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question:What is the 1314th digit after the decimal point in the decimal representation of frac{5}{14}?
answer:To find the decimal representation of frac{5}{14}, we perform long division: begin{array}{c|cccccccccc} & multicolumn{2}{r}{0.3571428571} cline{2-11} 14 & 5.0000000000 & & -4.9000000000 & cline{2-10} & 1.0000000000 & & -0.9800000000 & cline{2-9} & 0.0200000000 & & -0.0140000000 & cline{2-8} & 0.0060000000 & & -0.0056000000 & cline{2-7} & 0.0004000000 & & -0.0003920000 & cline{2-6} & 0.0000080000 & & -0.0000072800 & cline{2-5} & 0.0000007200 & & -0.0000006860 & cline{2-4} & 0.0000000340 & & -0.0000000280 & cline{2-3} & 0.0000000060 & end{array} We can see that the decimal representation of frac{5}{14} is 0.3571428571ldots To find the 1314th digit after the decimal point, we look at the repeating pattern of the decimal. The pattern is 357142857, and it repeats every 6 digits. Since 1314 div 6 = 219 with no remainder, the 1314th digit after the decimal point is the same as the 6th digit in the repeating pattern, which is boxed{2}. The answer is: 2
question:The function y=frac{x^3+8x^2+21x+18}{x+2} can be simplified into the function y=Ax^2+Bx+C, defined everywhere except at x=D. What is the sum of the values of A, B, C, and D?
answer:We can simplify the function using polynomial division or synthetic division: [ begin{array}{c|ccccc} multicolumn{2}{r}{x^2+6x+9} & cline{2-6} x+2 & x^3+8x^2+21x+18 & quad x multicolumn{2}{r}{x^3+2x^2} & cline{2-3} multicolumn{2}{r}{6x^2+21x} & quad 9 multicolumn{2}{r}{} & quad 6x^2 cline{3-4} multicolumn{2}{r}{} & 15x & quad 9 multicolumn{2}{r}{} & quad 15x & quad 30 cline{4-5} multicolumn{2}{r}{} & multicolumn{2}{r}{-21} & quad 18 end{array} ] Therefore, we have [ frac{x^3+8x^2+21x+18}{x+2} = x^2+6x+9 - frac{21}{x+2}. ] This is defined everywhere except at x=-2, so D=-2. The function is in the form y=Ax^2+Bx+C, so A=1, B=6, and C=9. Therefore, the sum of A, B, C, and D is 1+6+9+(-2)=boxed{14}. The answer is: 14
question:Tamara, Nora, and Lulu have been saving to pay off a 40 debt. So far, Nora has saved five times what Lulu has, but three times Tamara’s savings. Lulu has saved 6. If they pay off the debt and divide the remaining money equally, how many dollars will each girl get?
answer:If Lulu has saved 6, then Nora has saved 5*6 = 30 Tamara has saved 30/3 = 10 In total, they have saved 6 + 30 + 10 = 46 If they pay off the 40 debt, they will have 46 - 40 = 6 remaining If they divide the remaining money equally among the three girls, each girl will get 6/3 = 2 Therefore, each girl will get 2 2 The answer is: 2
question:f Gabe has a playlist with three songs, where "The Best Day" is 3 minutes long, "Raise the Roof" is 2 minutes long, and "Rap Battle" is 3 minutes long, how many times can he listen to his entire playlist during the 40-minute ride to his wrestling match?
answer:The total length of Gabe's playlist is 3 + 2 + 3 = 8 minutes. If the playlist is 8 minutes long and Gabe has a 40-minute ride, he can listen to his playlist 40 / 8 = 5 times. 5 The answer is: 5