2 bài tập So sánh các phân số (có lời giải)

a) \(\frac{3}{5}\) < 1 b) \(\frac{7}{6}\,\, > 1\)                            c) \(\frac{4}{5}\) < 1     d) \(\frac{9}{{14}}\) < 1

e) \(\frac{8}{3}\) > 1 f) \(\frac{{14}}{3}\, > \frac{{10}}{3}\)     g) \(\frac{5}{7} > \frac{3}{7}\)                      h) \(\frac{7}{5} < \frac{9}{5}\)

i) \(\frac{{218}}{{376}} < \frac{{218}}{{367}}\)                       j) \(\frac{{721}}{{218}} < \frac{{721}}{{215}}\)                         

k)

Vì: \(\frac{{1999}}{{2003}} < 1 < \frac{9}{8}\) Nên: \[\frac{{1999}}{{2003}} < \frac{9}{8}\]   

l)

Vì: \(\frac{{12}}{{31}} < 1 < \frac{{22}}{{21}}\) Nên: \[\frac{{12}}{{31}} < \frac{{22}}{{21}}\]

m)                            

Vì: \(\frac{{47}}{{52}} > \frac{{47}}{{58}} > \frac{{45}}{{58}}\) Nên: \(\frac{{47}}{{52}} > \frac{{45}}{{58}}\)

n)                             

\(\frac{{2015}}{{2011}} - 1 = \frac{4}{{2011}}\)

\(\frac{{1996}}{{1992}} - 1 = \frac{4}{{1992}}\)

Vì: \(\frac{4}{{2011}} < \frac{4}{{1992}}\) Nên: \(\frac{{2015}}{{2011}}\,\, < \frac{{1996}}{{1992}}\)

o)

Vì: \(\frac{{1998}}{{1995}} > 1 > \frac{{2015}}{{2016}}\) Nên: \(\frac{{1998}}{{1995}} > \frac{{2015}}{{2016}}\)

p)

Ta có:

\(\frac{{222}}{{333}} = \frac{{222:111}}{{333:111}} = \frac{2}{3} = 1 - \frac{1}{3}\)

\(\frac{{333}}{{444}} = \frac{{333:111}}{{444:111}} = \frac{3}{4} = 1 - \frac{1}{4}\)

Vì \(\frac{1}{3} > \frac{1}{4}\) nên \(\frac{2}{3} < \frac{3}{4}\) hay \(\frac{{222}}{{333}} < \frac{{333}}{{444}}\)

q)

Vì: \(\frac{{1313}}{{1515}} < \frac{{1326}}{{1515}} < \frac{{1326}}{{1428}}\) nên \(\frac{{1313}}{{1515}} < \frac{{1326}}{{1428}}\)

r)

\(\frac{{119}}{{120}} = 1 - \frac{1}{{120}}\)

\(\frac{{118}}{{119}} = 1 - \frac{1}{{119}}\)

Vì: \(\frac{1}{{120}} < \frac{1}{{119}}\) Nên: \(\frac{{119}}{{120}} > \frac{{118}}{{119}}\)

s)

\(\frac{{222}}{{555}} = \frac{{222:111}}{{555:111}} = \frac{2}{5}\)

\(\frac{{333}}{{444}} = \frac{{333:111}}{{444:111}} = \frac{3}{4}\)

Vì \(\frac{2}{5} < \frac{3}{5} < \frac{3}{4}\) nên \(\frac{2}{5} < \frac{3}{4}\) hay \(\frac{{222}}{{555}} < \frac{{333}}{{444}}\)

a)

Quy đồng mẫu số các phân số với mẫu số chung bằng 120

\(\frac{2}{3} = \frac{{2 \times 40}}{{3 \times 40}} = \frac{{80}}{{120}}\);   \(\frac{5}{4} = \frac{{5 \times 30}}{{4 \times 30}} = \frac{{150}}{{120}}\);                               \(\frac{3}{8} = \frac{{3 \times 15}}{{8 \times 15}} = \frac{{45}}{{120}}\);

\(\frac{7}{5} = \frac{{7 \times 24}}{{5 \times 24}} = \frac{{168}}{{120}}\);   \[\frac{6}{5} = \frac{{6 \times 24}}{{5 \times 24}} = \frac{{144}}{{120}}\]

So sánh: \[\frac{{45}}{{120}} < \frac{{80}}{{120}} < \frac{{144}}{{120}} < \frac{{150}}{{120}} < \frac{{168}}{{120}}\]

Vậy: Sắp xếp các phân số theo thứ tự tăng dần là: \[\frac{3}{8};\frac{2}{3};\frac{6}{5};\frac{5}{4};\frac{7}{5}\]

b)

Quy đồng mẫu số các phân số với mẫu số chung bằng 72

\[\frac{3}{4} = \frac{{3 \times 18}}{{4 \times 18}} = \frac{{24}}{{72}}\]; \[\frac{{11}}{8} = \frac{{11 \times 9}}{{8 \times 9}} = \frac{{99}}{{72}}\]; \[\frac{5}{6} = \frac{{5 \times 12}}{{6 \times 12}} = \frac{{60}}{{72}}\]; \[\frac{2}{9} = \frac{{2 \times 8}}{{9 \times 8}} = \frac{{16}}{{72}}\]; \[\frac{8}{3} = \frac{{8 \times 24}}{{3 \times 24}} = \frac{{192}}{{72}}\]

So sánh: \[\frac{{16}}{{72}} < \frac{{24}}{{72}} < \frac{{60}}{{72}} < \frac{{99}}{{72}} < \frac{{192}}{{72}}\]

Vậy: Sắp xếp các phân số theo thứ tự tăng dần là: \[\frac{2}{9};\frac{3}{4};\frac{5}{6};\frac{{11}}{8};\frac{8}{3}\]

c)

Nhận thấy: \[1 < \frac{{26}}{{15}} < \frac{{26}}{{11}}\]; \[1 > \frac{{215}}{{253}} > \frac{{162}}{{253}}\]

Do đó: \[\frac{{26}}{{11}} > \frac{{26}}{{15}} > \frac{{18}}{{18}} > \frac{{215}}{{253}} > \frac{{162}}{{253}}\]

Vậy: Sắp xếp các phân số theo thứ tự tăng dần là: \[\frac{{162}}{{253}};\frac{{215}}{{253}};\frac{{18}}{{18}};\frac{{26}}{{15}};\frac{{26}}{{11}}\]