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29 problems match “fractions”. Clear

Sum of fractions with shared denominators ●●○○○ 20 ptsFind the value of $\left(\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{20}\right)+\left(\frac{2}{3}+\cdots+\frac{2}{20}\right)+\cdots+\left(\frac{18}{19}+\frac{18}{20}\right)+\frac{19}{20}$ .
Telescoping sum of odd-numerator fractions ●●○○○ 20 ptsCalculate $\frac{3}{4}+\frac{5}{36}+\frac{7}{144}+\frac{9}{400}+\frac{11}{900}+\frac{13}{1764}+\frac{15}{3136}$ . Give your answer as a decimal.
Distinct candy flavors by percent red ●●○○○ 20 ptsFour red candies and three green candies can be combined to make many different flavors. Two flavors are different if their percent red is different, so 3 red / 0 green is the same flavor as 2 red / 0 green, and 4 red / 2 green is the same flavor as 2 red / 1 green. If a flavor is made using some or all of the seven candies (at least one candy), how many different flavors are possible?
Positive integers giving a hole ●●○○○ 20 ptsFind the number of positive integers $n$ for which the graph of $y = \frac{(x + 3)(x - 7)^5}{(x - 1)(x - 7)^n}$ has a hole at $x = 7$ .
Contour integral of 1/(z^2+1) ●●○○○ 20 ptsCompute the complex contour integral $\displaystyle\int_{|z|=2}\frac{1}{z^2+1}\,dz$ , where the circle $|z|=2$ is traversed once counterclockwise.
Share of total area from rose fractions ●●○○○ 20 ptsTwo fields are planted with roses and lavender. $65\%$ of the area of the first field is roses, $45\%$ of the area of the second field is roses, and $53\%$ of the area of both fields together is roses. What percentage of the total area (of both fields together) is the area of the first field?
Difference of fractions equals one ●●○○○ 20 ptsFind all values of $x$ (with $x \neq 0, 1, -1$ ) for which $\frac{1}{x^2 - x} - \frac{1}{x^3 - x} = 1$ . Round each value to two decimal places.
Digit sum of a sum of fractions ●●○○○ 20 ptsWhat is the value of $a+b+c+d+e+f$ , where the decimal representation of $\frac{4}{37}+\frac{3}{11}+\frac{23}{9}$ is $2.abcdef\ldots$ ?
Sum of conjugate fractions ●●○○○ 20 ptsSimplify $\frac{3+\sqrt{5}}{3-\sqrt{5}}+\frac{3-\sqrt{5}}{3+\sqrt{5}}$ .
Sum of four rational numbers ●●○○○ 20 ptsAdd $2\frac{3}{5}$ , $\frac{-15}{13}$ , $\frac{-13}{-15}$ and $-4\frac{3}{5}$ . Round the sum to two decimal places.