6. \(13\times99+135\times999+1357\times9999=13570000+135000+1300-13-135-1357=13704795\)
7. \(529\times8.08-529\times7.08=529\times(8.08-7.08)=529\)
8. \[2.1\times5.1\times159.1\div0.7\div1.7\div3.7=
\frac{7\times3\times17\times3\times37\times43}{7\times17\times37}=387\]
9. \[357\div357\frac{357}{358}=\frac{357\times358}{357(358+1)}=\frac{358}{359} \]
10. \((2+5+8+...+1997)-(1+4+7+...+1996)=\underbrace{1+1+\cdots+1}_{666 ตัว}=666\)
11. \[\frac{123456+234561+345612+456123+561234+612345}{111111+222222+333333+444444+555555}
=\frac{21\times111111}{15\times111111}=\frac{7}{5}\]
12. \(1.25\times5.6+2.25\times3.6=5\times1.4+9\times0.9=7+8.1=15.1\)
13. \(\ 134\times3.3-240\times0.12+158\times6.7
=134\times(3.3+6.7)+24(6.7-1.2)=1340+132=1472\)
14. \[126\frac{3}{4}\times4 + 326\frac{3}{8}\times8 +526\frac{3}{16}\times16=(126+326\times2+526\times4)\times4+9=11537\]
Variant1:LHS=\(126\times28+200\times8+400\times16+9=11537\)
Variant2:LHS=\(326\times28+200\times(16-4)+9=11537\)
15. \[37.5\times21.5\times0.112+35.5\times12.5\times0.112
=0.112(\frac{75\times43}{4}+\frac{71\times25}{4})=0.112\times\frac{25}{4}(129+71)
=\frac{5}{4}\times112=140\]
16. \[\frac{1}{819}+\frac{1}{2223}-\frac{1}{1197}
=\frac{1}{9\times13\times17}+\frac{1}{9\times13\times19}-\frac{1}{9\times7\times19}
=\frac{1}{9}(\frac{19+7-13}{7\times13\times19})=\frac{1}{1197}\]
17. \([2-(5.55\times1\frac{1}{3} - 2\frac{7}{10}\div0.4)]\div0.135=[2-(7.4-6.75)]\div0.135=10\)
18. \[14\frac{2345}{2907}\div\frac{7267}{9747}\times\frac{221}{4389}
={\frac{\overbrace{43043}^{=43\times13\times11\times7}}{9\times17\times19}}\times{\frac{3^3\times19^2}{13^2\times43}}
\times{\frac{13\times17}{11\times19\times3\times7}}=1\]
19. ให้ \(a=\frac{151}{315}, b=\frac{517}{751}, c=\frac{319}{537}\) จะได้
\[\frac{151}{315}(\frac{517}{751}-\frac{319}{537})+\frac{517}{751}(\frac{319}{537}-\frac{151}{315})+\frac{319}{537}(\frac{151}{315}-\frac{517}{751})
=a(b-c)+b(c-a)+c(a-b)=0\]
20. จาก \(\frac{a+(a-1)(a+1)}{a(a+1)-1}=1\) จะได้
\[\frac{1994+1993\times1995}{1994\times1995-1}+
\frac{1995+1994\times1996}{1995\times1996-1}+
\frac{1996+1995\times1997}{1996\times1997-1}=3\]