Курс:
01. Рационал функциялар
№ 1 Есептеңіз: ${3,7 - (1,133:1,1 - 1,2):}$ ${( - 0,02) \cdot 0,2}$
$${1,133:1,1 = \dfrac{{1133}}{{1000}} \cdot \dfrac{{10}}{{11}} = \dfrac{{1100 + 33}}{{100 \cdot 11}} = \dfrac{{103}}{{100}} = 1,03}$$
$${1,03 - 1,2 = - 0,17}$$
$${ - 0,17:( - 0,02) \cdot 0,2 = \dfrac{{17}}{{100}} \cdot \dfrac{{100}}{2} \cdot \dfrac{2}{{10}} = 1,7}$$
$${3,7 - 1,7 = 2}$$
№ 2 Есептеңіз: $\dfrac{{(4,3 - 0,64:1,6) \cdot 0,25}}{{\dfrac{{25}}{{16}}:2,5 + 0,375 \cdot \dfrac{1}{3}}}$
$$ = \dfrac{{(4,3 - 0,4) \cdot 0,25}}{{\dfrac{{25}}{{16}} \cdot \dfrac{2}{5} + \dfrac{3}{8} \cdot \dfrac{1}{3}}} $$
$$=\dfrac{{3,9 \cdot 0,25}}{{\dfrac{5}{8} + \dfrac{1}{8}}} = \dfrac{{3,9 \cdot \dfrac{1}{4}}}{{\dfrac{6}{8}}} $$
$$= \dfrac{{3,9 \cdot \dfrac{1}{4}}}{{\dfrac{3}{4}}} = 1,3$$
№ 3 Есептеңіз: ${2\dfrac{2}{3}:1\dfrac{7}{9} + \dfrac{{55}}{{84}}:\left( {\dfrac{{43}}{{63}} - \dfrac{{23}}{{36}}} \right)}$
${ = \dfrac{8}{3}:\dfrac{{16}}{9} + \dfrac{{55}}{{84}}:\left( {\dfrac{{43}}{{9 \cdot 7}} - \dfrac{{23}}{{9 \cdot 4}}} \right) }$ $$=\dfrac{8}{3} \cdot \dfrac{9}{{16}} + \dfrac{{55}}{{84}}:\dfrac{{43 \cdot 4 - 23 \cdot 7}}{{9 \cdot 7 \cdot 4}} = $$
$$ = \dfrac{3}{2} + \dfrac{{55}}{{84}}:\dfrac{{9 \cdot 7 \cdot 4}}{{172 - 161}} $$ $$= \dfrac{3}{2} + \dfrac{{55}}{3} \cdot \dfrac{9}{{11}} = \dfrac{3}{2} + 15 = 16,5$$
№ 4 Есептеңіз: ${5\dfrac{5}{7}:\dfrac{8}{{21}} + 1\dfrac{8}{{13}} \cdot \left( {\dfrac{{43}}{{56}} - \dfrac{{11}}{{24}}} \right)}$
$$ = \dfrac{{40}}{7} \cdot \dfrac{{21}}{8} + \dfrac{{21}}{{13}} \cdot \left( {\dfrac{{43}}{{8 \cdot 7}} - \dfrac{{11}}{{8 \cdot 3}}} \right) $$ $$= 15 + \dfrac{{21}}{{13}} \cdot \left( {\dfrac{{43 \cdot 3 - 7 \cdot 11}}{{8 \cdot 7 \cdot 3}}} \right) = $$
$$ = 15 + \dfrac{{21}}{{13}} \cdot \dfrac{{52}}{{8 \cdot 21}} = 15 + \dfrac{4}{8} = 15,5$$
№ 5 Есептеңіз: ${\left( {\dfrac{{41}}{{18}} - \dfrac{{17}}{{36}}} \right) \cdot \dfrac{{18}}{{65}} +}$ $ {\left( {\dfrac{8}{7} - \dfrac{{23}}{{49}}} \right):\dfrac{{99}}{{49}} + \dfrac{7}{6}}$
$$ = \dfrac{{41 \cdot 2 - 17}}{{36}} \cdot \dfrac{{18}}{{65}} + \dfrac{{8 \cdot 7 - 23}}{{49}} \cdot \dfrac{{49}}{{99}} + \dfrac{7}{6} $$ $$= \dfrac{{18}}{{36}} + \dfrac{{33}}{{99}} + \dfrac{7}{6} = $$
$$ = \dfrac{1}{2} + \dfrac{1}{3} + \dfrac{7}{6} = \dfrac{{3 + 2 + 7}}{6} $$ $$ =\dfrac{{12}}{6} = 2$$
№ 6 Есептеңіз: ${\left( {\dfrac{1}{2} + 0,125 - \dfrac{1}{6}} \right) }$ ${\cdot \left( {6,4:\dfrac{{80}}{3}} \right) + \dfrac{1}{8}}$
$$ = \left( {\dfrac{1}{2} + \dfrac{1}{8} - \dfrac{1}{6}} \right) \cdot \left( {\dfrac{{32}}{5} \cdot \dfrac{3}{{80}}} \right) + \dfrac{1}{8} $$ $$ = \dfrac{{24 + 6 - 8}}{{48}} \cdot \dfrac{6}{{25}} + \dfrac{1}{8} = \dfrac{{22}}{{48}} \cdot \dfrac{6}{{25}} + \dfrac{1}{8} $$
$$= \dfrac{{11}}{{4 \cdot 25}} + \dfrac{1}{8} = $$ $$ = \dfrac{{110}}{{1000}} + \dfrac{{125}}{{1000}} = 0,235$$
№ 7 Есептеңіз: ${\left( {6\dfrac{2}{3} + 2\dfrac{4}{{15}} + 5\dfrac{1}{2}} \right):\dfrac{1}{{15}} - }$ ${30:\dfrac{5}{{28}}}$
$$ = \left( {\dfrac{{20}}{3} + \dfrac{{34}}{{15}} + \dfrac{{11}}{2}} \right) \cdot 15 - 30 \cdot \dfrac{{28}}{5} $$ $$ = \dfrac{{20 \cdot 10 + 2 \cdot 34 + 15 \cdot 11}}{{30}} \cdot 15 - 6 \cdot 28 $$
$$\dfrac{{268 + 165}}{2} - 168 = 134 + \dfrac{{165}}{2} - 168 $$ $$= 82\dfrac{1}{2} - 34 = 48\dfrac{1}{2}$$
№ 8 Есептеңіз: ${\left( {13\dfrac{{13}}{{15}} - 12\dfrac{3}{{20}} - 5\dfrac{4}{{45}} - 0,85} \right) \cdot 3}$
$$ = \left( {3\dfrac{{13}}{{15}} - 5\dfrac{4}{{45}} - \left( {12\dfrac{3}{{20}} + \dfrac{{85}}{{100}}} \right)} \right) \cdot 3 $$ $$= \left( {3\dfrac{{39}}{{45}} - 4\dfrac{{49}}{{45}} - \left( {12\dfrac{3}{{20}} + \dfrac{{17}}{{20}}} \right)} \right) \cdot 3 $$
$$ = \left( { - 1\dfrac{{10}}{{45}} - 13} \right) \cdot 3 = - 14\dfrac{2}{9} \cdot 3 $$ $$= - \dfrac{{126 + 2}}{9} \cdot 3 = - \dfrac{{128}}{3} = - 42\dfrac{2}{3}$$
№ 9 Есептеңіз: ${ - 7,8 - 1,3 \cdot (19,6:1,4 - 20)}$
$$ = - 7,8 - 1,3 \cdot (14 - 20) $$ $$ =- 7,8 - 1,3 \cdot ( - 6) $$
$$= - 7,8 + 7,8 = 0$$
№ 10 Есептеңіз: ${\left( {3,24:\dfrac{9}{7} - 3\dfrac{1}{5}:1\dfrac{1}{3}} \right):0,9}$
$$ = \left( {3,24 \cdot \dfrac{7}{9} - \dfrac{{16}}{5} \cdot \dfrac{3}{4}} \right) \cdot \dfrac{{10}}{9} = $$
$$=\left( {\dfrac{{324}}{{100}} \cdot \dfrac{7}{9} - \dfrac{{12}}{5}} \right) \cdot \dfrac{{10}}{9} $$ $$= \left( {\dfrac{{81}}{{25}} \cdot \dfrac{7}{9} - \dfrac{{12}}{5}} \right) \cdot \dfrac{{10}}{9} $$
$$ =\left( {\dfrac{{63}}{{25}} - \dfrac{{12 \cdot 5}}{{25}}} \right) \cdot \dfrac{{10}}{9} $$ $$=\dfrac{3}{{25}} \cdot \dfrac{{10}}{9} = \dfrac{2}{{15}}$$
№ 11 Өрнектің мәніне кері санды табыңыз: $\dfrac{5}{{14}}:\dfrac{{25}}{{72}} \cdot \dfrac{7}{{12}}:\dfrac{5}{{36}} \cdot 2\dfrac{1}{{12}}$
$$ = \dfrac{5}{{14}} \cdot \dfrac{{72}}{{25}} \cdot \dfrac{7}{{72}} \cdot \dfrac{{36}}{5} \cdot \dfrac{{25}}{{12}} = \dfrac{{18}}{2} = 9$$
Кері сан : $\dfrac{1}{9}$
№ 12 Мына санды периодты ондық ондық бөлшек түрінде жазыңыз: $\dfrac{{53}}{{22}}$
$$\dfrac{{53}}{{22}} = 2,40909… = 2,4\left( {09} \right)$$
№ 13 ЕҮОБ (42; 140; 882) табыңыз.
№ 14 ЕҮОБ (42; 140; 882)
$$2 \cdot 7 = 14$$
№ 15 ЕКОЕ (54; 81; 135; 189) табыңыз.
$$27 \cdot 2 \cdot 3 \cdot 5 \cdot 7 = 54 \cdot 105 = 5670$$
№ 16 ЕКОЕ (156; 195; 1950)
$$39 \cdot 2 \cdot 2 \cdot 5 \cdot 5 = 3900$$
№ 17 Есептеңіз:
$80 - \left( {\left( {\dfrac{4}{5} \cdot 7 + 0,64} \right) \cdot \left( {1,25 \cdot 7 - \dfrac{4}{5} \cdot 1,25} \right) + 31,64} \right)$
$$ = 80 - \left( {\dfrac{4}{5} \cdot \left( {7 + \dfrac{4}{5}} \right) \cdot \dfrac{5}{4} \cdot \left( {7 - \dfrac{4}{5}} \right) + 31,64} \right) = $$
$$ = 80 - \left( {{7^2} - {{\left( {\dfrac{4}{5}} \right)}^2} + 31,64} \right) =$$ $$= 80 - 49 + \dfrac{{16}}{{25}} - 31\dfrac{{16}}{{25}} = 31 - 31 = 0$$
№ 18 Есептеңіз:
${\left( {22,385:3,7 - 2,9 \cdot 1\dfrac{2}{3}} \right):9\dfrac{{11}}{{15}}}$
$$\left( {\dfrac{{22385}}{{37000}} - \dfrac{{29}}{{10}} \cdot \dfrac{5}{3}} \right):\dfrac{{146}}{{15}} =$$ $$= \left( {\dfrac{{121}}{{20}} - \dfrac{{29}}{6}} \right) \cdot \dfrac{{15}}{{146}}$$
$$= \dfrac{{363 - 290}}{{60}} \cdot \dfrac{{15}}{{146}} =$$ $$= \dfrac{{73}}{4} \cdot \dfrac{1}{{146}} = \dfrac{1}{8}$$
№ 19 Есептеңіз:
$\dfrac{{45\dfrac{{10}}{{63}} - 44\dfrac{{25}}{{84}}}}{{\dfrac{3}{4} - \left( {2\dfrac{1}{3} - 1\dfrac{1}{9}} \right):4}}:31$
$$ = \dfrac{{45\dfrac{{10}}{{21 \cdot 3}} - 44\dfrac{{25}}{{21 \cdot 4}}}}{{\dfrac{3}{4} - \left( {2\dfrac{3}{9} - 1\dfrac{1}{9}} \right):4}}:31 =$$ $$= \dfrac{{1\dfrac{{40 - 75}}{{21 \cdot 12}}}}{{\dfrac{3}{4} - 1\dfrac{2}{9}:4}}:31 = $$
$$ = \dfrac{{\dfrac{{252 + 40 - 75}}{{252}}}}{{\dfrac{3}{4} - \dfrac{{11}}{9} \cdot \dfrac{1}{4}}}:31 = \dfrac{{\dfrac{{217}}{{252}}}}{{\dfrac{{27}}{{36}} - \dfrac{{11}}{{36}}}}:31= $$ $$ = \dfrac{{217}}{{252}} \cdot \dfrac{{36}}{{16}}:31 = \dfrac{{217}}{{7 \cdot 16}}:31 =$$ $$= \dfrac{{31}}{{16}} \cdot \dfrac{1}{{31}} = \dfrac{1}{{16}}$$
№ 20 Есептеңіз:
${\dfrac{{12\dfrac{4}{5} \cdot 3\dfrac{3}{4} - 4\dfrac{4}{{11}} \cdot 4,125}}{{2\dfrac{4}{7}:\dfrac{3}{{35}}}}}$
$$=\dfrac{{\dfrac{{64}}{5} \cdot \dfrac{{15}}{4} - \dfrac{{48}}{{11}} \cdot 4\dfrac{1}{8}}}{{\dfrac{{18}}{7} \cdot \dfrac{{35}}{3}}} $$ $$ = \dfrac{{48 - \dfrac{{48}}{{11}} \cdot \dfrac{{33}}{8}}}{{30}} $$
$$= \dfrac{{48 - 3\dfrac{4}{4} - 4\dfrac{4}{{11}} \cdot 4,125}}{{2\dfrac{4}{7}:\dfrac{3}{{35}}}} = 1$$
№ 21 ${71 \cdot 72 \cdot 73 \cdot \ldots \cdot 79}$ саны қандай цифрмен аяқталады?
${72 \cdot 75 = \ldots 0}$, 0-мен аяқталады.
№ 22 Пропорциядан $x$-ті табыңыз: $\dfrac{{3,2}}{{\dfrac{4}{9}}} = \dfrac{x}{{\dfrac{{216 - 21}}{{90}}}}$
$$x = \dfrac{{\dfrac{{195}}{{90}} \cdot 3,2}}{{\dfrac{4}{9}}} $$ $$= \dfrac{{\dfrac{{195 \cdot 3,2}}{{10}}}}{4} $$
$$= \dfrac{{195 \cdot 3,2}}{{40}} = \dfrac{{195 \cdot 0,8}}{{10}} $$ $$= 19,5 \cdot 0,8 = 15,6$$
№ 23 Пропорциядан $x$-ті табыңыз: $\dfrac{{\dfrac{7}{8}}}{{\dfrac{7}{9}}} = \dfrac{x}{{\dfrac{{316 - 31}}{{90}}}}$
$$x = \dfrac{{\dfrac{7}{8} \cdot \dfrac{{285}}{{90}}}}{{\dfrac{7}{9}}} $$
$$= \dfrac{1}{8} \cdot \dfrac{{285}}{{10}} = \dfrac{{57}}{{16}} = 3\dfrac{9}{{16}}$$
№ 24 Пропорциядан $x$-ті табыңыз: $\dfrac{{\dfrac{7}{9}}}{{\dfrac{{31 - 3}}{9}}} = \dfrac{{1\dfrac{7}{8}}}{x}$
$$x = \dfrac{{\dfrac{{28}}{9} \cdot \dfrac{{15}}{8}}}{{\dfrac{7}{9}}} $$
$$= \dfrac{{15}}{2} = 7,5$$
№ 25 Есептеңіз:
$\dfrac{{\left( {\dfrac{2}{3} + 0,(3)} \right):0,25}}{{0,12(3):0,0925}}$
$$\dfrac{{\left( {\dfrac{2}{3} + \dfrac{1}{3}} \right):\dfrac{1}{4}}}{{\dfrac{{123 - 12}}{{900}}:\dfrac{{37}}{{400}}}} $$
$$= \dfrac{4}{{\dfrac{{111}}{{900}} \cdot \dfrac{{400}}{{37}}}} = \dfrac{4}{{\dfrac{4}{3}}} = 3$$
№ 26 $m$ санының $72\% $-ын табыңыз:
$m = \dfrac{{\left( {13\dfrac{1}{4} - 2\dfrac{5}{{27}} - 10\dfrac{5}{6}} \right) \cdot 230,04 + 46,75}}{{0,01}}$
$$m = \left( {\left( {13\dfrac{1}{4} - 2\dfrac{5}{{27}} - 10\dfrac{5}{6}} \right) \cdot 230,04 + 46,75} \right) \cdot 100 = $$ $$ = \left( {13\dfrac{1}{4} - 2\dfrac{5}{{27}} - 10\dfrac{5}{6}} \right) \cdot 23004 + 4675 = $$ $$ = 1\dfrac{{27 - 20 - 90}}{{108}} \cdot 23004 + 4675 $$
$$=\dfrac{{25}}{{108}} \cdot 23004 + 4675 = 25 \cdot 213 + 4675$$ $${ = 5325 + 4675 = 10000}$$ $${m = 10000;\,\,\,m \cdot 72\% = 10000 \cdot \dfrac{{72}}{{100}} = 7200}$$
№ 27 Пропорциядан $x$-ті табыңыз:
$\dfrac{{3,6}}{{14 - 15\dfrac{1}{8}:2,2}} = \dfrac{x}{{1,5 + 2\dfrac{2}{3} + 3,75}}$
$$\dfrac{{3,6}}{{14 - \dfrac{{121}}{8} \cdot \dfrac{{10}}{{22}}}} = \dfrac{x}{{\dfrac{3}{2} + \dfrac{8}{3} + 3\dfrac{3}{4}}}$$ $$\dfrac{{\dfrac{{18}}{5}}}{{14 - \dfrac{{55}}{8}}} = \dfrac{x}{{1\dfrac{1}{2} + 2\dfrac{2}{3} + 3\dfrac{3}{4}}} = $$
$$x = \dfrac{{\dfrac{{18}}{5} + \dfrac{{11}}{{12}}}}{{13\dfrac{8}{8} - 6\dfrac{7}{8}}} = \dfrac{{\dfrac{{18}}{5} \cdot \dfrac{{95}}{{12}}}}{{7 - \dfrac{1}{8}}} $$ $$= \dfrac{{\dfrac{{19 \cdot 3}}{{\dfrac{2}{{57}}}}}}{{\dfrac{5}{8}}} = \dfrac{{157}}{2} \cdot \dfrac{8}{{57}} = 4$$
№ 28 $3,6\% $-ы $\dfrac{{3 + 4,2:0,1}}{{\left( {1:0,3 - 2\dfrac{1}{3}} \right) \cdot 0,3125}}$ өрнегінің мәніне тең санды табыңыз.
$$\dfrac{{3 + 42}}{{\left( {\dfrac{{10}}{3} - \dfrac{7}{3}} \right) \cdot \dfrac{5}{{16}}}} = \dfrac{{45}}{{\dfrac{5}{{16}}}} = 45 \cdot \dfrac{{16}}{5} = 144$$
$$144:3,6\% = 144 \cdot \dfrac{{1000}}{{36}} = 4000$$
№ 29 \( \text{ЕҮОБ} \left( {38;b} \right) = 4,\quad \text{ЕКОЕ} \left( {38;b} \right) = 1216\). $b$-ны табыңыз.
$$38 = 2 \cdot 19;\,\,\,\,b = 2 \cdot \dfrac{b}{2}$$
$$19 \cdot 2 \cdot \dfrac{b}{2} = 1216,$$ $$b = 1216:19 = 64$$
№ 30 ${\text{ЕҮОБ}}\left( {68;b} \right) = 4,\,\,\,{\text{ЕКОЕ}}\left( {68;b} \right) = 1292$. $b$-ны табыңыз.
$$68 = 4 \cdot 17;\,\,\,\,b = 4 \cdot \dfrac{b}{4}$$
$$17 \cdot 4 \cdot \dfrac{b}{4} = 1292,$$ $$b = 1292:17 = 76$$
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