Умножение и деление рациональных дробей

{"questions":[{"instruction":"Выполните умножение","content":"$$\\frac{3x}{5}\\cdot\\frac{10}{9x}$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$\\frac{2}{3}$","$\\frac{27x^2}{50}$","$\\frac{30x}{45}$","$\\frac{3}{5}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Чтобы умножить рациональные дроби, нужно перемножить числители и знаменатели:<br />$\\frac{3x}{5}\\cdot\\frac{10}{9x}=\\frac{3x\\cdot 10}{5\\cdot 9x}$.","Сократим общие множители:<br />$\\frac{3x\\cdot 10}{5\\cdot 9x}=\\frac{3x\\cdot 2\\cdot 5}{5\\cdot 3\\cdot 3x}$.","Сокращаем $5$, $3$ и $x$:<br />$\\frac{2}{3}$."]},{"instruction":"Выполните умножение","content":"$$\\frac{a+2}{a-1}\\cdot\\frac{3a-3}{a+2}$$[[fill_choice_big-19]]","widgets":{"fill_choice_big-19":{"type":"fill_choice_big","options":["$3$","$\\frac{1}{3}$","$3a^2$","$a-3$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим выражение $3a-3$ на множители:<br />$3a-3=3(a-1)$<br />Получаем:<br />$\\frac{a+2}{a-1}\\cdot\\frac{3(a-1)}{a+2}$.","Сокращаем одинаковые множители $a+2$ и $a-1$:<br />$\\frac{1}{1}\\cdot 3=3$."]},{"instruction":"Выполните умножение","content":"$$\\frac{x^2-9}{x+4}\\cdot\\frac{x+4}{x-3}$$[[fill_choice_big-59]]","widgets":{"fill_choice_big-59":{"type":"fill_choice_big","options":["$x+3$","$x-3$","$x+4$","$x+1$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов на множители:<br />$x^2-9=(x-3)(x+3)$<br />Получаем:<br />$\\frac{(x-3)(x+3)}{x+4}\\cdot\\frac{x+4}{x-3}$.","Сокращаем одинаковые множители $x+4$ и $x-3$:<br />$x+3$."]},{"instruction":"Выполните умножение","content":"$$\\frac{2m}{m+5}\\cdot\\frac{m^2+5m}{4m^2}$$[[fill_choice_big-125]]","widgets":{"fill_choice_big-125":{"type":"fill_choice_big","options":["$\\frac{1}{2}$","$\\frac{m+5}{2}$","$\\frac{1}{m^2}$","$2$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим выражение $m^2+5m$ на множители:<br />$m^2+5m=m(m+5)$<br />Получаем:<br />$\\frac{2m}{m+5}\\cdot\\frac{m(m+5)}{4m^2}$.","Сокращаем одинаковые множители $m+5$ и $m$:<br />$\\frac{2m}{1}\\cdot\\frac{1}{4m}$<br />Сокращаем $2m$ и $4m$:<br />$\\frac{1}{2}$."]},{"instruction":"Выполните умножение","content":"$$\\frac{y^2-4}{6y}\\cdot\\frac{3y}{y+2}$$[[fill_choice_big-170]]","widgets":{"fill_choice_big-170":{"type":"fill_choice_big","options":["$\\frac{y-2}{2}$","$y-2$","$\\frac{y}{y-2}$","$\\frac{1}{2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов на множители:<br />$y^2-4=(y-2)(y+2)$<br />Получаем:<br />$\\frac{(y-2)(y+2)}{6y}\\cdot\\frac{3y}{y+2}$.","Сокращаем одинаковые множители $y+2$ и $y$:<br />$\\frac{y-2}{6}\\cdot 3$<br />Сокращаем $3$ и $6$:<br />$\\frac{y-2}{2}$."]}]}

{"questions":[{"instruction":"Выполните умножение","content":"$$\\frac{4x}{7}\\cdot\\frac{21}{8x}$$[[fill_choice_big-1]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$\\frac{3}{2}$","$\\frac{2}{3}$","$\\frac{84x}{56x}$","$\\frac{1}{2x}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Перемножим числители и знаменатели:<br />$\\frac{4x}{7}\\cdot\\frac{21}{8x}=\\frac{4x\\cdot 21}{7\\cdot 8x}$.","Разложим множители для сокращения:<br />$\\frac{4x\\cdot 21}{7\\cdot 8x}=\\frac{4x\\cdot 3\\cdot 7}{7\\cdot 8x}$.","Сократим $7$ и $x$, затем сократим $4$ и $8$:<br />$\\frac{4\\cdot 3}{8}=\\frac{12}{8}=\\frac{3}{2}$."]},{"instruction":"Выполните умножение","content":"$$\\frac{p-5}{2p}\\cdot\\frac{6p}{p-5}$$[[fill_choice_big-2]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$3$","$\\frac{1}{3}$","$\\frac{p-5}{p}$","$\\frac{6p}{2p(p-5)}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Перемножим дроби:<br />$\\frac{p-5}{2p}\\cdot\\frac{6p}{p-5}=\\frac{(p-5)\\cdot 6p}{2p\\cdot (p-5)}$.","Сократим одинаковые множители $(p-5)$ и $p$:<br />$\\frac{6}{2}$.","Выполним деление:<br />$\\frac{6}{2}=3$."]},{"instruction":"Выполните умножение","content":"$$\\frac{x^2-16}{5x}\\cdot\\frac{10x}{x+4}$$[[fill_choice_big-3]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$2(x-4)$","$2(x+4)$","$\\frac{x-4}{2}$","$\\frac{2}{x-4}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов:<br />$x^2-16=(x-4)(x+4)$<br />Тогда:<br />$\\frac{(x-4)(x+4)}{5x}\\cdot\\frac{10x}{x+4}$.","Сократим одинаковые множители $(x+4)$ и $x$:<br />$\\frac{x-4}{5}\\cdot 10$.","Сократим $10$ и $5$:<br />$2(x-4)$."]},{"instruction":"Выполните умножение","content":"$$\\frac{3a}{a-2}\\cdot\\frac{a^2-4}{9a^2}$$[[fill_choice_big-4]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$\\frac{a+2}{3a}$","$\\frac{a-2}{3a}$","$\\frac{a+2}{a}$","$\\frac{3(a+2)}{a}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов:<br />$a^2-4=(a-2)(a+2)$<br />Получаем:<br />$\\frac{3a}{a-2}\\cdot\\frac{(a-2)(a+2)}{9a^2}$.","Сократим $(a-2)$:<br />$\\frac{3a\\cdot (a+2)}{9a^2}$.","Сократим $3$ и $9$, затем $a$ и $a^2$:<br />$\\frac{a+2}{3a}$."]},{"instruction":"Выполните умножение","content":"$$\\frac{2y+6}{y}\\cdot\\frac{y}{4y+12}$$[[fill_choice_big-5]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$\\frac{1}{2}$","$2$","$\\frac{y+3}{2}$","$\\frac{2}{y+3}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Вынесем общий множитель в числителе и знаменателе:<br />$2y+6=2(y+3)$, $4y+12=4(y+3)$.<br />Тогда:<br />$\\frac{2(y+3)}{y}\\cdot\\frac{y}{4(y+3)}$.","Сократим $y$ и $(y+3)$:<br />$\\frac{2}{4}$.","Сократим дробь:<br />$\\frac{2}{4}=\\frac{1}{2}$."]},{"instruction":"Выполните умножение","content":"$$\\frac{5m}{2(m-1)}\\cdot\\frac{4(m-1)}{15m}$$[[fill_choice_big-6]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$\\frac{2}{3}$","$\\frac{3}{2}$","$\\frac{4}{15}$","$\\frac{2(m-1)}{3m}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Перемножим дроби:<br />$\\frac{5m}{2(m-1)}\\cdot\\frac{4(m-1)}{15m}=\\frac{5m\\cdot 4(m-1)}{2(m-1)\\cdot 15m}$.","Сократим одинаковые множители $(m-1)$ и $m$:<br />$\\frac{5\\cdot 4}{2\\cdot 15}$.","Сократим $4$ и $2$, затем $5$ и $15$:<br />$\\frac{10}{15}=\\frac{2}{3}$."]},{"instruction":"Выполните умножение","content":"$$\\frac{b^2-1}{3b}\\cdot\\frac{9b}{b+1}$$[[fill_choice_big-7]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$3(b-1)$","$3(b+1)$","$\\frac{b-1}{3}$","$\\frac{3}{b-1}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов:<br />$b^2-1=(b-1)(b+1)$<br />Получаем:<br />$\\frac{(b-1)(b+1)}{3b}\\cdot\\frac{9b}{b+1}$.","Сократим $(b+1)$ и $b$:<br />$\\frac{b-1}{3}\\cdot 9$.","Сократим $9$ и $3$:<br />$3(b-1)$."]},{"instruction":"Выполните умножение","content":"$$\\frac{2t}{t+7}\\cdot\\frac{t^2+7t}{6t^2}$$[[fill_choice_big-8]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$\\frac{1}{3}$","$\\frac{t+7}{3t}$","$\\frac{t}{3}$","$\\frac{1}{3t}$"],"placeholder":0,"answer":3}},"step":1,"hints":["Разложим $t^2+7t$ на множители:<br />$t^2+7t=t(t+7)$<br />Тогда:<br />$\\frac{2t}{t+7}\\cdot\\frac{t(t+7)}{6t^2}$.","Сократим $(t+7)$ и один множитель $t$:<br />$\\frac{2t\\cdot t}{6t^2}=\\frac{2}{6}$.","Сократим дробь:<br />$\\frac{2}{6}=\\frac{1}{3}$."]},{"instruction":"Выполните умножение","content":"$$\\frac{3k-3}{k}\\cdot\\frac{k}{9k-9}$$[[fill_choice_big-9]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$\\frac{1}{3}$","$3$","$\\frac{k-1}{3}$","$\\frac{1}{k-1}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Вынесем общий множитель:<br />$3k-3=3(k-1)$, $9k-9=9(k-1)$.<br />Тогда:<br />$\\frac{3(k-1)}{k}\\cdot\\frac{k}{9(k-1)}$.","Сократим $k$ и $(k-1)$:<br />$\\frac{3}{9}$.","Сократим дробь:<br />$\\frac{3}{9}=\\frac{1}{3}$."]},{"instruction":"Выполните умножение","content":"$$\\frac{z^2-25}{4z}\\cdot\\frac{8z}{z-5}$$[[fill_choice_big-10]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$2(z+5)$","$\\frac{z+5}{2}$","$2(z-5)$","$\\frac{2}{z+5}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов:<br />$z^2-25=(z-5)(z+5)$<br />Тогда:<br />$\\frac{(z-5)(z+5)}{4z}\\cdot\\frac{8z}{z-5}$.","Сократим $(z-5)$ и $z$:<br />$\\frac{z+5}{4}\\cdot 8$.","Сократим $8$ и $4$:<br />$2(z+5)$."]}]}

{"questions":[{"instruction":"Выполните умножение","content":"$$\\frac{6x}{5}\\cdot\\frac{10}{9x}$$[[fill_choice_big-11]]","widgets":{"fill_choice_big-11":{"type":"fill_choice_big","options":["$\\frac{4}{3}$","$\\frac{3}{4}$","$\\frac{60x}{45x}$","$\\frac{2}{3x}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Перемножим числители и знаменатели:<br />$\\frac{6x}{5}\\cdot\\frac{10}{9x}=\\frac{6x\\cdot 10}{5\\cdot 9x}$.","Сократим $x$, затем сократим $10$ и $5$:<br />$\\frac{6\\cdot 2}{9}$.","Сократим $6$ и $9$ на $3$:<br />$\\frac{12}{9}=\\frac{4}{3}$."]},{"instruction":"Выполните умножение","content":"$$\\frac{2a-8}{3a}\\cdot\\frac{9a}{a-4}$$[[fill_choice_big-12]]","widgets":{"fill_choice_big-12":{"type":"fill_choice_big","options":["$6$","$3(a-4)$","$\\frac{2}{3}$","$\\frac{a-4}{6}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Вынесем общий множитель в числителе:<br />$2a-8=2(a-4)$. Тогда:<br />$\\frac{2(a-4)}{3a}\\cdot\\frac{9a}{a-4}$.","Сократим $(a-4)$ и $a$:<br />$\\frac{2\\cdot 9}{3}$.","Выполним деление:<br />$\\frac{18}{3}=6$."]},{"instruction":"Выполните умножение","content":"$$\\frac{y^2-9}{4y}\\cdot\\frac{12y}{y+3}$$[[fill_choice_big-13]]","widgets":{"fill_choice_big-13":{"type":"fill_choice_big","options":["$3(y-3)$","$3(y+3)$","$\\frac{y-3}{3}$","$\\frac{3}{y-3}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов:<br />$y^2-9=(y-3)(y+3)$. Тогда:<br />$\\frac{(y-3)(y+3)}{4y}\\cdot\\frac{12y}{y+3}$.","Сократим $(y+3)$ и $y$:<br />$\\frac{y-3}{4}\\cdot 12$.","Сократим $12$ и $4$:<br />$3(y-3)$."]},{"instruction":"Выполните умножение","content":"$$\\frac{5}{2n}\\cdot\\frac{6n^2}{15}$$[[fill_choice_big-14]]","widgets":{"fill_choice_big-14":{"type":"fill_choice_big","options":["$n$","$\\frac{1}{n}$","$\\frac{n}{2}$","$\\frac{2}{3}n$"],"placeholder":0,"answer":0}},"step":1,"hints":["Перемножим дроби:<br />$\\frac{5}{2n}\\cdot\\frac{6n^2}{15}=\\frac{5\\cdot 6n^2}{2n\\cdot 15}$.","Сократим $5$ и $15$, затем сократим $n^2$ и $n$:<br />$\\frac{6n}{2\\cdot 3}$.","Сократим $6$ и $6$:<br />$n$."]},{"instruction":"Выполните умножение","content":"$$\\frac{r^2-2r}{7r}\\cdot\\frac{21r}{r-2}$$[[fill_choice_big-15]]","widgets":{"fill_choice_big-15":{"type":"fill_choice_big","options":["$3r$","$3(r-2)$","$\\frac{3}{r}$","$\\frac{r}{3}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим числитель на множители:<br />$r^2-2r=r(r-2)$. Тогда:<br />$\\frac{r(r-2)}{7r}\\cdot\\frac{21r}{r-2}$.","Сократим $(r-2)$ и один множитель $r$:<br />$\\frac{21r}{7}$.","Сократим $21$ и $7$:<br />$3r$."]}],"mix":1}

{"questions":[{"instruction":"Выполните деление","content":"$$\\frac{8x}{15}\\div\\frac{4}{9x}$$[[fill_choice_big-16]]","widgets":{"fill_choice_big-16":{"type":"fill_choice_big","options":["$\\frac{6x^2}{5}$","$\\frac{5}{6x^2}$","$\\frac{2x}{5}$","$\\frac{18x}{60}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{8x}{15}\\div\\frac{4}{9x}=\\frac{8x}{15}\\cdot\\frac{9x}{4}$.","Перемножим и сократим:<br />$\\frac{8x\\cdot 9x}{15\\cdot 4}=\\frac{72x^2}{60}$.","Сократим $72$ и $60$ на $12$. Получим:<br />$\\frac{6x^2}{5}$."]},{"instruction":"Выполните деление","content":"$$\\frac{3a-9}{10a}\\div\\frac{6}{a+2}$$[[fill_choice_big-17]]","widgets":{"fill_choice_big-17":{"type":"fill_choice_big","options":["$\\frac{3(a-3)(a+2)}{60a}$","$\\frac{(a-3)(a+2)}{20a}$","$\\frac{20a}{(a-3)(a+2)}$","$\\frac{a-3}{20a(a+2)}$"],"placeholder":0,"answer":1}},"step":1,"hints":["Разложим числитель на множители и заменим деление умножением:<br />$3a-9=3(a-3)$, поэтому<br />$\\frac{3(a-3)}{10a}\\div\\frac{6}{a+2}=\\frac{3(a-3)}{10a}\\cdot\\frac{a+2}{6}$.","Сократим $3$ и $6$ на $3$:<br />$\\frac{(a-3)(a+2)}{10a\\cdot 2}$ = $\\frac{(a-3)(a+2)}{20a}$."]},{"instruction":"Выполните деление","content":"$$\\frac{p^2-16}{3p}\\div\\frac{p-4}{6}$$[[fill_choice_big-18]]","widgets":{"fill_choice_big-18":{"type":"fill_choice_big","options":["$\\frac{2(p+4)}{p}$","$\\frac{p}{2(p+4)}$","$\\frac{p+4}{2p}$","$\\frac{2p}{p+4}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов и заменим деление умножением:<br />$p^2-16=(p-4)(p+4)$, поэтому<br />$\\frac{(p-4)(p+4)}{3p}\\div\\frac{p-4}{6}=\\frac{(p-4)(p+4)}{3p}\\cdot\\frac{6}{p-4}$.","Сократим $(p-4)$:<br />$\\frac{p+4}{3p}\\cdot 6$.","Сократим $6$ и $3$:<br />$\\frac{2(p+4)}{p}$."]},{"instruction":"Выполните деление","content":"$$\\frac{5}{2m^2}\\div\\frac{15}{8m}$$[[fill_choice_big-19]]","widgets":{"fill_choice_big-19":{"type":"fill_choice_big","options":["$\\frac{4}{3m}$","$\\frac{3m}{4}$","$\\frac{4m}{3}$","$\\frac{15}{16m^3}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{5}{2m^2}\\div\\frac{15}{8m}=\\frac{5}{2m^2}\\cdot\\frac{8m}{15}$.","Сократим $5$ и $15$ на $5$, затем сократим $8$ и $2$ на $2$, а также $m$ с $m^2$:<br />$\\frac{1\\cdot 4}{m\\cdot 3}$ = $\\frac{4}{3m}$."]},{"instruction":"Выполните деление","content":"$$\\frac{2y^2-8y}{5y}\\div\\frac{4y}{y-4}$$[[fill_choice_big-20]]","widgets":{"fill_choice_big-20":{"type":"fill_choice_big","options":["$\\frac{(y-4)^2}{10}$","$\\frac{2(y-4)}{5}$","$\\frac{y-4}{10}$","$\\frac{10}{(y-4)^2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим числитель на множители и заменим деление умножением:<br />$2y^2-8y=2y(y-4)$, поэтому<br />$\\frac{2y(y-4)}{5y}\\div\\frac{4y}{y-4}=\\frac{2y(y-4)}{5y}\\cdot\\frac{y-4}{4y}$.","Сократим $y$ в первой дроби, затем сократим $2$ и $4$ на $2$:<br />$\\frac{(y-4)}{5}\\cdot\\frac{y-4}{2y}$.","Перемножим:<br />$\\frac{(y-4)^2}{10y}$."]}]}

{"questions":[{"instruction":"Выполните деление","content":"$$\\frac{7x}{12}\\div\\frac{14}{3x}$$[[fill_choice_big-21]]","widgets":{"fill_choice_big-21":{"type":"fill_choice_big","options":["$\\frac{x^2}{8}$","$\\frac{8}{x^2}$","$\\frac{x}{8}$","$\\frac{7x^2}{56}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{7x}{12}\\div\\frac{14}{3x}=\\frac{7x}{12}\\cdot\\frac{3x}{14}$.","Сократим $7$ и $14$ на $7$:<br />$\\frac{x}{12}\\cdot\\frac{3x}{2}$.","Перемножим и сократим $3$ и $12$ на $3$:<br />$\\frac{x^2}{8}$."]},{"instruction":"Выполните деление","content":"$$\\frac{4a^2}{9b}\\div\\frac{8a}{3b}$$[[fill_choice_big-22]]","widgets":{"fill_choice_big-22":{"type":"fill_choice_big","options":["$\\frac{a}{6}$","$\\frac{6}{a}$","$\\frac{2a}{3}$","$\\frac{a}{2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{4a^2}{9b}\\div\\frac{8a}{3b}=\\frac{4a^2}{9b}\\cdot\\frac{3b}{8a}$.","Сократим $b$, затем сократим $a^2$ и $a$:<br />$\\frac{4a}{9}\\cdot\\frac{3}{8}$.","Сократим $4$ и $8$ на $4$, затем $3$ и $9$ на $3$:<br />$\\frac{a}{6}$."]},{"instruction":"Выполните деление","content":"$$\\frac{t^2-9}{5t}\\div\\frac{t-3}{10}$$[[fill_choice_big-23]]","widgets":{"fill_choice_big-23":{"type":"fill_choice_big","options":["$\\frac{2(t+3)}{t}$","$\\frac{t}{2(t+3)}$","$\\frac{t+3}{2t}$","$\\frac{2t}{t+3}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов и заменим деление умножением:<br />$t^2-9=(t-3)(t+3)$, поэтому<br />$\\frac{(t-3)(t+3)}{5t}\\div\\frac{t-3}{10}=\\frac{(t-3)(t+3)}{5t}\\cdot\\frac{10}{t-3}$.","Сократим $(t-3)$:<br />$\\frac{t+3}{5t}\\cdot 10$.","Сократим $10$ и $5$ на $5$:<br />$\\frac{2(t+3)}{t}$."]},{"instruction":"Выполните деление","content":"$$\\frac{6m}{5n^2}\\div\\frac{3m}{10n}$$[[fill_choice_big-24]]","widgets":{"fill_choice_big-24":{"type":"fill_choice_big","options":["$\\frac{4}{n}$","$\\frac{n}{4}$","$\\frac{4n}{1}$","$\\frac{2}{n}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{6m}{5n^2}\\div\\frac{3m}{10n}=\\frac{6m}{5n^2}\\cdot\\frac{10n}{3m}$.","Сократим $m$, затем сократим $6$ и $3$ на $3$:<br />$\\frac{2}{5n^2}\\cdot 10n$.","Сократим $10$ и $5$ на $5$, затем $n$ с $n^2$:<br />$\\frac{4}{n}$."]},{"instruction":"Выполните деление","content":"$$\\frac{3y^2-12y}{7y}\\div\\frac{6y}{y-4}$$[[fill_choice_big-25]]","widgets":{"fill_choice_big-25":{"type":"fill_choice_big","options":["$\\frac{(y-4)^2}{14y}$","$\\frac{(y-4)^2}{14}$","$\\frac{14y}{(y-4)^2}$","$\\frac{y-4}{14y}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим числитель на множители и заменим деление умножением:<br />$3y^2-12y=3y(y-4)$, поэтому<br />$\\frac{3y(y-4)}{7y}\\div\\frac{6y}{y-4}=\\frac{3y(y-4)}{7y}\\cdot\\frac{y-4}{6y}$.","Сократим $y$ в первой дроби, затем сократим $3$ и $6$ на $3$:<br />$\\frac{y-4}{7}\\cdot\\frac{y-4}{2y}$.","Перемножим:<br />$\\frac{(y-4)^2}{14y}$."]},{"instruction":"Выполните деление","content":"$$\\frac{9}{4k}\\div\\frac{3}{8k^2}$$[[fill_choice_big-26]]","widgets":{"fill_choice_big-26":{"type":"fill_choice_big","options":["$6k$","$\\frac{6}{k}$","$\\frac{k}{6}$","$\\frac{3k}{2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{9}{4k}\\div\\frac{3}{8k^2}=\\frac{9}{4k}\\cdot\\frac{8k^2}{3}$.","Сократим $9$ и $3$ на $3$:<br />$\\frac{3}{4k}\\cdot 8k^2$.","Сократим $8$ и $4$ на $4$, затем $k^2$ и $k$:<br />$6k$."]},{"instruction":"Выполните деление","content":"$$\\frac{2r-8}{3r}\\div\\frac{r-4}{9}$$[[fill_choice_big-27]]","widgets":{"fill_choice_big-27":{"type":"fill_choice_big","options":["$\\frac{6}{r}$","$\\frac{r}{6}$","$\\frac{6(r-4)}{r}$","$\\frac{6}{r-4}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим числитель и заменим деление умножением:<br />$2r-8=2(r-4)$, поэтому<br />$\\frac{2(r-4)}{3r}\\div\\frac{r-4}{9}=\\frac{2(r-4)}{3r}\\cdot\\frac{9}{r-4}$.","Сократим $(r-4)$:<br />$\\frac{2}{3r}\\cdot 9$.","Сократим $9$ и $3$ на $3$:<br />$\\frac{6}{r}$."]},{"instruction":"Выполните деление","content":"$$\\frac{5b}{6c}\\div\\frac{25}{12bc}$$[[fill_choice_big-28]]","widgets":{"fill_choice_big-28":{"type":"fill_choice_big","options":["$\\frac{2b^2}{5}$","$\\frac{5}{2b^2}$","$\\frac{2b}{5}$","$\\frac{5b^2}{2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{5b}{6c}\\div\\frac{25}{12bc}=\\frac{5b}{6c}\\cdot\\frac{12bc}{25}$.","Сократим $c$, затем сократим $5$ и $25$ на $5$:<br />$\\frac{b}{6}\\cdot\\frac{12b}{5}$.","Сократим $12$ и $6$ на $6$:<br />$\\frac{2b^2}{5}$."]},{"instruction":"Выполните деление","content":"$$\\frac{z^2-25}{2z}\\div\\frac{z+5}{4}$$[[fill_choice_big-29]]","widgets":{"fill_choice_big-29":{"type":"fill_choice_big","options":["$\\frac{2(z-5)}{z}$","$\\frac{z}{2(z-5)}$","$\\frac{z-5}{2z}$","$\\frac{2z}{z-5}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов и заменим деление умножением:<br />$z^2-25=(z-5)(z+5)$, поэтому<br />$\\frac{(z-5)(z+5)}{2z}\\div\\frac{z+5}{4}=\\frac{(z-5)(z+5)}{2z}\\cdot\\frac{4}{z+5}$.","Сократим $(z+5)$:<br />$\\frac{z-5}{2z}\\cdot 4$.","Сократим $4$ и $2$ на $2$:<br />$\\frac{2(z-5)}{z}$."]},{"instruction":"Выполните деление","content":"$$\\frac{8u}{15v}\\div\\frac{4u^2}{9v}$$[[fill_choice_big-30]]","widgets":{"fill_choice_big-30":{"type":"fill_choice_big","options":["$\\frac{6}{5u}$","$\\frac{6u}{5}$","$\\frac{5u}{6}$","$\\frac{6}{5u^2}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{8u}{15v}\\div\\frac{4u^2}{9v}=\\frac{8u}{15v}\\cdot\\frac{9v}{4u^2}$.","Сократим $v$, затем сократим $8$ и $4$ на $4$:<br />$\\frac{2u}{15}\\cdot\\frac{9}{u^2}$.","Сократим $9$ и $15$ на $3$, затем сократим $u$ и $u^2$:<br />$\\frac{6}{5u}$."]}]}

{"questions":[{"instruction":"Выполните деление","content":"$$\\frac{5p^2}{12q}\\div\\frac{15p}{8q}$$[[fill_choice_big-31]]","widgets":{"fill_choice_big-31":{"type":"fill_choice_big","options":["$\\frac{2p}{9}$","$\\frac{9}{2p}$","$\\frac{2p}{3}$","$\\frac{p}{9}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{5p^2}{12q}\\div\\frac{15p}{8q}=\\frac{5p^2}{12q}\\cdot\\frac{8q}{15p}$.","Сократим $q$, затем сократим $p^2$ и $p$:<br />$\\frac{5p}{12}\\cdot\\frac{8}{15}$.","Сократим $5$ и $15$ на $5$, затем $8$ и $12$ на $4$:<br />$\\frac{2p}{9}$."]},{"instruction":"Выполните деление","content":"$$\\frac{x^2-16}{6x}\\div\\frac{x-4}{9}$$[[fill_choice_big-32]]","widgets":{"fill_choice_big-32":{"type":"fill_choice_big","options":["$\\frac{3(x+4)}{2x}$","$\\frac{2x}{3(x+4)}$","$\\frac{3(x-4)}{2x}$","$\\frac{3x}{2(x+4)}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим разность квадратов и заменим деление умножением:<br />$x^2-16=(x-4)(x+4)$, поэтому<br />$\\frac{(x-4)(x+4)}{6x}\\div\\frac{x-4}{9}=\\frac{(x-4)(x+4)}{6x}\\cdot\\frac{9}{x-4}$.","Сократим $(x-4)$:<br />$\\frac{x+4}{6x}\\cdot 9$.","Сократим $9$ и $6$ на $3$:<br />$\\frac{3(x+4)}{2x}$."]},{"instruction":"Выполните деление","content":"$$\\frac{7a}{10b^2}\\div\\frac{14a^2}{5b}$$[[fill_choice_big-33]]","widgets":{"fill_choice_big-33":{"type":"fill_choice_big","options":["$\\frac{b}{4a}$","$\\frac{4a}{b}$","$\\frac{b}{2a}$","$\\frac{1}{4ab}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{7a}{10b^2}\\div\\frac{14a^2}{5b}=\\frac{7a}{10b^2}\\cdot\\frac{5b}{14a^2}$.","Сократим $7$ и $14$ на $7$, затем сократим $a$ и $a^2$, а также $b$ и $b^2$:<br />$\\frac{a}{10b^2}\\cdot\\frac{5b}{2a^2}=\\frac{1}{10b}\\cdot\\frac{5}{2a}$.","Сократим $5$ и $10$ на $5$:<br />$\\frac{1}{2b}\\cdot\\frac{1}{2a}=\\frac{1}{4ab}=\\frac{b}{4a}\\cdot\\frac{1}{b^2}$.<br />Запишем в упрощенном виде: $\\frac{b}{4a}$."]},{"instruction":"Выполните деление","content":"$$\\frac{3c^2-6c}{8c}\\div\\frac{3c}{c-2}$$[[fill_choice_big-34]]","widgets":{"fill_choice_big-34":{"type":"fill_choice_big","options":["$\\frac{(c-2)^2}{8c}$","$\\frac{8c}{(c-2)^2}$","$\\frac{c-2}{8c}$","$\\frac{(c-2)^2}{8}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Разложим числитель на множители и заменим деление умножением:<br />$3c^2-6c=3c(c-2)$, поэтому<br />$\\frac{3c(c-2)}{8c}\\div\\frac{3c}{c-2}=\\frac{3c(c-2)}{8c}\\cdot\\frac{c-2}{3c}$.","Сократим $c$ в первой дроби и сократим $3$:<br />$\\frac{c-2}{8}\\cdot\\frac{c-2}{c}$.","Перемножим:<br />$\\frac{(c-2)^2}{8c}$."]},{"instruction":"Выполните деление","content":"$$\\frac{10}{3s}\\div\\frac{5}{12s^2}$$[[fill_choice_big-35]]","widgets":{"fill_choice_big-35":{"type":"fill_choice_big","options":["$8s$","$\\frac{8}{s}$","$\\frac{s}{8}$","$\\frac{5s}{6}$"],"placeholder":0,"answer":0}},"step":1,"hints":["Заменим деление умножением на обратную дробь:<br />$\\frac{10}{3s}\\div\\frac{5}{12s^2}=\\frac{10}{3s}\\cdot\\frac{12s^2}{5}$.","Сократим $10$ и $5$ на $5$:<br />$\\frac{2}{3s}\\cdot 12s^2$.","Сократим $12$ и $3$ на $3$, затем $s^2$ и $s$:<br />$8s$."]}]}