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# Recent problems solved by 'bucky'

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 Numbers_Word_Problems/494823: i found a two digit number less than 50 the sum of its digit was 12 their difference was 4 what number did i find1 solutions Answer 335806 by bucky(2189)   on 2011-09-10 08:00:20 (Show Source): You can put this solution on YOUR website!Let x equal one of the digits. And let y equal the other digit. . You are told that the sum of the two digits is 12. So you can write an equation to express this as: . x + y = 12 . Next you are told that the difference of the two digits is 4. In equation form this is: . x - y = 4 . At this point you have a system of two independent equations that can be used to solve for the two unknowns. . x + y = 12 x - y = 4 . If you add the two equations vertically, note that the +y and the -y will cancel each other and you are left with: . 2x = 16 . Solve for x by dividing both sides by 2 to get x = 8 . If one of the digits is 8 and the sum of the two digits is 12, then by subtraction you know that the other digit must be 4. So the two digits are 8 and 4. . From your answers for the digits you know that the answer formed from the two digits of 8 and 4 must be either 48 or 84. But you were told that the two-digit number you are looking for is less than 50. So the answer to this problem must be 48. . Hope this helps you.
 real-numbers/494026: PLEASE HELP...IF YOU DOUBLE A NUMBER AND THEN ADD 12, YOU GET ONE HALF OF THE ORIGINAL NUMBER. WHAT IS THE ORIGINAL NUMBER?1 solutions Answer 335694 by bucky(2189)   on 2011-09-09 15:15:42 (Show Source): You can put this solution on YOUR website!Let x be the original unknown number. . Double that number means taking 2 times x which is written as 2x. . Then add 12 to that and you have 2x + 12 . The problem tells you that this should equal 1/2 of the original number, or (1/2)x . Setting these two quantities equal results in the equation: . 2x + 12 = (1/2)x . You can get rid of the fraction on the right side by doubling both sides of this equation ... Do that by multiplying both sides (all terms) by 2 and the equation becomes: . 4x + 24 = x . Subtract x from both sides: . 3x + 24 = 0 . Then subtract 24 from both sides: . 3x = -24 . Finally solve for x (the original number) by dividing both sides of this equation by 3 to get: . x = -24/3 = -8 . So the answer is that the original unknown number is -8. . Check it out. Double it to get -16 then add 12 to the -16 and the result is -4. This checks because it is supposed to equal half the original number which is one-half of -8. And it does. . This checks the answer. The original number is -8. . Hope this helps you to understand the problem. .
 Quadratic_Equations/493431: find all solutions of the equation... 27x^3-512=0 i know i have to get the cube or something like that or the GCF?1 solutions Answer 335510 by bucky(2189)   on 2011-09-08 17:39:29 (Show Source): You can put this solution on YOUR website!Given: . . First, move the -512 to the other side of the equation by adding +512 to both sides as follows: . . On the left side the -512 and the +512 total zero. Therefore, you are left with the following equation: . . Solve for by dividing both sides by 27 and you get: . . Now note that 512 is equal to 8 cubed. Also note that 27 is equal to 3 cubed. You can verify these two statements by using a calculator to first multiply 8 times 8 times 8 to get 512. Then use it to multiply 3 times 3 times 3 and you get 27. So by substituting 8 cubed for 512 and 3 cubed for 27, you can write the equation as: . . By the power rule of exponents you can furthermore write the right side of this equation as shown: . . Now solve for x by taking the cube root of both sides to get: . . which simplifies to: . . You can check this answer by returning to the equation that you were originally given and substitute for x. You should then see that: . . becomes (after cubing the ): . . and by cancelling the 27 in the numerator with the 27 in the denominator, this equation reduces to: . . Since the left side does equal the right side, the answer that is correct. . Hope this helps you to see how to do this problem.
 Numbers_Word_Problems/489517: find three consecutive odd integers such that the sum of the first integer, twice the second integer, and four times the third is 97. 1 solutions Answer 333691 by bucky(2189)   on 2011-09-01 01:27:48 (Show Source): You can put this solution on YOUR website!Each consecutive odd integer is separated by adding 2 to its preceding odd integer. For example: odd integers 3, 5, 7, and 9 are consecutive odd integers. Note that 5 is 3 + 2 and 7 is 5 + 2 and 9 is 7 + 2. . Therefore, if N is an odd integer, the next consecutive odd integer is N+2 and the next consecutive odd integer after that is (N+2) + 2 and this can be simplified to N + 2 + 2 which is N+4. So we can say that three consecutive odd numbers are N, N+2, and N+4. . The problem then describes three terms as follows: . The first term is the first odd integer or N. . The second term is twice the second odd integer or twice N+2. If you multiply N+2 by 2 the result is 2N+4. . The third term is 4 times the third odd integer. This would be 4 times N+4. If you multiply 4*(N+4) the result is 4N+ 16. . You are then to add these three terms and you are told that the answer is 97. When written in equation form this is: . N + (2N + 4) + (4N + 16) = 97 . Group the terms containing N and separately group the constants and the equation becomes: . (N + 2N + 4N) + (4 + 16) = 97 . Add the terms containing N and also add the constants and the equation simplifies to: . 7N + 20 = 97 . Get rid of the 20 on the left side of the equation by subtracting 20 from both sides to get: . 7N + 20 - 20 = 97 - 20 . and the subtractions simplify this equation to: . 7N = 77 . Solve for N by dividing both sides by 7 and the result is: . N = 11 . So now we know that N, the first odd integer, is 11. This means that the second consecutive odd integer is 2 greater than 11 or it is 13 and the third consecutive odd integer is 2 greater than 13 or it is 15. . So the three consecutive odd integers you were to find are 11, 13, and 15. . Check this by adding the first odd integer (11) to two times 13 or (26) to 4 times 15 or (60). . 11 + 26 + 60 . If you add these three numbers, the sum is 97 just as the problem said it should be. Therefore, the answer is correct. The three consecutive odd integers the problem was looking for are 11, 13, and 15. . Hope this helps you to understand the problem.
 Unit_Conversion_Word_Problems/488479: Good computation method to use to solve number 3 down (10 letter word) 3 Down A cheetah can run 60 miles an hour. If it could maintain that speed for half an hour, how many miles could it run? I think the answer is 30 miles, to this question, but don't know the 10 letter work for the computation method.1 solutions Answer 333497 by bucky(2189)   on 2011-08-31 10:40:05 (Show Source): You can put this solution on YOUR website!You are correct that the answer to the question about distance is 30 miles. . In solving that problem, you can use a PROPORTION as follows: . . You can solve this by cross multiplying and setting the products equal. . Cross multiplying means first to multiply the numerator on the left side times the denominator on the right side. So multiply 60 (the left side numerator) times 1/2 (the right side denominator) to get 30. . The second step in cross multiplying is to multiply the denominator on the left side times the numerator on the right side. So multiply 1 (the left side denominator) times x (the right side numerator). The product is 1x or just x for short. . Finally, set the two products equal and solve for the unknown. So in setting the two products equal you have: . . No other steps are needed because this is already the value for x. . Notice the PROPORTION we set up has miles as the units for both the left and right side numerators and hours as the units for both the left and right side denominators. That's a check. The numerators on both sides should be in the same units and the denominators on both sides should also have units that agree as well. The location of the unknown can appear in any of the 4 locations - left numerator, left denominator, right numerator, or right denominator. It all depends on what you are trying to solve for. In this case we were trying to find the miles for 1/2 hour. So the x had to be in the numerator because we set up miles to be the two numerators. Then the x also had to be on the right side because it was related to the 1/2 hour which is the right side denominator. . As far as the ten-letter word ... a clue is to look above for the word that is in all capital letters. (It appears twice.) I think that this is the 10-letter word you are looking for. . Hope this helps you to understand the problem and how to solve problems that involve the 10-letter word.
 Linear-equations/488099: How do you graph the following? y=-3/4x-1 x=-2 y=1.5 x=1 y=-1.75 x=2 y=-2.5 x=-3 y=1.25 The graph points don't see to line up. 1 solutions Answer 333434 by bucky(2189)   on 2011-08-30 20:57:16 (Show Source): You can put this solution on YOUR website!There are three different forms of graphs in this problem. . The first form is represented by the equation . This is called the slope intercept form and the standard general form of this equation is written as . In this general form, m which is the multiplier of the x is the slope of the graphed line. If m is positive, the graphed line goes upward as you move toward the right on the graph. But if m is negative, the graphed line goes downward as you move toward the right on the graph. Regardless of the sign, the amount of the slope is determined by saying "as I move horizontally to the right the number of units in the denominator of m, the graph goes up or down (depending on the sign) the number of units in the numerator of m. (If m is a whole number, the denominator is 1 and the numerator is the whole number.) The second thing to notice is the term . The value of + b is the point on the y-axis where the graph crosses the axis. . In the first problem, m (which is the multiplier of x) is . The minus sign tells us that the graph slants downward as you move toward the right. It also tells us that as you move horizontally 4 units from any point on the graph, at the place you reach the end of the 4 units you go down 3 units and at that location you have another point on the graph. Where is a point on this graph. Look at + (-1). The -1 is the b that is added to the x term in the equation you were given. b is the point on the graph where the graph crosses the y-axis. Therefore, this graph crosses the y-axis at -1 on the axis. From this point move 4 units (slope denominator) horizontally and then down 3 units vertically (slope numerator) and mark that location. This location is a second point on the graph. From that second point move horizontally 4 units and then vertically down 3 units and you have the third point on the graph. When you finish you should have a graph that looks like this: . . Notice that from the y-axis intercept at -1, if you move 4 units horizontally to the right you will be at the point (+4, -1). Then if you move vertically down 3 units you will be at the point (4, -4) and that is a second point on the graph. Then you can move 4 more units horizontally to the right from this second point and you will be at (8, -4). Then go vertically down 3 units and you will be at the point (8, -7) and that is a third point on the graph. . That pretty much is the first problem. All the other problems involve X = a constant and y = another constant. . x = a constant means that no matter what value y may take, x will always be the same. The graph of this will be a vertical line that crosses the x-axis at the constant. Similarly, for y = a constant, that means that no matter what value x is, the corresponding value of y is the constant. The graph of this will be a horizontal line that crosses the y-axis at the constant. . Second problem: x = -2 and y = 1.5. The two graphs are as shown: . . Notice that the vertical red line is the graph of x = -2 and the horizontal green line is the graph of y = +1.5. The common solution for these two graphs is the point where the red graph and the green graph cross at the point (-2, 1.5). . Third problem: x=1 y=-1.75. The two graphs are as shown: . . Notice that the red graph is the graph of x = +1 and the green graph is the graph of y = -1.75. . The pair of graphs has as a common solution the point (1, -1.75) . Fourth problem: x=2 y=-2.5. The two graphs are: . . Make sure that you understand why these graphs look as they do and what the common solution for these two graphs is. . Finally, the Fifth problem: x=-3 y=1.25 . . Study this graph also so that you thoroughly understand how to graph x = constant and y = constant. . Hope this helps you with understanding these problems.
 Length-and-distance/488296: 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 slope-intercept form of an equation of the line that passes through the given point and is parallel to the equation given. (5,-1), y= -3/4x+11 solutions Answer 333345 by bucky(2189)   on 2011-08-30 12:44:34 (Show Source): You can put this solution on YOUR website!The slope intercept form, as you can see, is: . . m, the multiplier of the x is the slope of the line that the slope intercept equation establishes. . In order for a line to be parallel to a given line, it must have the same slope as the given line. The line that you were given is: . . Since -3/4 is the multiplier of x, it is the slope of the graph for given line. Therefore, what you are being asked to do is to find the slope intercept line that has a slope of and passes through the point (5, -1). So let's begin with the general slope intercept form: . . Just for your info, a slope of can be interpreted as follows: The minus sign tells you that as the line moves to the right the graphed line drops downward. Had the sign been + the graphed line would go upward as you moved to the right. The fraction can be determined as follows. The denominator (4) tells you that for every 4 units the graph moves to the right, the drop or rise is the numerator (in this problem it is a drop of 3 units). If the multiplier of x is a whole number, for example 5, think of it as a numerator of 5 and a denominator of 1 (that is so that for every 1 unit moved horizontally to the right, the change upward is 5 (if +) or downward 5 (if minus)). . We've already determined that the graph of the line that we need has a slope of . So substitute this into the slope intercept form and you have: . . What else do you know? You know that the given point (5, -1) must satisfy this equation if it is to be on the line. So we can substitute x = 5 and y = -1 into the equation to get: . . Now all you have to do is a little math and equation solving to find what b (the point where the graph crosses the y-axis) needs to be. Multiply out the first term on the right side and you have: . . Add to both sides and this reduces the equation to: . . Convert -1 to -4/4 so that you can combine the two left hand terms: . . . which simplifies to: . . Since you now have both m and b, you can write the equation of this new line as: . . . and the graph of these two parallel lines looks like: . . The red line is the graph that you were given. (Note the negative slope (down and to the right) and that graph crosses the y-axis at y = +1 and indicated by the +1 value for b in the original equation.) The green line is the graph that we developed as given by the slope intercept form: . . Note here that the slope (multiplier of x) is again , but this time the value of b (the intercept on the y-axis) is . . A great big thanks for taking the time to work with your daughter. More parents should be like you !!! . Hope this helps you in some small way.
 logarithm/487642: In 2000, there were about 202 million vehicles and about 283 million people in a certain country. The number of vehicles has been growing at 4.5% a year, while the population has been growing at 1% a year. (a) Write a formula for the number of vehicles (in millions) as a function of t, the number of years since 2000. V(t) = (b) Write a formula for the number of people (in millions) as a function of t, the number of years since 2000. P(t) = (c) If the growth rates remain constant, when is there, on average, one vehicle per person? Give your answer in exact form and decimal form. Exact form: ______years since 2000 Decimal form (nearest tenth): _______years since 2000 Thank you so much and God bless you.1 solutions Answer 333340 by bucky(2189)   on 2011-08-30 11:54:46 (Show Source): You can put this solution on YOUR website!Let's see if we can't figure this problem out. . Since it's not very well defined by the problem, I'm going to assume that the number of vehicles for the year 2000 represents the number of vehicles on December 31st of that year. . First for the number of vehicles. Each year that goes by, the number of vehicles increases by 4.5% (that is by the decimal 0.045). So that on December 31st of the year 2001 the increase in the number of cars is 202M times 0.045. That means that at the end of the first year (the end of 2001) the number of cars is the 202M at the end of 2000 plus the increase of 202M times 0.045. In algebraic form this is: . 202M + (202M*0.045) . Factor out the 202M and this expression for the number of cars at the end of the first year (that is at the end of 2001) becomes: . 202M*(1 + 0.045) = 202M*1.045 . Next what happens during the second year? You start the second year with the number of vehicles at the end of the year 2001. We just determined it to be: . 202M*(1.045) . So the increase by the end of the second year (the end of 2002) will be 0.045 times 202M*(1.045) and at the end of the second year the total number of cars will be what you had at the end of 2001 plus the increase of 0.045 times the number number of cars at the end of 2001. In algebraic form this is: . 202M*(1.045) + (202M*(1.045))*(0.045) . Factor out (202M*1.045) and you have: . (202M*1.045)*(1 + 0.045) = (202M*1.045)*(1.045) = 202M*(1.045)^2 . If you try this analysis for another year or two, it will become apparent that for each passing year, the number of cars increases by a factor of 1.045. This means that the number of cars at the end of a given year (call it V(t)) can be determined from the equation: . V(t) = 202M*(1.045)^(t) . Where t is the number of years after the year 2000. So for example in the year 2003, t would be 2003 minus 2000 or t would equal 3. . So on December 31 of the year 2005 the number of cars would be: . V(t) = 202M*(1.045)^t = 202M*(1.045)^(2005 - 2000) = 202M*(1.045)^5 . You can use a calculator to determine that 1.045 raised to the exponent 5 is 1.246181938 and when you multiply this by 202M the answer becomes 251.728751M. At the end of the year 2005 the number of cars will be 251,728,751 . The same type of analysis can be done for the population. (Again assume that the population each year is for the last day of that year.) The difference is that each year the increase is 1% or 0.01. And the population in the year 2000 is 258M on December 31st 2000. So by introducing these changes into the equation for V(t) we can say that the equation for the population at the end of a given year is: . P(t) = 258M*(1.01^t) . where t is again defined as the year of interest minus 2000. If you want to find the population on December 31st of 2004, t would be 2004 - 2000 or t would be 4. . Now to find the year when the number of vehicles equals the population so that there is one vehicle per person on average. This is done by setting the right side of the equation for V(t) equal to the right side of the equation for P(t) and then solving for t. In other words: . 202M*(1.045^t) = 258M*(1.01^t) . Our goal is to get terms containing t on the left side of the equation, and all other terms on the right side. Begin by dividing both sides of this equation by 202M and you get: . 1.045^t = (258M/202M)*(1.01^t) . Divide the 258M by 202M and you get 1.277227723. Substitute this into the equation and it becomes: . 1.045^t = (1.277227723)*(1.01^t) . Divide both sides by 1.01^t and the equation becomes: . (1.045^t)/(1.01^t) = 1.277227723 . Note on the left side that the exponent in the numerator is the same as the exponent in the denominator. Therefore, by the rules of exponents we can say: . (1.045/1.01)^t = 1.277227723 . to simplify this a little, divide 1.045 by 1.01 and you have 1.034653465. Substitute this and the equation becomes: . (1.034653465)^t = 1.277227723 . Any time you have a variable in an exponent, you should consider taking the log of both sides so that the variable can be brought out as a multiplier of the log. Let's use log base 10 since we can readily use a scientific calculator to determine the logarithm. Take log base 10 of both sides: . log((1.034653465)^t) = log(1.277227723) . bring the t out as the multiplier of the log: . t*log(1.034653465)= log(1.277227723) . Use a calculator to find the two logs: . t*(0.014794916) = 0.106268336 . Solve for t by dividing both sides of the equation by 0.014794916: . t = 0.106268336/0.014794916 = 7.182760136 . Since t is the year of interest minus 2000, we know that t occurs exactly 7.182760136 years after 2000 or to the nearest tenth, 7.2 years after 2000 which would be 2 tenths of the way into the year 2008. (Since a tenth of a year is 1.2 months, 2 tenths of the year should be 2.4 months into 2008 which would be around the middle of March 2008). . Check my work to ensure that I didn't make some dumb error or a "fat finger" calculator mistake. . Hope this helps you to understand the problem.
 Percentage-and-ratio-word-problems/486216: After being discounted 35%, a radio sells for \$38.58. Find the original price. (Round your answer to the nearest cent.)1 solutions Answer 332889 by bucky(2189)   on 2011-08-27 19:56:31 (Show Source): You can put this solution on YOUR website!Call the original price P. Then the store subtracts 35% of P from the price. Recall that 35% of P is the same as 0.35 times P. So the new price is: . P - 0.35P . and the problem says that this new price is \$38.58. . So in equation form this becomes: . P - 0.35P = 38.58 . Combine the two terms on the left side by performing the subtraction to get: . 0.65P = 38.58 . Solve for P by dividing both sides by 0.65 to get: . P = 38.58/0.65 = 59.35384615 . And rounding off the answer to the nearest cent tells us that the original price of the radio was: . P = \$59.35 . Hope this helps you with learning how prices are marked down.
 Linear-equations/486960: find the slope of the line containing (-1,-3)and (-4,-6) this is how i started: m=y2-y1/x2-x1 -6-(-3)/-4-(-1) =-3/-3 or 1 That cant be could it? 1 solutions Answer 332872 by bucky(2189)   on 2011-08-27 18:31:37 (Show Source): You can put this solution on YOUR website!Yes it can be. Here's the graph: . . This graph shows that the ordered pairs (-1,-3)and (-4,-6) are on the line and the slope is 1 because for every unit the graph moves horizontally to the right it also moves up one unit in the vertical direction. . Good job!!!
 Age_Word_Problems/486921: a father is 32 years old and his son is 5. How many years will pass before the father is 10 years older than the son1 solutions Answer 332865 by bucky(2189)   on 2011-08-27 18:20:33 (Show Source): You can put this solution on YOUR website!This problem does not make sense at all. . Think about it. At present the father is 32 and his son is 5. That means there is 27 years difference in their ages. . There will always be 27 years difference in their ages ... now and forever more. . This problem implies that as time goes by the difference in their ages gets smaller. In this case it implies that as time passes the difference in their ages reduces until it is only 10 years. Stretching that a little more, one might as well write the problem as: . "a father is 32 years old and his son is 5. How many years will pass before the father is the same age as his son?" . Do you really think that can happen?
 Coordinate-system/486127: Find all points on the x-axis that are 5 units from the point (4,-3).1 solutions Answer 332842 by bucky(2189)   on 2011-08-27 16:59:28 (Show Source): You can put this solution on YOUR website!The locus of all the points that are located 5 units away from the point (4, -3) is a circle with the center located at the point (4, -3) and having a radius of 5. The equation for such a circle is: . . The two points where this circle intersects with the x-axis will have "y" values of zero. (Any point on the x-axis has zero as its corresponding "y" value.) And these are the two points that we are looking for. . That being the case, set y = 0 in the circle equation and the equation reduces to: . . Square out each of the terms in this equation to get: . . Combine the two constants on the left side and this equation becomes: . . Subtract 25 from both sides: . . On the left side the +25 and -25 sum to zero, and the same thing happens on the right side. This reduces the equation to: . . Factor out an x on the left side and the equation becomes: . . Note that this equation will be true if either of the factors is equal to zero. This is because when one of the factors is equal to zero, the left side gets multiplied by zero, and this makes the left side equal to the zero on the right side. . Therefore, the equation will be correct if either: . . or . . and in this second equation when you add + 8 to both sides it becomes: . . This means that the points 0 and +8 on the x-axis are 5 units from the point (4, -3). In the form of ordered pairs the answers will be that the points (0, 0) and (8, 0) are the two points that are on the x-axis and are 5 units from the given ordered pair (4, -3). . Hope this helps you to understand this way of using the distance formula in the form of the equation for a circle to solve this problem.
 Graphs/486093: can someone please tell me how to graph 4x + y =01 solutions Answer 332813 by bucky(2189)   on 2011-08-27 15:19:41 (Show Source): You can put this solution on YOUR website!Sometimes the answer is a little easier to find if you put the equation into the slope intercept form. The slope intercept form is: . . In this slope intercept form the value of m, that is the multiplier of x, is the slope of the graph line. And the value of +b is the value on the y-axis where the graph crosses. . The equation you were given is: . . Let's move the 4x to the other side of this equation by subtracting 4x from both sides of the equation as follows: . . On the left side the 4x and the -4x cancel each other out and we are left with the equation: . . Notice that if you compare this with the slope intercept form, it is in exactly the same format. For this equation m, the multiplier of the x, is -4. The minus sign tells you that the slope is negative, meaning that as you move to the right on the graph, the line goes downward. The downward slope is 4. That means that every unit you move to the right along the x-axis the corresponding change in the value of y is 4 units. Since the slope is downward for this problem, that means that every unit you move to the right along the x-axis the value of y goes down 4 units. . One point on the graph is +b or the constant term in the slope intercept form. It is the point on the y-axis where the graph crosses. In your problem, the value of +b is +0. That means that the graph crosses the y-axis at y = 0 and this is the origin. . So you can start with a point at the origin, and then as you move 1 unit to the right along the x-axis the graph drops 4 units in the value of y. So when x = -1, the value of y drops to -4. Then as you move another unit to the right along the x-axis (you are now at x = 2), the value of y drops another 4 units from y = -4 down to y = -8. This is the point x = 2 and y = -8. . You now have the following three ordered pairs on the graph: (0, 0), (1, -4), and (2, -8). Plot these three points and draw an extended line through them to get the graph. When you do that it should look like this: . . Hope this helps you to understand graphing of linear functions a little better. .
 expressions/486101: How can I rewrite the expression using parentheses to get the given value of 28-3x3+4 with the value being 23? 1 solutions Answer 332795 by bucky(2189)   on 2011-08-27 14:29:26 (Show Source): You can put this solution on YOUR website!I notice that your mathematical expression uses "x" as meaning multiplication. . You can set this up as an equation. When you do it looks like this: . 28 - 3 x 3 + 4 = 23 . You can make this an equality (both sides are equal) by inserting one set of parentheses as follows: . 28 - (3 x 3) + 4 = 23 . Inside of the parentheses the 3 x 3 equals 9. Substituting 9 for the contents inside the parentheses changes the left side to: . 28 - 9 + 4 = 23 . If you algebraically total the three numbers on the left side by subtracting 9 from 28 and then adding 4 to that answer, the equation becomes: . 23 = 23 . So the answer to your problem is to insert parentheses as shown below and the result will equal 23: . 28 - (3 x 3) + 4 . Hope this helps you understand the problem better.
 test/486663: If 2a/a + 1/a = 4, then a = ?1 solutions Answer 332650 by bucky(2189)   on 2011-08-27 00:00:07 (Show Source): You can put this solution on YOUR website!You can get rid of the "a"s in the denominator by multiplying all terms on both sides by a as follows: . . The a in the numerator cancels with the a in the denominator as follows: . . and what's left is: . . Subtract 4a from both sides: . . and this simplifies to: . . Next subtract 1 from both sides: . . On the left side the +1 and -1 cancel each other out and this reduces the equation to: . . Solve for a by dividing both sides by -2: . . and the answer becomes: . . That's the answer to this problem. . Hope this helps you to see your way through the problem.
 Sequences-and-series/486660: Twelve consecutive integers are arranged in order from least to greatest. If the sum of the first six integers is 381, what is the sum of the last six integers?1 solutions Answer 332647 by bucky(2189)   on 2011-08-26 23:44:06 (Show Source): You can put this solution on YOUR website!The integers are consecutive. Therefore, each integer is obtained by adding 1 to the immediately preceding integer. . So if n+0 is the first integer, then n+1 is the second integer and the third integer is n+1+1 or n+2, and so on. The first six of the integers are: . n + 0 n + 1 n + 2 n + 3 n + 4 n + 5 . The sum of these 6 numbers is 6n + (0+1+2+3+4+5) which simplifies to 6n + 15. . The problem states that the sum of the first 6 numbers is 381. So we can set up the equation: . 6n + 15 = 381 . Get rid of the 15 on the left side by subtracting 15 from both sides as follows: . 6n + 15 - 15 = 381 - 15 . On the left side the + 15 and the -15 cancel each other out and on the right side 381 - 15 = 366. So the equation simplifies to: . 6n = 366 . solve for n by dividing both sides by 6 to get: . n = 366/6 = 61 . We now know that the first number is 61. Therefore, because they are consecutive, the first 6 numbers are 61, 62, 63, 64, 65, and 66. That means that the seventh through twelfth numbers are 67, 68, 69, 70, 71, and 72. Adding these 6 numbers: . 67 + 68 + 69 + 70 + 71 + 72 . results in the total of 417 and that's the answer to this problem. . Hope that this helps you to understand the problem.