Đề cương ôn tập giữa kì 2 Toán 6 Kết nối tri thức cấu trúc mới (Tự luận) có đáp án - Phần 1

a) \(\frac{4}{5} + \frac{{ - 6}}{5} = \frac{{4 + \left( { - 6} \right)}}{5} = \frac{{ - 2}}{5}.\)

b) \(\frac{{ - 1}}{8} + \frac{{ - 5}}{3}\)\( = \frac{{ - 3}}{{24}} + \frac{{ - 40}}{{24}}\)\( = \frac{{ - 43}}{{24}}\).

c) \(\frac{{ - 6}}{{35}} \cdot \frac{{ - 49}}{{54}}\)

\( = \frac{{ - 6 \cdot \left( { - 49} \right)}}{{35 \cdot 54}}\)

\( = \frac{7}{{45}}\).

d) \(\frac{{ - 4}}{5}:\frac{3}{4}\)

\( = \frac{{ - 4}}{5}.\frac{4}{3}\)

\( = \frac{{ - 16}}{{15}}\).

e) \(\frac{{ - 8}}{{13}} - \left( {\frac{3}{7} + \frac{5}{{13}}} \right)\)

\( = \frac{{ - 8}}{{13}} - \frac{3}{7} - \frac{5}{{13}}\)

\( = \frac{{ - 8}}{{13}} - \frac{5}{{13}} - \frac{3}{7}\)\( = - 1 - \frac{3}{7}\)

\( = \frac{{ - 7 - 3}}{7}\)\( = \frac{{ - 10}}{7}\).

f) \(\frac{1}{2} + \frac{{ - 3}}{8} + \frac{5}{9}\)

\( = \frac{{36}}{{72}} + \frac{{ - 27}}{{72}} + \frac{{40}}{{72}}\)

\( = \frac{{49}}{{72}}.\)

g) \(6\frac{2}{5} - \left( {2\frac{3}{7} + 3\frac{2}{5}} \right)\)

\( = \frac{{32}}{5} - \frac{{17}}{7} - \frac{{17}}{5}\)

\( = \frac{{32}}{5} - \frac{{17}}{5} - \frac{{17}}{7}\)

\( = 3 - \frac{{17}}{7}\)\( = \frac{{21 - 17}}{7}\)\( = \frac{4}{7}\).

h) \(\frac{2}{3} - \frac{5}{3} \cdot \frac{9}{{25}}\)

\( = \frac{2}{3} - \frac{{5 \cdot 9}}{{3 \cdot 25}} = \frac{2}{3} - \frac{3}{5}\)

\( = \frac{{10}}{{15}} - \frac{9}{{15}} = \frac{1}{{15}}.\)

i) \[\frac{4}{5} + \frac{9}{{15}}:\frac{1}{{15}}\]

\( = \frac{4}{5} + \frac{9}{{15}} \cdot 15\)

\( = \frac{4}{5} + 9\)

\( = \frac{4}{5} + \frac{{45}}{5}\)

\( = \frac{{49}}{5}.\)

j) \( - \frac{{10}}{{21}} \cdot \left[ {\frac{9}{{15}} + {{\left( {\frac{3}{5}} \right)}^2}} \right]\)

\( = - \frac{{10}}{{21}} \cdot \left[ {\frac{3}{5} + \frac{9}{{25}}} \right]\)

\( = \frac{{ - 10}}{{21}} \cdot \left[ {\frac{{15}}{{25}} + \frac{9}{{25}}} \right]\)

\( = \frac{{ - 10}}{{21}} \cdot \frac{{24}}{{25}}\)

\( = \frac{{ - 10 \cdot 24}}{{21 \cdot 25}} = \frac{{ - 16}}{{35}}.\)

k) \(\left( {\frac{1}{4} + \frac{{ - 5}}{{13}}} \right) + \left( {\frac{2}{{11}} + \frac{{ - 8}}{{13}} + \frac{3}{4}} \right)\)

\( = \frac{1}{4} + \frac{{ - 5}}{{13}} + \frac{2}{{11}} + \frac{{ - 8}}{{13}} + \frac{3}{4}\)

\( = \left( {\frac{1}{4} + \frac{3}{4}} \right) + \left( {\frac{{ - 5}}{{13}} + \frac{{ - 8}}{{13}}} \right) + \frac{2}{{11}}\)

\( = \frac{4}{4} + \frac{{ - 13}}{{13}} + \frac{2}{{11}}\)

\( = 1 + \left( { - 1} \right) + \frac{2}{{11}}\)\( = \frac{2}{{11}}.\)

l) \(\frac{{ - 3}}{{13}} + \frac{{ - 7}}{{17}} + \frac{{ - 5}}{{19}} + \frac{{ - 10}}{{13}} + \frac{{24}}{{17}} + \frac{{ - 14}}{{19}}\)

\[ = \left( {\frac{{ - 3}}{{13}} + \frac{{ - 10}}{{13}}} \right) + \left( {\frac{{ - 7}}{{17}} + \frac{{24}}{{17}}} \right) + \left( {\frac{{ - 5}}{{19}} + \frac{{ - 14}}{{19}}} \right)\]

\[ = \frac{{ - 13}}{{13}} + \frac{{17}}{{17}} + \frac{{ - 19}}{{19}}\]

\[ = \left( { - 1} \right) + 1 + \left( { - 1} \right)\]

\( = - 1.\)

m) \(\left( {\frac{5}{{17}} - \frac{8}{{33}}} \right) - \left( {\frac{{25}}{{33}} - \frac{{12}}{{17}}} \right) - {2021^0}\)

\( = \frac{5}{{17}} - \frac{8}{{33}} - \frac{{25}}{{33}} + \frac{{12}}{{17}} - 1\)

\[ = \left( {\frac{5}{{17}} + \frac{{12}}{{17}}} \right) - \left( {\frac{8}{{33}} + \frac{{25}}{{33}}} \right) - 1\]

\[ = \frac{{17}}{{17}} - \frac{{33}}{{33}} - 1\]

\[ = 1 - 1 - 1 = - 1.\]

n) $\frac{-4}{12}+\frac{18}{45}+\frac{-6}{9}+\frac{-21}{35}+\frac{6}{30}.$

n) \(\frac{{ - 4}}{{12}} + \frac{{18}}{{45}} + \frac{{ - 6}}{9} + \frac{{ - 21}}{{35}} + \frac{6}{{30}}\)

\( = \frac{{ - 1}}{3} + \frac{2}{5} + \frac{{ - 2}}{3} + \frac{{ - 3}}{5} + \frac{1}{5}\)\( = \left( {\frac{{ - 1}}{3} + \frac{{ - 2}}{3}} \right) + \left( {\frac{2}{5} + \frac{{ - 3}}{5} + \frac{1}{5}} \right)\)

\( = \frac{{ - 3}}{3} + \frac{0}{5} = - 1.\)

a) \(\frac{{ - 5}}{7} \cdot \frac{2}{{11}} + \frac{{ - 5}}{7} \cdot \frac{9}{{11}} + \frac{5}{7}\)

\[ = \frac{5}{7} \cdot \frac{{ - 2}}{{11}} + \frac{5}{7} \cdot \frac{{ - 9}}{{11}} + \frac{5}{7} \cdot 1\]

\[ = \frac{5}{7} \cdot \left( {\frac{{ - 2}}{{11}} + \frac{{ - 9}}{{11}} + 1} \right)\]

\[ = \frac{5}{7} \cdot \left( {\frac{{ - 11}}{{11}} + 1} \right) = \frac{5}{7} \cdot \left( { - 1 + 1} \right)\]

\( = \frac{5}{7} \cdot 0 = 0.\)

b) \(\frac{5}{9} \cdot \frac{{ - 4}}{7} - \frac{5}{9} \cdot \left( {\frac{{ - 18}}{7}} \right)\)

\( = \frac{5}{9} \cdot \frac{{ - 4}}{7} + \frac{5}{9} \cdot \frac{{18}}{7}\)

\( = \frac{5}{9} \cdot \left( {\frac{{ - 4}}{7} + \frac{{18}}{7}} \right)\)

\( = \frac{5}{9} \cdot \frac{{14}}{7}\)

\( = \frac{5}{9} \cdot 2 = \frac{{10}}{9}.\)

c) \[\frac{{11}}{{12}} \cdot \frac{7}{{15}} + \frac{{11}}{{12}} \cdot \frac{8}{{15}} - \frac{{23}}{{12}}\]

\( = \frac{{11}}{{12}} \cdot \left( {\frac{7}{{15}} + \frac{8}{{15}}} \right) - \frac{{23}}{{12}}\)

\( = \frac{{11}}{{12}} \cdot \frac{{15}}{{15}} - \frac{{23}}{{12}}\)

\( = \frac{{11}}{{12}} - \frac{{23}}{{12}}\)

\( = \frac{{ - 12}}{{12}} = - 1.\)

d) \( - \frac{3}{7} \cdot \frac{{12}}{{13}} + \frac{3}{7} \cdot \left( { - \frac{1}{{13}}} \right) + \frac{5}{{14}}\)

\( = \frac{3}{7} \cdot \frac{{ - 12}}{{13}} + \frac{3}{7} \cdot \left( { - \frac{1}{{13}}} \right) + \frac{5}{{14}}\)

\[ = \frac{3}{7} \cdot \left[ {\frac{{ - 12}}{{13}} + \left( { - \frac{1}{{13}}} \right)} \right] + \frac{5}{{14}}\]

\[ = \frac{3}{7} \cdot \frac{{ - 13}}{{13}} + \frac{5}{{14}}\]

\[ = \frac{3}{7} \cdot \left( { - 1} \right) + \frac{5}{{14}}\]\[ = \frac{{ - 3}}{7} + \frac{5}{{14}}\]

\[ = \frac{{ - 6}}{{14}} + \frac{5}{{14}} = \frac{{ - 1}}{{14}}.\]

e) \(\frac{{ - 9}}{4}:\frac{7}{2} + \frac{9}{4} \cdot \frac{{16}}{7} - \frac{2}{3}\)

\( = \frac{{ - 9}}{4} \cdot \frac{2}{7} + \frac{9}{4} \cdot \frac{{16}}{7} - \frac{2}{3}\)

\( = \frac{9}{4} \cdot \frac{{ - 2}}{7} + \frac{9}{4}.\frac{{16}}{7} - \frac{2}{3}\)

\( = \frac{9}{4} \cdot \left( {\frac{{ - 2}}{7} + \frac{{16}}{7}} \right) - \frac{2}{3}\)

\( = \frac{9}{4} \cdot \frac{{14}}{7} - \frac{2}{3} = \frac{9}{4} \cdot 2 - \frac{2}{3}\)

\( = \frac{9}{2} - \frac{2}{3} = \frac{{27}}{6} - \frac{4}{6} = \frac{{23}}{6}.\)

f) \(\frac{3}{{11}} \cdot \frac{9}{{17}} - \frac{{12}}{{11}}:\frac{{17}}{9} - \frac{{13}}{{11}} \cdot \frac{9}{{17}}\)

\[ = \frac{3}{{11}} \cdot \frac{9}{{17}} - \frac{{12}}{{11}} \cdot \frac{9}{{17}} - \frac{{13}}{{11}} \cdot \frac{9}{{17}}\]

\( = \frac{9}{{17}} \cdot \left( {\frac{3}{{11}} - \frac{{12}}{{11}} - \frac{{13}}{{11}}} \right)\)

\( = \frac{9}{{17}} \cdot \frac{{ - 22}}{{11}}\)

\( = \frac{9}{{17}} \cdot \left( { - 2} \right) = \frac{{ - 18}}{{17}}.\)

g) \(\frac{{17}}{8}:\frac{{ - 11}}{3} + \frac{{31}}{8} \cdot \frac{{ - 3}}{{11}} + \frac{4}{{11}}\)

\( = \frac{{17}}{8} \cdot \frac{{ - 3}}{{11}} + \frac{{31}}{8} \cdot \frac{{ - 3}}{{11}} + \frac{4}{{11}}\)

\( = \frac{{ - 3}}{{11}} \cdot \left( {\frac{{17}}{8} + \frac{{31}}{8}} \right) + \frac{4}{{11}}\)

\( = \frac{{ - 3}}{{11}} \cdot \frac{{48}}{8} + \frac{4}{{11}}\)

\( = \frac{{ - 3}}{{11}} \cdot 6 + \frac{4}{{11}}\)

\( = \frac{{ - 18}}{{11}} + \frac{4}{{11}} = \frac{{ - 14}}{{11}}.\)

h) \[\frac{7}{{13}} \cdot 1\frac{{14}}{{31}} - \frac{{37}}{{31}}:\frac{{13}}{7} + \frac{7}{{13}}\]

\( = \frac{7}{{13}} \cdot \frac{{45}}{{31}} - \frac{{37}}{{31}} \cdot \frac{7}{{13}} + \frac{7}{{13}}\)

\[ = \frac{7}{{13}} \cdot \left( {\frac{{45}}{{31}} - \frac{{37}}{{31}} + 1} \right)\]

\[ = \frac{7}{{13}} \cdot \frac{{39}}{{31}} = \frac{{21}}{{31}}.\]

i) \(\frac{4}{5}:\frac{1}{2} + \frac{1}{5}:\frac{1}{2} - \frac{7}{8}\)

\[ = \left( {\frac{4}{5} + \frac{1}{5}} \right):\frac{1}{2} - \frac{7}{8}\]

\[ = 1.2 - \frac{7}{8}\]

\[ = \frac{{16}}{8} - \frac{7}{8}\]\[ = \frac{9}{8}.\]

j) \[1\frac{1}{8}:\frac{{ - 9}}{2} + 2\frac{7}{8}:\frac{{ - 9}}{2} + \frac{5}{9}\]

\[ = \frac{9}{8} \cdot \frac{{ - 2}}{9} + \frac{{23}}{8} \cdot \frac{{ - 2}}{9} + \frac{5}{9}\]

\[ = \frac{{ - 2}}{9} \cdot \left( {\frac{9}{8} + \frac{{23}}{8}} \right) + \frac{5}{9}\]

\[ = \frac{{ - 2}}{9} \cdot \frac{{32}}{8} + \frac{5}{9}\]

\[ = \frac{{ - 2}}{9} \cdot 4 + \frac{5}{9}\]

\[ = \frac{{ - 8}}{9} + \frac{5}{9} = \frac{{ - 3}}{9} = \frac{{ - 1}}{3}.\]

k) \(\left( {\frac{4}{5} + \frac{{ - 9}}{7}} \right):\frac{{2024}}{{2025}} + \left( {\frac{{ - 5}}{7} - \frac{{ - 6}}{5}} \right):\frac{{2024}}{{2025}}\)

\[ = \left( {\frac{4}{5} + \frac{{ - 9}}{7} + \frac{{ - 5}}{7} - \frac{{ - 6}}{5}} \right):\frac{{2024}}{{2025}}\]

\[ = \left[ {\left( {\frac{4}{5} - \frac{{ - 6}}{5}} \right) + \left( {\frac{{ - 9}}{7} + \frac{{ - 5}}{7}} \right)} \right]:\frac{{2024}}{{2025}}\]

\[ = \left[ {\frac{{10}}{5} + \frac{{ - 14}}{7}} \right]:\frac{{2024}}{{2025}}\]

\[ = \left[ {2 + \left( { - 2} \right)} \right]:\frac{{2024}}{{2025}}\]

\[ = 0:\frac{{2024}}{{2025}} = 0.\]

a) \[\frac{{\frac{3}{4} + \frac{3}{5} + \frac{3}{7} - \frac{3}{{11}}}}{{\frac{6}{4} + \frac{6}{5} + \frac{6}{7} - \frac{6}{{11}}}} = \frac{{3 \cdot \left( {\frac{1}{4} + \frac{1}{4} + \frac{1}{7} - \frac{1}{{11}}} \right)}}{{6 \cdot \left( {\frac{1}{4} + \frac{1}{4} + \frac{1}{7} - \frac{1}{{11}}} \right)}} = \frac{3}{6} = \frac{1}{2}.\]

b) \[\frac{{\frac{1}{3} - \frac{1}{5} + \frac{1}{{10}}}}{{\frac{6}{3} - \frac{6}{5} + \frac{3}{5}}} + \frac{5}{6}\]\[ = \frac{{\frac{1}{3} - \frac{1}{5} + \frac{1}{{10}}}}{{6 \cdot \left( {\frac{1}{3} - \frac{1}{5} + \frac{1}{{10}}} \right)}} + \frac{5}{6} = \frac{1}{6} + \frac{5}{6} = \frac{6}{6} = 1.\]

c) \[\frac{{\frac{1}{2}.\frac{5}{{17}} - \frac{{13}}{{14}}.\frac{5}{{17}} + \frac{{15}}{{238}}}}{{\frac{{ - 20}}{{68}} + \frac{{26}}{{14}}.\frac{5}{{17}} - \frac{{15}}{{119}}}}\]\[ = \frac{{\frac{5}{{34}} - \frac{{65}}{{14.17}} + \frac{{15}}{{238}}}}{{ - 2 \cdot \left( {\frac{5}{{34}} - \frac{{65}}{{14.17}} + \frac{{15}}{{238}}} \right)}} =  - \frac{1}{2}\]

d) \(\frac{{\frac{1}{{29}} \cdot \frac{3}{2} - \frac{{26}}{{11}} \cdot \frac{3}{{23}} + \frac{9}{{238}}}}{{\frac{{ - 3}}{{29}} + \frac{{13}}{{11}} \cdot \frac{3}{{23}} - \frac{9}{{119}}}}\)\[ = \frac{{ - 2 \cdot \left( {\frac{3}{{58}} - \frac{{39}}{{11.23}} + \frac{9}{{238}}} \right)}}{{\frac{3}{{58}} - \frac{{39}}{{11.23}} + \frac{9}{{238}}}} =  - 2\]

e) \[\frac{1}{{2 \cdot 3}} + \frac{1}{{3 \cdot 4}} + \frac{1}{{4 \cdot 5}} + \frac{1}{{5 \cdot 6}} + \frac{1}{{6 \cdot 7}} + \frac{1}{{7 \cdot 8}}\]

\( = \frac{1}{2} - \frac{1}{3} + \frac{1}{3} - \frac{1}{4} + \frac{1}{4} - \frac{1}{5} + \frac{1}{5} - \frac{1}{6} + \frac{1}{6} - \frac{1}{7} + \frac{1}{7} - \frac{1}{8}\)

\( = \frac{1}{2} - \frac{1}{8} = \frac{4}{8} - \frac{1}{8} = \frac{3}{8}.\)

f) \(\frac{2}{{3.7}} + \frac{2}{{7.11}} + \frac{2}{{11.15}} + ... + \frac{2}{{63.67}}\)

\( = \frac{1}{2} \cdot \left( {\frac{4}{{3.7}} + \frac{4}{{7.11}} + \frac{4}{{11.15}} + ... + \frac{4}{{63.67}}} \right)\)

\( = \frac{1}{2} \cdot \left( {\frac{1}{3} - \frac{1}{7} + \frac{1}{7} - \frac{1}{{11}} + \frac{1}{{11}} - \frac{1}{{15}} + ... + \frac{1}{{63}} - \frac{1}{{67}}} \right)\)

\( = \frac{1}{2} \cdot \left( {\frac{1}{3} - \frac{1}{{67}}} \right) = \frac{1}{2} \cdot \left( {\frac{{67}}{{201}} - \frac{3}{{201}}} \right)\)

\[ = \frac{1}{2} \cdot \frac{{64}}{{201}} = \frac{1}{2} \cdot \frac{{32}}{{201}}\]

g) \(\frac{4}{{15}} + \frac{4}{{35}} + \frac{4}{{63}} + ... + \frac{4}{{399}}\)

\[ = \frac{4}{{3 \cdot 5}} + \frac{4}{{5 \cdot 7}} + \frac{4}{{7 \cdot 9}} + ... + \frac{4}{{19 \cdot 21}}\]

\[ = 2 \cdot \left( {\frac{2}{{3 \cdot 5}} + \frac{2}{{5 \cdot 7}} + \frac{2}{{7 \cdot 9}} + ... + \frac{2}{{19 \cdot 21}}} \right)\]

\[ = 2 \cdot \left( {\frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + ... + \frac{1}{{19}} - \frac{1}{{21}}} \right)\]

\[ = 2 \cdot \left( {\frac{1}{3} - \frac{1}{{21}}} \right)\]\[ = 2 \cdot \left( {\frac{7}{{21}} - \frac{1}{{21}}} \right)\]               

\[ = 2 \cdot \frac{6}{{21}} = 2 \cdot \frac{2}{7} = \frac{4}{7}.\]

h) \[\frac{1}{3} + \frac{1}{{15}} + \frac{1}{{35}} + \frac{1}{{63}} + \frac{1}{{99}} + \frac{1}{{143}} + \frac{1}{{195}}\]

\( = \frac{1}{{1 \cdot 3}} + \frac{1}{{3 \cdot 5}} + \frac{1}{{5 \cdot 7}} + \frac{1}{{7 \cdot 9}} + \frac{1}{{9 \cdot 11}} + \frac{1}{{11 \cdot 13}} + \frac{1}{{13 \cdot 15}}\)

\( = \frac{1}{2} \cdot \left( {\frac{2}{{1 \cdot 3}} + \frac{2}{{3 \cdot 5}} + \frac{2}{{5 \cdot 7}} + \frac{2}{{7 \cdot 9}} + \frac{2}{{9 \cdot 11}} + \frac{2}{{11 \cdot 13}} + \frac{2}{{13 \cdot 15}}} \right)\)

\[ = \frac{1}{2} \cdot \left( {\frac{1}{1} - \frac{1}{3} + \frac{1}{3} - \frac{1}{5} + \frac{1}{5} - \frac{1}{7} + \frac{1}{7} - \frac{1}{9} + \frac{1}{9} - \frac{1}{{11}} + \frac{1}{{11}} - \frac{1}{{13}} + \frac{1}{{13}} - \frac{1}{{15}}} \right)\]

\[ = \frac{1}{2} \cdot \left( {\frac{1}{1} - \frac{1}{{15}}} \right) = \frac{1}{2} \cdot \left( {\frac{{15}}{{15}} - \frac{1}{{15}}} \right)\]

\[ = \frac{1}{2} \cdot \frac{{14}}{{15}} = \frac{7}{{15}}.\]

i) \[\left( {1 - \frac{1}{2}} \right) \cdot \left( {1 - \frac{1}{3}} \right) \cdot \left( {1 - \frac{1}{4}} \right) \cdot \left( {1 - \frac{1}{5}} \right) \cdot \left( {1 - \frac{1}{6}} \right) \cdot \left( {1 - \frac{1}{7}} \right) \cdot \left( {1 - \frac{1}{8}} \right)\]

\[ = \left( {\frac{2}{2} - \frac{1}{2}} \right) \cdot \left( {\frac{3}{3} - \frac{1}{3}} \right) \cdot \left( {\frac{4}{4} - \frac{1}{4}} \right) \cdot \left( {\frac{5}{5} - \frac{1}{5}} \right) \cdot \left( {\frac{6}{6} - \frac{1}{6}} \right) \cdot \left( {\frac{7}{7} - \frac{1}{7}} \right) \cdot \left( {\frac{8}{8} - \frac{1}{8}} \right)\]

\[ = \frac{1}{2} \cdot \frac{2}{3} \cdot \frac{3}{4} \cdot \frac{4}{5} \cdot \frac{5}{6} \cdot \frac{6}{7} \cdot \frac{7}{8}\]\[ = \frac{1}{8}.\]

Hướng dẫn giải:

j) \(\left( {1 + \frac{1}{{10}}} \right) \cdot \left( {1 + \frac{1}{{11}}} \right) \cdot \left( {1 + \frac{1}{{12}}} \right) \cdot ... \cdot \left( {1 + \frac{1}{{59}}} \right) \cdot \left( {1 + \frac{1}{{60}}} \right)\)

\[ = \left( {\frac{{10}}{{10}} + \frac{1}{{10}}} \right) \cdot \left( {\frac{{11}}{{11}} + \frac{1}{{11}}} \right) \cdot \left( {\frac{{12}}{{12}} + \frac{1}{{12}}} \right) \cdot ... \cdot \left( {\frac{{59}}{{59}} + \frac{1}{{59}}} \right) \cdot \left( {\frac{{60}}{{60}} + \frac{1}{{60}}} \right)\]

\[ = \frac{{11}}{{10}} \cdot \frac{{12}}{{11}} \cdot \frac{{13}}{{12}} \cdot ... \cdot \frac{{60}}{{59}} \cdot \frac{{61}}{{60}}\]\[ = \frac{{61}}{{10}}.\]