💡To find the remainder when the sum (2101 + 2103 + 2105 + 2107 + 2109) is divided by (12), I can start by determining the remainder of each individual number when divided by (12). First, I calculate (2101 mod 12). Dividing (2101) by (12) gives a quotient of (175) with a remainder of (1), so (2101 mod 12 = 1).Next, for (2103), dividing by (12) results in a quotient of (175) and a remainder of (3), so (2103 mod 12 = 3).Similarly, (2105 mod 12) is calculated by dividing (2105) by (12), yielding a quotient of (175) and a remainder of (5), hence (2105 mod 12 = 5).For (2107), dividing by (12) gives a quotient of (175) and a remainder of (7), so (2107 mod 12 = 7).Lastly, (2109 mod 12) is found by dividing (2109) by (12), resulting in a quotient of (175) and a remainder of (9), thus (2109 mod 12 = 9).Now, I sum these remainders: (1 + 3 + 5 + 7 + 9 = 25).Finally, I find the remainder of (25) when divided by (12). Dividing (25) by (12) gives a quotient of (2) and a remainder of (1), so (25 mod 12 = 1).Therefore, the remainder when (2101 + 2103 + 2105 + 2107 + 2109) is divided by (12) is (1).

💡Okay, so I have this problem here: If the line ax + 2y + a = 0 is parallel to the line 3ax + (a - 1)y + 7 = 0, then I need to find the value of the real number a. Hmm, okay. Let me think about how to approach this.First, I remember that two lines are parallel if their slopes are equal. So, maybe I should find the slopes of both lines and set them equal to each other. But wait, how do I find the slope of a line given in standard form like ax + by + c = 0?Oh, right! I can convert each equation into slope-intercept form, which is y = mx + b, where m is the slope. That should make it easier to compare the slopes.Let me start with the first line: ax + 2y + a = 0. I need to solve for y.So, subtract ax and a from both sides: 2y = -ax - a. Then, divide both sides by 2: y = (-a/2)x - a/2. Okay, so the slope (m1) of the first line is -a/2.Now, moving on to the second line: 3ax + (a - 1)y + 7 = 0. Again, I'll solve for y.Subtract 3ax and 7 from both sides: (a - 1)y = -3ax - 7. Then, divide both sides by (a - 1): y = (-3a/(a - 1))x - 7/(a - 1). So, the slope (m2) of the second line is -3a/(a - 1).Since the lines are parallel, their slopes must be equal. Therefore, I can set m1 equal to m2:-a/2 = -3a/(a - 1)Hmm, okay, let's solve this equation for a. First, I can multiply both sides by 2(a - 1) to eliminate the denominators:(-a/2) * 2(a - 1) = (-3a/(a - 1)) * 2(a - 1)Simplifying both sides:-a(a - 1) = -6aLet me expand the left side:-a^2 + a = -6aNow, bring all terms to one side to form a quadratic equation:-a^2 + a + 6a = 0Combine like terms:-a^2 + 7a = 0Factor out a:a(-a + 7) = 0So, the solutions are a = 0 or -a + 7 = 0, which gives a = 7.Wait a second, let me check if these solutions make sense. If a = 0, let's substitute back into both lines.First line becomes 0x + 2y + 0 = 0, which simplifies to 2y = 0 or y = 0. That's the x-axis.Second line becomes 0x + (0 - 1)y + 7 = 0, which simplifies to -y + 7 = 0 or y = 7. That's a horizontal line parallel to the x-axis. So, yes, they are parallel.Now, let's check a = 7.First line: 7x + 2y + 7 = 0. Let's write it in slope-intercept form:2y = -7x - 7 => y = (-7/2)x - 7/2. So, slope is -7/2.Second line: 3*7x + (7 - 1)y + 7 = 0 => 21x + 6y + 7 = 0. Solving for y:6y = -21x - 7 => y = (-21/6)x - 7/6 => y = (-7/2)x - 7/6. So, slope is -7/2.Yep, the slopes are equal, so they are parallel. That checks out.Wait, but I should also make sure that when I divided by (a - 1) earlier, I didn't divide by zero. So, if a = 1, then (a - 1) would be zero, but in our solutions, a is 0 or 7, so that's fine. No division by zero issues here.So, both a = 0 and a = 7 are valid solutions. Therefore, the real number a can be either 0 or 7.**Final Answer**The value of the real number ( a ) is boxed{0} or boxed{7}.**Final Answer**The value of the real number ( a ) is boxed{0} or boxed{7}.

💡Okay, so I have this problem about two workers, A and B, completing a task. It seems a bit complicated, but I'll try to break it down step by step. First, let me read the problem again to make sure I understand all the details."Two workers, A and B, completed a task in the following manner. First, A worked alone for 2/3 of the time it would take B to complete the entire task alone. Then B took over and finished the task. This method took 2 hours longer than if they had worked together and completed the task jointly. If they had worked together in this manner, A would have done half of the work that he actually left for B to do alone. How many hours would it take for A and B to each complete the task alone?"Alright, so let's parse this out.1. A works alone for (2/3) of the time it would take B to complete the task alone. So, if B takes y hours to do the task alone, then A works for (2/3)y hours.2. Then B takes over and finishes the task. So, the total time taken in this method is (2/3)y + time B takes to finish the remaining work.3. This method took 2 hours longer than if they had worked together. So, if they worked together, they would have completed the task in T hours, and the method described above took T + 2 hours.4. Additionally, if they had worked together in this manner, A would have done half of the work that he actually left for B to do alone. Hmm, this part is a bit confusing. Let me think.So, when A works alone for (2/3)y hours, he completes some portion of the work, and the remaining portion is done by B. If instead, they had worked together, A would have done half of that remaining portion. So, the work A would have done when working together is half of the work he left for B when working alone.Let me try to formalize this.Let’s denote:- Let the total work be 1 unit.- Let A's work rate be a (work per hour), so A takes 1/a hours to complete the task alone.- Let B's work rate be b (work per hour), so B takes 1/b hours to complete the task alone.Wait, actually, the problem says "the time it would take B to complete the entire task alone." So, if B takes y hours alone, then A works for (2/3)y hours alone.So, let me define:- Let y be the time B takes to complete the task alone. So, B's work rate is 1/y.- Then, A works alone for (2/3)y hours. So, A's work rate is 1/x, where x is the time A takes to complete the task alone.Wait, maybe I should define:- Let x be the time A takes to complete the task alone, so A's work rate is 1/x.- Let y be the time B takes to complete the task alone, so B's work rate is 1/y.Then, A works alone for (2/3)y hours. So, the amount of work A completes is (1/x)*(2/3)y = (2y)/(3x).Then, the remaining work is 1 - (2y)/(3x). This remaining work is completed by B. So, the time B takes to finish the remaining work is [1 - (2y)/(3x)] / (1/y) = y*(1 - 2y/(3x)) = y - (2y^2)/(3x).Therefore, the total time taken in this method is (2/3)y + y - (2y^2)/(3x) = (5/3)y - (2y^2)/(3x).Now, if they had worked together, their combined work rate is (1/x + 1/y). So, the time taken to complete the task together is 1 / (1/x + 1/y) = xy/(x + y).According to the problem, the method where A works alone for (2/3)y hours and then B finishes took 2 hours longer than working together. So:(5/3)y - (2y^2)/(3x) = xy/(x + y) + 2.That's one equation.Now, the second condition: If they had worked together in this manner, A would have done half of the work that he actually left for B to do alone.So, when working alone, A left (1 - 2y/(3x)) work for B. If they had worked together, the work done by A would be half of that, which is (1 - 2y/(3x))/2.But when working together, the total work is done in xy/(x + y) hours. So, the work done by A when working together is (1/x)*(xy/(x + y)) = y/(x + y).According to the problem, this is half of the work he left for B. So:y/(x + y) = (1 - 2y/(3x))/2.Let me write that down:y/(x + y) = (1 - 2y/(3x))/2.So, now we have two equations:1. (5/3)y - (2y^2)/(3x) = xy/(x + y) + 2.2. y/(x + y) = (1 - 2y/(3x))/2.Let me try to solve these equations.First, let's work on equation 2:y/(x + y) = (1 - 2y/(3x))/2.Multiply both sides by 2:2y/(x + y) = 1 - 2y/(3x).Let me rearrange this:2y/(x + y) + 2y/(3x) = 1.Let me find a common denominator for the left side. The denominators are (x + y) and 3x. So, the common denominator is 3x(x + y).So:[2y * 3x + 2y * (x + y)] / [3x(x + y)] = 1.Simplify numerator:6xy + 2xy + 2y^2 = 8xy + 2y^2.So:(8xy + 2y^2) / [3x(x + y)] = 1.Multiply both sides by denominator:8xy + 2y^2 = 3x(x + y).Simplify right side:3x^2 + 3xy.So:8xy + 2y^2 = 3x^2 + 3xy.Bring all terms to left side:8xy + 2y^2 - 3x^2 - 3xy = 0.Simplify:(8xy - 3xy) + 2y^2 - 3x^2 = 0.5xy + 2y^2 - 3x^2 = 0.Let me write this as:-3x^2 + 5xy + 2y^2 = 0.Multiply both sides by -1:3x^2 - 5xy - 2y^2 = 0.This is a quadratic equation in terms of x and y. Let me try to solve for x in terms of y.Let me write it as:3x^2 - 5xy - 2y^2 = 0.This is a quadratic in x, so using quadratic formula:x = [5y ± sqrt(25y^2 + 24y^2)] / 6.Because discriminant D = (5y)^2 - 4*3*(-2y^2) = 25y^2 + 24y^2 = 49y^2.So,x = [5y ± 7y]/6.So, two solutions:x = (5y + 7y)/6 = 12y/6 = 2y.Or,x = (5y - 7y)/6 = (-2y)/6 = -y/3.Since time cannot be negative, we discard the negative solution.Thus, x = 2y.So, A takes twice as long as B to complete the task alone.So, x = 2y.Now, let's substitute x = 2y into equation 1.Equation 1:(5/3)y - (2y^2)/(3x) = xy/(x + y) + 2.Substitute x = 2y:Left side:(5/3)y - (2y^2)/(3*(2y)) = (5/3)y - (2y^2)/(6y) = (5/3)y - (y)/3 = (5/3 - 1/3)y = (4/3)y.Right side:(2y * y)/(2y + y) + 2 = (2y^2)/(3y) + 2 = (2y)/3 + 2.So, equation becomes:(4/3)y = (2y)/3 + 2.Subtract (2y)/3 from both sides:(4/3)y - (2/3)y = 2.(2/3)y = 2.Multiply both sides by 3/2:y = 3.So, y = 3 hours. Therefore, B takes 3 hours to complete the task alone.Since x = 2y, x = 6 hours. So, A takes 6 hours to complete the task alone.Let me verify if these values satisfy both equations.First, equation 2:y/(x + y) = 3/(6 + 3) = 3/9 = 1/3.(1 - 2y/(3x))/2 = (1 - 2*3/(3*6))/2 = (1 - 6/18)/2 = (1 - 1/3)/2 = (2/3)/2 = 1/3.So, both sides equal 1/3. Good.Now, equation 1:Left side: (5/3)y - (2y^2)/(3x) = (5/3)*3 - (2*9)/(3*6) = 5 - (18)/(18) = 5 - 1 = 4.Right side: xy/(x + y) + 2 = (6*3)/(6 + 3) + 2 = 18/9 + 2 = 2 + 2 = 4.Both sides equal 4. Good.So, the solution is consistent.Therefore, A takes 6 hours and B takes 3 hours to complete the task alone.

💡Okay, so I have this problem here: I need to find the base of a parallelogram when I know the area and the height. The area is 612 square centimeters, and the height is 18 centimeters. Alright, let's see how to approach this.First, I remember that the area of a parallelogram is calculated by multiplying the base by the height. So, the formula is:[ text{Area} = text{Base} times text{Height} ]In this case, I know the area and the height, but I need to find the base. That means I should rearrange the formula to solve for the base. If I divide both sides of the equation by the height, I get:[ text{Base} = frac{text{Area}}{text{Height}} ]Okay, so plugging in the numbers I have:[ text{Base} = frac{612 , text{cm}^2}{18 , text{cm}} ]Now, let's do the division. 612 divided by 18. Hmm, 18 times 30 is 540, and 18 times 34 is 612 because 18 times 30 is 540, and 18 times 4 is 72, so 540 plus 72 is 612. So, 18 times 34 equals 612.Therefore, the base is 34 centimeters.Wait, let me double-check that. If I multiply 34 cm by 18 cm, do I get 612 cm²? Let's see:34 times 18. 30 times 18 is 540, and 4 times 18 is 72. Adding those together, 540 plus 72 is indeed 612. So, yes, that checks out.I think that's it. The base of the parallelogram is 34 centimeters.

💡Okay, so Seth and his brother want to see their favorite band in concert. They've got a few things to consider for the total cost. Let me try to break this down step by step.First, they need to buy tickets. There are three tiers: General Admission for 50, Premium Seating for 75, and VIP Access for 150. They decide on Premium Seating. Since there are two of them, that would be 2 tickets at 75 each. So, 2 times 75 is 150 for the tickets alone.But wait, there's a 15% processing fee on the tickets. I need to calculate that. 15% of 150 is... let's see, 0.15 times 150 equals 22.50. So, adding that to the ticket cost, the total for the tickets with processing fee is 150 plus 22.50, which is 172.50.Next, they have to pay for parking. The parking fee is 10.00. Adding that to the previous total: 172.50 plus 10 is 182.50.Then, there's an additional entrance fee of 5.00 per person. Since there are two people, that's 2 times 5, which is 10. Adding that to the total: 182.50 plus 10 equals 192.50.During the concert, they buy refreshments totaling 20.00. Adding that: 192.50 plus 20 is 212.50.Lastly, they decide to buy two band t-shirts costing a total of 40.00. Adding that to the total: 212.50 plus 40 equals 252.50.So, putting it all together, the total cost for Seth and his brother to go to the concert, including tickets, processing fee, parking, entrance fees, refreshments, and merchandise, is 252.50.

💡First, I need to determine the annual interest rate for Tim's investment. I'll start by calculating the interest earned by Lana's investment over two years.Lana invested 900 at a 5% annual interest rate, compounded annually. The formula for compound interest is:[ A = P(1 + r)^n ]where:- ( A ) is the amount after ( n ) years,- ( P ) is the principal amount,- ( r ) is the annual interest rate,- ( n ) is the number of years.Plugging in Lana's values:[ A_L = 900(1 + 0.05)^2 ][ A_L = 900(1.05)^2 ][ A_L = 900 times 1.1025 ][ A_L = 992.25 ]The interest earned by Lana is:[ I_L = A_L - P ][ I_L = 992.25 - 900 ][ I_L = 92.25 ]According to the problem, Tim's interest earned is 12.75 more than Lana's:[ I_T = I_L + 12.75 ][ I_T = 92.25 + 12.75 ][ I_T = 105 ]Now, using Tim's investment details:[ I_T = A_T - P ][ 105 = 500(1 + r)^2 - 500 ]Solving for ( r ):[ 105 + 500 = 500(1 + r)^2 ][ 605 = 500(1 + r)^2 ][ frac{605}{500} = (1 + r)^2 ][ 1.21 = (1 + r)^2 ]Taking the square root of both sides:[ sqrt{1.21} = 1 + r ][ 1.1 = 1 + r ][ r = 0.1 ]Therefore, Tim's annual interest rate is 10%.