.2093772
 Rectangles/488537: Suppose the length and width of the rectangle are doubled. what effect would this have on the area? Justify your answer. rectangle dimensions = 17ft L 4 ft W1 solutions Answer 333400 by Theo(3464)   on 2011-08-30 18:27:51 (Show Source): You can put this solution on YOUR website!A = L * W this means area equals length times width. if you double the length and you double the width, the formula becomes: A = 2 * L * 2 * W which is the same as: A = 2 * 2 * L * W which is the same as: A = 4 * L * W if you double the length and the width, you are quadrupling the area. this is because 2 * 2 = 4. an example: L = 2 and W = 4 Area = 2 * 4 = 8 double the length and double the width to get: L = 4 and W = 8 Area = 4 * 8 = 32 32 is 4 times as large as 8. you doubled the length and the width and you quadrupled the area. this is because 2 * 2 = 4 if you tripled the length and doubled the width, then the area would be 6 times as large because 2 * 3 = 6 an example: L = 2 and W = 4 Area = 2 * 4 = 8 double the length and triple the width to get: L = 4 and W = 12 Area = 4 * 12 = 48 48 is 6 times as big as 8. what you did was: L * W = Area you doubled the length and tripled the width to get: 2 * L * 3 * W = Area this becomes: 2 * 3 * L * W = Area which becomes: 6 * L * W = Area the area becomes 6 times as large because 2 * 3 = 6.
 Length-and-distance/488297: Hi, I am trying to help my daughter with her Algebra and forgot how to solve for the following (y=mx+b) statement. I hope you can help as I have been out of this since my late twenties! Write the slope-intercept form of an equation of the line that passes through the given point and is perpendicular to the equation given. (2,2), y= -1/5 x +5 1 solutions Answer 333338 by Theo(3464)   on 2011-08-30 11:47:12 (Show Source): You can put this solution on YOUR website!the slope intercept form of the equation of a straight line is y = mx + b m is the slope b is the y intercept the point you are given is: (x,y) = (2,2) the equation you are given is: y = (-1/5)x + 5 the slope is (-1/5) the y intercept is 5 the line perpendicular to this line will have a slope that is a negative reciprocal of the slope of this line. if the slope is a/b, then the negative reciprocal of the slope is -b/a since our slope is (-1/5), then the negative reciprocal is (5/1) which is equal to 5. the slope of our line is 5 and it passes through the point (2,2) the equation of our line will be y = mx + b we replace the m with 5 and we get: y = 5x + b to find b, we replace y with 2 and x with 2 and solve for b. this is because the point (x,y) = (2,2) which means that x = 2 and y = 2 at that point. we get: 2 = 5*2 + b which becomes: 2 = 10 + b subtract 10 from both sides of this equation to get: -8 = b which becomes: b = -8 our equation becomes: y = 5x - 8 a graph of both equations will show that our new equation is perpendicular to the original equation and will pass through the point (2,2). i have created a horizontal line at y = 2 and a vertical line at x = 2 in order to show you that our new line passes through that point. for your additional information, the 2 lines will intersect at the point where the two equations are equal to each other. that point will be solved for below. the 2 equations are: y = (-1/5)x + 5 y = 5x - 8 we set both these equations equal to each other to get: (-1/5)x + 5 = 5x - 8 add 8 to both sides of this equation to get: (-1/5)x + 13 = 5x add (1/5)x to both sides of this equation to get: 13 = 5x + (1/5)x which becomes: 13 = 26/5x that makes x = (5*13)/26 = 5/2) this is the same as x = 2.5 we'll solve for y in both equations. the answers should be the same. y = -(1/5)x + 5 becomes y = -(1/5)*2.5 + 5 which becomes y = 4.5 y = 5x - 8 becomes y = 5*2.5 - 8 which becomes y = 4.5 we have our point of intersection of the 2 lines at x = 2.5 and y = 4.5 this can be expressed as (x,y) = 2.5,4.5) the graph is shown below with the vertical and horizontal lines removed so you can see the intersection point easier. i also made the x and y axis symmetrical so you can see that the lines are really perpendicular to each other easier.
 Travel_Word_Problems/488292: Please help me solve this word problem, I do not understand what to do. It takes John 30 minutes to clean his house, When his wife helps him, it only takes him 18 minutes. How long would it take his wife by herself?1 solutions Answer 333336 by Theo(3464)   on 2011-08-30 11:23:49 (Show Source): You can put this solution on YOUR website!rate * time = units of work the unit of work is equal to 1 which is the house. john takes 30 minutes to clean the house. if time is in minutes, then the equation for john is: rate * 30 = 1 divide both sides of this equation by 30 and you get: rate = 1/30 this means that john can clean 1/30 of the house in 1 minute. when john's wife helps him, they both can clean the house in 18 minutes. rate * time = units of work. the unit of work is still the house so it is equal to 1 the rate of work is the combined rate of john and his wife. the time is takes is now 18 minutes. the formula of rate * time becomes: (john's rate + john's wife's rate) * 18 = 1 we know john's rate because we just solved for it. we'll make john's wife's rate = x our formula becomes: (1/30 + x) * 18 = 1 we multiply this out to get: 18/30 + 18x = 1 we subtract 18/30 from both sides of this equation to get: 18x = 1 - 18/30 we simplify this to get: 18x = 12/30 we divide both sides of this equation by 18 to get: x = (12/30) / 18 = 1/45 this means that john's wife's rate is equal to 1/45. john's rate is 1/30 of the house in one minute. john's wife's rate is 1/45 of the house in one minute. i believe we will see that it will take john's wife 45 minutes to clean the hours by herself. the formula is rate * time = units of work. the units of work is still equal to 1. the wife's rate is equal to 1/45. the formula becomes: 1/45 * time = 1 multiply both sides of this equation by 45 to get: time = 45 minutes. when they work together, their rates are additive. the formula becomes: (1/30 + 1/45) * time = 1 this simplifies to: (3/90 + 2/90) * time = 1 which simplifies further to: 5/90 * time = 1 which simplifies further to: 1/18 * time = 1 multiply both sides of this equation by 18 to get: time = 18 the number work out ok so we have the answer. the answer is: john's wife takes 45 minutes to clean the house by herself.
 Equations/488286: Need help please. The area of a rectangular swimming pool is given by 3x^2+11x+6 ft^2. One side length of the pool is given by 3x + 2 feet. What is an algebraic expression for the other side length of the pool? Simplify this, and include correct units as part of your answer.1 solutions Answer 333335 by Theo(3464)   on 2011-08-30 11:10:58 (Show Source): You can put this solution on YOUR website!area of rectangular swimming pool is equal to 3x^2 + 11x + 6 square feet. one side of the pool is equal to 3x + 2 feet. the formula for the area of a rectangle is L * W = A L is the length W is the width. A is the area. you have A = 3x^2 + 11x + 6 you have L = 3x + 2 if A equals L * W, then: W = A / L this would be: (3x^2 + 11x + 6) / (3x + 2) we can do the division, or we can determine that this is a quadratic equation whose factors will be (3x + 2) * (x + something). since 6 is equal to 2 * 3, that something is probably 3. let's try the factors of (3x + 2) * (x + 3) we multiply those factors together to get: 3x^2 + 9x + 2x + 6 combine like terms to get 3x^2 + 11x + 6 since this is the quadratic equation that represents the area of the pool, then we have found the width of the pool. A = L * W = (3x+2) * (x+3) L = 3x + 2 feet W = x + 3 feet we were given the length of the pool as 3x + 2 feet. the other side of the pool is the width of the pool which is equal to x + 3 feet if you multiply L * W you will get the area of the pool which is equal to 3x^2 + 11x + 6 square feet
 Quadratic_Equations/488256: I cannot figure out the answer to this problem. ax^2+bx+c=0 the given solution is -squareroot5,4squareroot5 can you please help me? thank u1 solutions Answer 333333 by Theo(3464)   on 2011-08-30 10:57:59 (Show Source): You can put this solution on YOUR website!your given solution is or the equation is ax^2 + bx + c = 0 if the solutions to the equation are: then the factors of the equation are: if you multiply these factors together you will get the equation. the following picture shows you the mutiplication of the factors. your equation is:
 Linear-equations/488247: If we choose x from the set of values bigger than 0, is y = x² a function of x? explain If we choose y from the set of values bigger than 0, is y = x² a function of y? explain 1 solutions Answer 333328 by Theo(3464)   on 2011-08-30 10:11:06 (Show Source): You can put this solution on YOUR website!y = x^2 makes y a function of x. the independent variable is x and the dependent variable is y. x^2 = y makes x^2 a function of y. the independent variable is y and the dependent variable is x^2. if we let z = x^2, then: x^2 = y becomes z = y z is considered a function of y because it depends on the value of y. you would solve this like you would solve any equation. if y = 5, then z = 5 since z = x^2, this means that x^2 = 5. suppose you had: x^2 = y-3 x^2 is still a function of y because it depends on the value of y. if y = 5, x^2 = y - 3 = 5 - 3 = 2 you get x^2 = 2 when y = 5 i believe this to be true, but i won't swear to it. there may be some rule that says you can't do this, but i haven't seen it. in practice, you can set anything equal to anything else. the dependent variable is a function of the independent variable. the equation determines the value of the dependent variable based on the value of the independent variable. x^2 is a function of y meets that definition. the value of x^2 is dependent on the value of y. the right side of that equation doesn't have to be just y. it can be any expression involving y. x^2 = y^3 + 5y - 3 would be an example. x^2 is a function of y. when y = 3, x^3 = 27 + 15 - 3 which becomes: x^2 = 39
 Geometry_Word_Problems/488212: The perimeter of this block of land is 72 m. Find: a the length of the block b the area of the block.1 solutions Answer 333306 by Theo(3464)   on 2011-08-30 05:38:15 (Show Source): You can put this solution on YOUR website!you don't have enough information to determine the length of the block or the area of the block. the area of the block will differ depending on the length of the block and the width of the block. the length of the block requires you to know the width of the block before determining what it is. here's some examples: 72 is the perimeter. you are given that. the perimeter is determined by the equation 2L + 2W = perimeter. you have 2L + 2W = 72 if you let W = 8, then 2W = 16 and the formula becomes: 2L + 16 = 72 subtract 16 from both sides of this equation to get: 2L = 56 divide this equation by 2 to get: L = 28 your dimensions are: L = 28 W = 8 the area is L * W which equals 28 * 8 which equals 224 square meters. now, assume W = 16. this means that 2W = 32. this means that 2L + 2W = 72 becomes 2L + 32 = 72 subtract 32 from both sides of that equation to get: 2L = 40 divide both sides of that equation to get L = 20 your dimensions are: L = 20 W = 16 the area is now L * W = 20 * 16 = 320 square units. Given that the perimeter is 72 meters, the length is dependent on the width and the area is dependent on the length times the width. without knowing the width, you can have many possible lengths depending on the dimension of the width that you choose. here's a table that shows you the possible lengths and widths with a perimeter of 72. these are all integers in multiples of 4. if you assume real numbers that could include fractional parts of a whole number, then the list can be endless. ```Perimeter Length Width Area 72 4 68 272 72 8 64 512 72 12 60 720 72 16 56 896 72 20 52 1040 72 24 48 1152 72 28 44 1232 72 32 40 1280 72 36 36 1296 72 40 32 1280 72 44 28 1232 72 48 24 1152 72 52 20 1040 72 56 16 896 72 60 12 720 72 64 8 512 72 68 4 272 ```