北大奥数少年 括号与分配律

1. Maths Foundation

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. 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. 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. 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. 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. 7.5×45＋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. 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. 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. （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