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# Recent problems solved by 'Theo'

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 Inequalities/258719: The temperature range for a week in a mountain town was between 0-30 degrees Celsius. Find the temerature range in Fahrenheit degrees. C=5/9F-321 solutions Answer 190436 by Theo(3504)   on 2010-01-20 04:38:29 (Show Source): You can put this solution on YOUR website!you have the formula wrong for conversion from Fahrenheit to Celsius. the correct formula is: C = (5/9) * (F-32) when F is 32 (freezing), C = 0 as it should be. when F is 212 (boiling), C = 100 as it should be. now that we have the correct formula for Centigrade, we can derive the correct formula for fahrenheit. start with: C = (5/9) * (F-32) multiply both sides of this formula by (9/5) to get: (9/5) * C = F - 32 add 32 to both sides of this formula to get: (9/5) * C + 32 = F formula for F is: F = (9/5) * C + 32 to test this formula we use freezing and boiling points again. F = (9/5) * 0 + 32 = 32 (freezing point). F = (9/5) * 100 + 32 = 180 + 32 = 212 (boiling point). formula is good and we can use it to solve the problem. when C = 0, F = 32 which we already solved. when C = 30, F = (9/5) * 30 + 32 = 54 + 32 = 86. your answer is temperature range in celsius is 0 to 30. equivalent temperature range in fahrenheit is 32 to 86.
 Travel_Word_Problems/258735: An airplane flew for 8 hours at an air speed of x miles per hour (mph), and for 7 more hours at 325 mph. If the average airspeed for the entire flight was 350 mph, which of the following equations could be used to find x ? A. x + 325 = 2(350) B. x + 7(325) = 15(350) C. 8x – 7(325) = 350 D. 8x + 7(325) = 2(350) E 8x + 7(325) = 15(350) 1 solutions Answer 190435 by Theo(3504)   on 2010-01-20 04:21:17 (Show Source): You can put this solution on YOUR website!An airplane flew for 8 hours at an air speed of x miles per hour (mph), and for 7 more hours at 325 mph. If the average airspeed for the entire flight was 350 mph, which of the following equations could be used to find x ? general formula is rate * time = distance. let r = rate let h = time in hours let d = distance total time it took was 8 + 7 = 15 hours. average speed for the whole flight was 350 miles per hour. formula for the entire flight becomes: 350 * 15 = d formula for the first leg of the trip is equal to: x * 8 = d1 where d1 is the distance flown for the first leg of the trip. formula for the second leg of the trip is equal to : 325 * 7 = d2 where d2 is the distance flown for the second leg of the trip. since we know that d1 + d2 = d, then we have a general formula that states x * 8 + 325 * 7 = 350 * 15 that looks very much like selection E which is: 8x + 7(325) = 15(350) your answer is selection E.
 Miscellaneous_Word_Problems/258736: A hiking group will go from a certain town to a certain village by van on 1 of 4 roads, from the village to a waterfall by riding bicycles on 1 of 2 bicycle paths, and then from the waterfall to their campsite by hiking on 1 of 6 trails. How many routes are possible for the hiking group to go from the town to the village to the waterfall to their campsite? a) 6 b) 12 c) 24 d) 48 e) 220 1 solutions Answer 190434 by Theo(3504)   on 2010-01-20 04:11:57 (Show Source): You can put this solution on YOUR website!total possible number of way is: 4 * 2 * 6 = 8 * 6 = 48 that would be selection d.
 Probability-and-statistics/258730: A candidate got interview calls from6 Institutes of Management. Assume that his chance of getting admission in each of the Institute is 0.25. Assume each institute conducts separate interview process and results are independent of each other. What is the probability that the candidate will get admission in at least 3 institutes?1 solutions Answer 190433 by Theo(3504)   on 2010-01-20 04:08:29 (Show Source): You can put this solution on YOUR website!p(admission) = .25 interviews from 6 institutions probability that the candidate will get admission in at 3 institutions? this would be equal to: probability of getting admission in exactly 3 plus: probability of getting admission in exactly 4 plus: probability of getting admission in exactly 5 plus: probability of getting admission in exactly 6. this is also equal to: 1 minus: probability of getting admission in exactly 0 plus: probability of getting admission in eacttly 1 plus: probability of getting admission in exactly 2. we'll do both to show you how it works, but in practice, you would normally choose the option that takes less work to get the result from. probability of getting exactly 0 admissions = 1 * .75^6 probability of getting exactly 1 admission = 6 * .25 * .75^5 probability of getting exactly 2 admissions = 15 * .25^2 * .75^4 probability of getting exactly 3 admissions = 20 * .25^3 * .75^3 probability of getting exactly 4 admissions = 15 * .25^4 * .75^2 probability of getting exactly 5 admissions = 6 * .25^5 * .75 probability of getting exactly 6 admissions = 1 * .25^6 the multipliers are derived as follows: you have a total of 6 to draw from. you want to know how many ways you can draw different sets from this number. the number of ways you can draw 0 is 1 based on the combination formula of c(6,0) = 6! / (6!*0!) the number of ways you can draw sets of 1 is 6 based on the combination formula of c(6,1) = 6! / (5!*1!) the number of ways you can draw sets of 2 is 15 based on the combination formula of c(6,2) = 6! / (4!*2!) the number of ways you can draw sets of 3 is 20 based on the combination formula of c(6,3) = 6! / (3!*3!) the number of ways you can draw sets of 4 is 15 based on the combination formula of c(6,4) = 6! / (2!*4!) the number of ways you can draw sets of 5 is 6 based on the combination formula of c(6,5) = 6! / (1!*5!) the number of ways you can draw sets of 6 is 1 based on the combination formula of c(6,6) = 6! / (0!*6!) note that 0! is always equal to 1. the total probability should always be equal to 1. the numbers from all the probabilities come out as follows: p(0) = 1 * .75^6 = .177978516 p(1) = 6 * .25 * .75^5 = .355957031 p(2) = 15 * .25^2 * .75^4 = .296630859 p(3) = 20 * .25^3 * .75^3 = .131835938 p(4) = 15 * .25^4 * .75^2 = .032958984 p(5) = 6 * .25^5 * .75 = .004394531 p(6) = 1 * .25^6 = .000244141 probability of getting all possible combinations is equal to 1 as it should be. probability of getting 0 or 1 or 2 = p(0) + p(1) + p(2) = .830566406 probability of getting 3 or 4 or 5 or 6 = p(3) + p(4) + p(5) + p(6) = .169433594 your answer is .169433594. this was achieved by taking p(0) + p(1) + p(2) and subtracting the result from 1. 1 minus (p(0) + p(1) + p(2) = 1 - (.17... + .35... + .29...) = 1 - .830566406 = .169433594. it was also achieved by taking p(3) + p(4) + p(5) + p(6) = .13... + .03... + .004... + .0002... = .169433594. either way you get the same answer. as an example of the ways that a particlar probability can happen, consider the following example: what is the probability of getting exactly 2 admissions from 6 interviews if the probability of getting an admission is .25. the probability of getting exactly 2 admissions is equal to .25^2 * .75^4 = .019775391. this can happen in c(6,2) ways = 6! / (4!*2!) = 15 ways. those possible ways are: sn = school number (ex: s1 = school number 1) 1 = admitted 0 = not admitted ```way s1 s2 s3 s4 s5 s6 1 1 1 0 0 0 0 2 1 0 1 0 0 0 3 1 0 0 1 0 0 4 1 0 0 0 1 0 5 1 0 0 0 0 1 6 0 1 1 0 0 0 7 0 1 0 1 0 0 8 0 1 0 0 1 0 9 0 1 0 0 0 1 10 0 0 1 1 0 0 11 0 0 1 0 1 0 12 0 0 1 0 0 1 13 0 0 0 1 1 0 14 0 0 0 1 0 1 15 0 0 0 0 1 1 ``` the probability for each one of these happening is .25^2 * .75^4. they are all mutually exclusive (either one way or the other, never both at the same time). the total probability is therefore equal to the sum of each individual one happening which is p(way 1) + p(way 2) ..... + p(way 25) which is eaual to: 15 * .24^2 * .75^4.
 Miscellaneous_Word_Problems/258729: The town of Dover currently has a population of 6500. If its population is increasing at a rate of 1200 people per year, how long will it take for the population to reach 20,600?1 solutions Answer 190432 by Theo(3504)   on 2010-01-20 03:24:37 (Show Source): You can put this solution on YOUR website!p = 6500 f = 20600 g = 1200 formula would be: f = p + (n * g) f = future value p = present value g = growth per year n = number of years we are solving for n. formula becomes: 20600 = 6500 + (1200 * n) subtract 6500 from both sides of the equation to get 20600 - 6500 = 1200*n simplify to get: 14100 = 1200*n divide both sides by 1200 to get: 14100 / 1200 = n simplify to get: 11.75 = n it will take 11.75 years for the population to reach 20600. .75 years * 12 months per year = 9 months that's equivalent to 11 years and 9 months. 11.75 * 1200 = 14100 + 6500 = 20600 confirming the answer is good. the answer is: it will take 11.75 years for a population of 6500 to reach 20600 with a growth rate of 1200 people per year.
 Points-lines-and-rays/258370: my question states to locate a position in triangle (SAB), that is equidistant from all the angles. Given that: SA is 8 cm, AB is 8 cm and SB is 11.4 cm. Angle S is 28 degrees, angle A is 90 degrees and angle B is 45 degrees. Thank you for your help in advance. billy-bob 1 solutions Answer 190214 by Theo(3504)   on 2010-01-19 10:28:27 (Show Source): You can put this solution on YOUR website!you want to find the point inside the triangle that is equidistance from all the vertex of the triangle. I believe you want the circumcenter of the triangle. this is the center of a circle that circumscribes the triangle. as such, all vertices of the triangle are on the circumference of the circle which means they are the same distance to the center of the circle. that point is equidistant from each vertex of the triangle. here's a link that shows you what I mean. http://www.mathopenref.com/trianglecircumcenter.html that point is the intersection of the perpendicular bisectors of each side of the triangle. that point can be inside or outside the triangle. just play with the figure and you can see how that happens. here's another look at it. http://faculty.evansville.edu/ck6/tcenters/class/ccenter.html in your triangle SAB, you have: SA is 8 cm. AB is 8 cm. SB is 11.4 cm. this should be an isosceles triangle since 2 sides of it are equal. the legs are SA and AB and the base is SB. you state that angle S is 28 degrees and angle A is 90 degrees and angle B is 45 degrees. that can't be. sum of the interior angles of a triangle is 180 degrees. sum of the angles you provided is 163 degrees. also, if the sides of a triangle are equal, then their corresponding angles are also equal. This does not ring true with your statement about the sides and the angles. something is wrong with your problem setup. check it again and correct it and resubmit the problem. in any case, if you want the point that is equidistance from the vertices of the triangle, then you are looking for the intersection of the perpendicular bisectors of each of the sides because that give you the center of the circle that circumscribes the triangle. that point is equidistant from the vertices of the triangle.
 Travel_Word_Problems/258233: two cars drive the 120 miles from town A to town B. The first car goes 10 mph faster than the second, and completes the journey in 1 hour less time. Let s be the speed of the slower car. (A) In terms of s, what is the speed of the faster car? (B) In terms of S, how long does the slower car take to complete the trip? (C) In terms of S, how long does the faster car take to complete the trip? (D) Express the conditions "the faster car completes the journey in one hour less than the slower car" as an equation involving s using the answers to part b and c. (E) Solve the equation in part d to find the speed of the slower car.1 solutions Answer 190178 by Theo(3504)   on 2010-01-19 08:27:12 (Show Source): You can put this solution on YOUR website!(A) In terms of s, what is the speed of the faster car? faster car travels at s + 10 miles per hour. (B) In terms of S, how long does the slower car take to complete the trip? rate * time = distance. this makes time = distance / rate. slower car takes (120/s) hours to complete the trip. (C) In terms of s, how long does the faster car take to complete the trip? faster car takes (120 / (s+10)) hours to complete the trip. (D) Express the conditions "the faster car completes the journey in one hour less than the slower car" as an equation involving s using the answers to part b and c. the time it takes the slower car is expressed by the equation: h[s] = 120/s where h represents the time in hours. the time it takes the faster car is expressed by the equation: h[f] = 120/(s+10) where h represents the time in hours. the statement that it takes the faster car one less hour to complete the trip can be expressed by the equation h[f] = h[s] - 1. replace h[f] and h[s] with their respective equivalents and you get: 120/(s+10) = 120/s - 1 (E) Solve the equation in part d to find the speed of the slower car. the equation in part D is: 120/(s+10) = 120/s - 1 multiply both sides of this equation by s * (s+10) to get: 120 * s = 120 * (s+10) - (1 * (s+10) * s) simplify to get: 120 * s = 120 * s + 1200 - (s^2 + 10*s)) simplify further to get: 120 * s = 120 * s + 1200 - s^2 - 10*s subtract 120 * s from both sides of this equation to get: 0 = 120 * s - 120 * s + 1200 - s^2 - 10*s combine like terms to get: 0 = 1200 - s^2 - 10*s add s^2 to both sides of this equation and add 10*s to both sides of this equation and subtract 1200 from both sides of this equation to get: s^2 + 10*s - 1200 = 0 factor this quadratic equation to get: (s+40) * (s-30) = 0 solve for s to get s = -40 or s = 30 since s can't be negative, then s must equal 30. speed of the slower car is 30 miles per hour. speed of the faster car is 40 miles per hour. since time = distance / rate, then: the time it takes the slower car = 120/30 = 4 hours. the time it takes the faster car = 120/40 = 3 hours. the faster car takes 1 hour less than the slower car so the equation is satisfied. answer to part E is that the speed of the slower car is 30 miles per hour.
 Geometry_Word_Problems/258341: A circular piece of paper has a 5 cm radius. It is cut once through to the center and curled up to form a model of a cone with a height of 4 cm. Find the volume and surface area of the cone. 1 solutions Answer 190176 by Theo(3504)   on 2010-01-19 07:54:25 (Show Source): You can put this solution on YOUR website!the length of the side of this cone will be 5 cm. the length of the altitude of this cone will be 4 cm. this should form a 3/4/5 right triangle. this makes the radius of the base of the cone equal to 3 cm. volume of the cone is equal to 1/3 * pi * r^2 * h. this becomes 1/3 * pi * 9 * 4 = 1/3 * pi * 36 = 12 * pi cubic cm. surface area of the cone is equal to pi * r * (r + s). this becomes pi * 3 * (3 + 5) = pi * 3 * 8 = 24 * pi square cm.
 Miscellaneous_Word_Problems/258342: A Solid sphere of radius 3cm is melted and then cast into small spherical balls each of diameter 0.6cm. Find number of balls thus obtained.1 solutions Answer 190168 by Theo(3504)   on 2010-01-19 07:38:31 (Show Source): You can put this solution on YOUR website!volume of sphere is 4/3 * pi * r^3. r = 3 cm volume of sphere is 4/3 * pi * (3)^3 = 4/3 * 27 * pi = 4*9 * pi = 36 * pi cubic cm. each small spherical ball has a diameter of .6 cm. this means that each small spherical ball has a radius of .3 cm because radius = diameter / 2. volume of each small spherical ball is equal to 4/3 * pi * r^2. since r = .3, then volume of each small spherical ball is equal to 4/3 * pi * (.3)^3 = 4/3 * pi * .027 = .036 * pi cubic cm. divide 36 * pi by .036 * pi and you get: 36 * pi / .036 * pi = 36 / .036 = 1000 you can make 1000 small spherical balls out of the one big spherical ball.
 Volume/258330: Larry's ice cream is installing a giant ice cone above the shop. The plans for the ice cream cone, which is constructed by connecting a hemisphere to a cone. The cone is 6 ft and the entire ice cream cone is 9 ft. What is the volume of the ice cream cone1 solutions Answer 190165 by Theo(3504)   on 2010-01-19 07:30:37 (Show Source): You can put this solution on YOUR website!volume of a sphere is equal to 4/3 * pi * r^3 volume of a cone is equal to 1/3 * pi * r^2 * h the cone is 6 feet high and the entire ice cream cone is 9 feet. this means that the radius of the sphere is 3 feet. this is also the radius of the base of the cone. volume of the sphere is 4/3 * pi * 3^3 = 4/3 * pi * 27 = 4 * pi * 9 = 36 * pi. take half of that to get the top half of the sphere only and you get the volume of the icream on top of the cone is equal to 18 * pi. volume of the cone is 1/3 * pi * r^2 * h r = 3 and h = 6 volume of the cone is 1/3 * pi * (3)^2 * 6 = 18 * pi. volume of the ice cream cone is equal to 18 * pi + 18 * pi = 36 * pi. 36 * pi = 113.0973355 cubic feet. click on the following link to see a picture of the cone. http://theo.x10hosting.com/problems/258330.jpg
 decimal-numbers/258329: how do we rename decimal as percent?1 solutions Answer 190159 by Theo(3504)   on 2010-01-19 06:57:40 (Show Source): You can put this solution on YOUR website!you multiply the decimal by 100% example 1 * 100% = 100% .5 * 100% = 50% .1 * 100% = 10%
 decimal-numbers/258328: how do we rename percent as decimal?1 solutions Answer 190153 by Theo(3504)   on 2010-01-19 05:55:03 (Show Source): You can put this solution on YOUR website!divide by 100% examples: 100% / 100% = 1 50% / 100% = .5 10% / 100% = .1
 Average/258316: The average (arithmetic mean) of a and b is 10, whereas the average of b and 10 is c/2. What is the average of a and c?1 solutions Answer 190151 by Theo(3504)   on 2010-01-19 05:27:04 (Show Source): You can put this solution on YOUR website!(a+b)/2 = 10 (first equation) (b+10)/2 = (c/2) (second equation) first equation states that the average of a and b is 10. second equation states that the average of b and 10 is c/2 question is what is the average of a and c. take the second equation and solve for b. second equation is: (b+10)/2 = c/2 multiply both sides of this equation by 2 to get: b + 10 = c subtract 10 from both sides of this equation to get: b = c - 10 substitute for b in the first equation. first equation is: (a+b)/2 = 10 multiply both sides of this equation by 2 to get: a + b = 20 subsstitute (c-10) for b to get: a + c - 10 = 20 add 10 to both sides of this equation to get: a + c = 30 divide both sides of this equation by 2 to get: (a+c)/2 = 15 average of (a+c) is equal to 15.
 Probability-and-statistics/258309: Airlines find that each passenger who reserves a seat fails to turn up with probability 1/10 independently of the other passengers. So Teeny Weeny Airlines always sell 10 tickets for their 9 seat aeroplane while Blockbuster Airways always sell 20 tickets for their 18 seat aeroplane. Which is more often over-booked? 1 solutions Answer 190150 by Theo(3504)   on 2010-01-19 05:14:51 (Show Source): You can put this solution on YOUR website!probability that a passenger will not show up 10% of the time means the passenger will show up 90% of the time. any plane is overbooked when at least one passenger is denied. for the smaller plane this happens when all 10 passengers show up for the 9 seats. the probability of this happening is .9^10 = .34867844. for the larger plan this happens when all 20 passengers show up or when 19 of the 20 passengers show up for the 18 seats. probability of all 20 passengers showing up for the larger plane is .9^20 = .121576655. probability that exactly 19 passengers show up for the larger plane is .1^1 * .9^19 * C(20,1). C(20,1) is the number of possible combinations of selecting 1 out of 20. C(20,1) = 20!/(19!*1!) = (20*19!)/19! = 20 proability that exactly 19 passengers will show up for the larger plane is therefore .1^1 * .9^19 * 20 = .270170344. probability that all 18 passengers or exactly 19 passengers will show up for the larger plane is equal to .121576655 + .270170344 = .391746998 based on the probabilities, the larger plane should be overbooked more often than the smaller plane.
 Functions/258299: is y2=3x A FUNCTION1 solutions Answer 190147 by Theo(3504)   on 2010-01-19 03:57:46 (Show Source): You can put this solution on YOUR website!y^2 = 3x is not a function. solve for y to get: y = +/- sqrt(3x) for it to be a function there has to be one and only one value of y for each value of x. if there is at least one instance where there are more than 1 value of y for each value of x, then it is a relation and not a function. you give the graph of this equation the vertical line test. if the vertical line cuts through more than one point on the graph of the equation, then it is not a function. graph of this equation is shown below: on this graph, you can see that a vertical line passing through the graph of the equation will cut through the equation twice for each value of x at all points except at the vertex.
 Expressions-with-variables/257890: Apply the order of operations to simplify the following expression. 2x[(5x^2-3x+2)-(6x^3+6x^2)/3x^2]1 solutions Answer 189789 by Theo(3504)   on 2010-01-18 07:07:04 (Show Source): You can put this solution on YOUR website!expression is: you process operations within parentheses first starting from the inner parentheses and working to the outer parentheses. your expressions within the inner parentheses can't be processed any further. they are: and and your terms in the outer parentheses need to be processed next. they are: you would need to divide by first. that gets you the expression within the outer parenthese becomes: you process this next by removing the inner parentheses from the outer parentheses to get: which becomes: you replace the original expression within the outer parentheses by this to make your original equation of: becomes: you multiply each term within the parentheses by 2*x to get: if you did this right, then you should be able to get the same results in the original equation and the final equation for any value of x. take x = 5 final equation of becomes: original equation of becomes: which becomes: which becomes: which becomes: which becomes: 1000 since they both come up with the same answer, the final equation is equivalent to the original equation which indicates we simplified the equation properly. the simplified equation is:
 Sequences-and-series/257916: In an arithmetic series the sum of the first 9 terms is 162 and the sum of the first 12 terms is 288. Determine the series.1 solutions Answer 189785 by Theo(3504)   on 2010-01-18 05:14:36 (Show Source): You can put this solution on YOUR website!sum of the terms of an arithmetic series is given by the formula: s[n] = n * (a[1] + a[n])/2 s[n] = sum of the n terms of the sequence. a[1] = first term in the sequence. a[n] = nth term in the sequence. n = number of terms in the sequence. the formuula for the nth term in the sequence is: a[n] = a[1] + (n-1) * d d is the common difference. you will use both these formulas to find the answer. you are given: sum of the first 9 terms in a sequence is 162 sum of the first 12 terms in the same sequence is 288. the formnula for the sum of the first n terms in a sequence is: s[n] = n * (a[1] + a[n])/2 substitute for the first 9 terms to get: 162 = 9 * (a[1] + a[9])/2 (first equation) substitute for the first 12 terms to get: 288 = 12 * (a[1] + a[12])/2 (second equation) multiply both sides of the first equation by 2 and divide both sides of the first equation by 9 to get: a[1] + a[9] = 36 (third equation) multiply both sides of the second equation by 2 and divide both sides of the second equation by 12 to get: a[1] + a[12] = 48 (fourth equation) subtract third equation from fourth equation to get: a[12] - a[9] = 12 since 12 - 9 = 3, there are 3 terms between a[9] and a[12]. 12/3 = 4 which means the common difference between each term is 4. the nth term in each sequence is given by the formula: a[n] = a[1] + (n-1) * d we now know that d = 4 we solve for a[9] to get: a[9] = a[1] + 8*4 = a[1] + 32 we take our first equation and plug in our new found values. first equation is: 162 = 9 * (a[1] + a[9])/2 (first equation) substitute a[1] + 32 for a[9] to get: 162 = 9 * (a[1] + a[1] + 32)/2 combine like terms to get: 162 = 9 * (2* a[1] + 32)/2 multiply both sides of this equation by 2 and divide both sides of this equation by 9 to get: 36 = 2 * a[1] + 32 subtract 32 from both sides of this equation to get: 36-32 = 2 * a[1] which becomes: 4 = 2 * a[1]. divide both sides of this equation by 2 to get: a[1] = 2. we now know a[1] and we know d so we can construct the sequence. formula for nth term in the sequence is: a[n] = a[1] + (n-1) * d since d = 4,... the 9th term in the sequence is: a[9] = 2 + 8*4 = 2 + 32 = 34 the 12th term in the sequence is: a[12] = 2 + 11*4 = 2 + 44 = 46 the sequence is: ``` term: 1 2 3 4 5 6 7 8 9 10 11 12 sequence: 2 6 10 14 18 22 26 30 34 38 42 46 ``` 2 is the 1st term. 34 is the 9th term. 46 is the 12th term. common difference is 4.
 Numbers_Word_Problems/257912: What is the sum of all the digits of all the positive integers that are less than 100?1 solutions Answer 189781 by Theo(3504)   on 2010-01-18 03:26:36 (Show Source): You can put this solution on YOUR website!the positive digits are 1 through 9. 0 is a non-negative digits. 0 is not a positive digit. when we sum up the digits it won't matter if we include 0 or not because the sum of all 0's is still 0. when you count to 100, you have 1 set of tens digits and 1 set of units digits. the total set of digits is shown in the following table: ```00 10 20 30 40 50 60 70 80 90 01 11 21 31 41 51 61 71 81 91 02 12 22 32 42 52 62 72 82 92 03 13 23 33 43 53 63 73 83 93 04 14 24 34 44 54 64 74 84 94 05 15 25 35 45 55 65 75 85 95 06 16 26 36 46 56 66 76 86 96 07 17 27 37 47 57 67 77 87 97 08 18 28 38 48 58 68 78 88 98 09 19 29 39 49 59 69 79 89 99 ``` the ones digits are on the right of each number. the tens digits are on the left of each number. example: number 49: 9 is the ones digit 4 is the tens digit to sum up all the digits, we sum up all the ones digits and all the tens digits separately and then add them together. within each column, the ones digits add up to 45. that would be the sum of 1 through 9. within each column, the tens digits add up to 10 * the digit shown. that would be 10 * 1, 10 * 2, etc. you get the following number: ``` sum of tens digit sum of units digit total from 0 to 9 0 45 45 from 10 to 19 10 45 55 from 20 to 29 20 45 65 from 30 to 39 30 45 75 from 40 to 49 40 45 85 from 50 to 59 50 45 95 from 60 to 69 60 45 105 from 70 to 79 70 45 115 from 80 to 89 80 45 125 from 90 to 99 90 45 135 total 450 450 900 ``` sume of all the positive digits less than 100 is equal to 900.
 Miscellaneous_Word_Problems/257640: There are four cottages on a straight road. The distance between Ted's and Alice's cottages is 3 km. Both Bob's and Carol's cottages are twice as far from Alice's as they are from Ted's. Find the distance between Bob's and Carol's cottages in kilometers.1 solutions Answer 189526 by Theo(3504)   on 2010-01-17 07:35:54 (Show Source): You can put this solution on YOUR website!they are on the road in the following sequence: ``` B --- T - C -- A ``` each dash is a kilometer. Bob is 3 km away from Ted and 6 km away from Alice. Carol is 1 km away from Ted and 2 km away from Alice. Ted is 4 km away from Carol.