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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