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北大奥数少年 括号与分配律

北大奥数少年 括号与分配律

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北大奥数少年 括号与分配律

  1. 1. Maths Foundation
  2. 2. Adding or removing a bracket after “+”, no changes to the “+” and “–” sign in the bracket 146+73+27 = 146 + (73 + 27) = 146 + 100 = 246 3.57 + 7.76 – 4.33 = 3.57 + (7.76 – 4.33) = 3.57 + 3.43 = 7 6.93 + (3.7 – 1.83) = 6.93 + 3.7 – 1.83 = 6.93 – 1.83 + 3.7 = 5.1 + 3.7 = 8.8
  3. 3. Adding or removing a bracket after “–”, the sign in the bracket will be changed; “+” becomes “–” and “–” becomes “+” 8724 – 398 + 98 = 8724 – (398 – 98) = 8724 – 300 = 8424 3.94 – 0.875 – 1.125 = 3.94 – (0.875 + 1.125) = 3.94 – 2 = 1.94 27.26 – (4.5 – 2.84) = 27.26 – 4.5 + 2.84 = 27.26 + 2.84 – 4.5 = 30.1 – 4.5 = 25.6 121 – 68 – 85 + 53 = 121 – 68 – (85 – 53) = 121 – 68 – 32 = 121 – (68 + 32) = 121 – 100 = 21
  4. 4. Adding or removing a bracket after “×”, no changes to the “×” and “÷” sign in the bracket (No “+” and “–” sign in the bracket ) 38×25×6 = 19×2×25×2×3 = 19×(2×25×2)×3 = 19×100×3 = 5700 124×25 = 124×(100÷4) = 124×100÷4 = 124÷4×100 = 3100 Also: a×(b+c) = a×b+a×c 4.25×18 = (4 + 0.25)×18 = 4×18 + 0.25×18 = 72 + 18÷4 = 72 + 4.5 = 76.5 From the above example: ×0.25  ÷4 ×0.5  ÷2 ×0.125  ÷8
  5. 5. Adding or removing a bracket after “÷”, the sign in the bracket will be changed; “×” becomes “÷” and “÷” becomes “×” (No “+” and “–” sign in the bracket ) 8.4÷5÷6 = 8.4÷(5×6) = 8.4÷30 = 0.28 27000÷125 = 27000÷(1000÷8) = 27000÷1000×8 = 27×8 = 216 From the above example: ÷ 0.25  ×4 ÷ 0.5  ×2 ÷ 0.125  ×8
  6. 6. 17×2.1 = 17×(2+0.1) = 17× 2 + 17× 0.1 = 34+1.7 = 35.7 132476×111 = 132476×(100+10+1) = 13247600 + 1324760 + 132476 = 14704836 0.6×1.6+0.6×26.4 = 0.6×(1.6+ 26.4) = 0.6×28 = 16.8
  7. 7. 7.5×45+17×2.5 First method: = 7.5×(28+17)+17×2.5 = 7.5×28 + 7.5×17+17×2.5 = 7.5×28+17×(7.5+2.5) = 210+170 = 380 7.5×45+17×2.5 Second method: = 2.5×3×45+17×2.5 = 2.5×135 + 2.5×17 = 2.5×(135+17) = 2.5 ×152 = 380
  8. 8. 1991+199.1+19.91+1.991 = 1991×(1+0.1+0.01+0.001) = 1991×1.111 = (2000 – 9 )×1.111 = 2000×1.111 – 9×1.111 = 2222 – 9.999 = 2222 – 10+0.001 = 2212.001
  9. 9. 41.2×8.1+11×9.25+537×0.19 = 412×0.81+537×0.19+11×9.25 = 412×0.81+(412+125)×0.19+11×9.25 = 412×0.81+412×0.19+125×0.19+11×9.25 = 412×(0.81+0.19)+125×0.19+11×9.25 = 412×1 +125×0.19+11×9.25 = 412 +1.25×19+11×(1.25+8) = 412+1.25× 19 + 11 × 1.25 + 11× 8 = 412+1.25×(19+11)+88 = 500+1.25×30 = 500+37.5 = 537.5
  10. 10. (1+0.23+0.34)×(0.23+0.34+0.65)– (1 + 0.23+ 0.34+0.65)×(0.23+0.34) = (1+(0.23+0.34))×(0.23+0.34+0.65)– (1 + (0.23 +0.34+0.65))×(0.23+0.34) = 1×(0.23+0.34+0.65)+(0.23+0.34)×(0.23+0.34 +0.65) – 1 ×(0.23+0.34) –( 0.23+0.34+0.65)× (0.23+0.34) = (0.23+0.34+0.65)+(0.23+0.34)×(0.23+0.34+ 0.65) –(0.23+0.34) –( 0.23+0.34+0.65)×(0.23+ 0.34) = (0.23+0.34+0.65)+(0.23+0.34)×(0.23+0.34+ 0.65) –(0.23+0.34) –( 0.23+0.34+0.65)×(0.23+ 0.34) = (0.23+0.34+0.65) –(0.23+0.34) = 0.23+0.34+0.65 – 0.23 – 0.34 = 0.65

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