With the vigorous development of mathematical competition, numerical calculation is full of vitality. In addition to following the operation sequence of elementary arithmetic, breaking local considerations, establishing overall analysis, and skillfully applying laws and methods, it is often twice the result with half the effort when dealing with some seemingly complicated calculation problems. There are ten common clever calculation methods.
First, rounding method.
Algorithm is the scaffolding and theoretical basis of clever calculation. According to the characteristics of formula problems, the application of laws and properties to "round off" the operational data can make the calculation simpler.
1, add "round". Use additive commutative law and the law of association to "round off", for example:
4673+27689+5327+223 1 1
=(4673+5327)+(27689+223 1 1)
= 10000+50000
= 60000
2. Subtract "rounding". Use subtraction attributes to "round up", for example:
50- 13-7
= 50-( 13+7)
= 30
3. Multiplication "rounding". Use multiplicative commutative law, associative law and distributive law to "round off", for example:
125×4×8×25×78
=( 125×8)×(4×25)×78
= 1000× 100×78
= 7800000
4. The number of supplements shall be rounded up. Numbers with one or more zeros at the end are relatively simple to operate. If the end of the number is not 0, but 98,51,and so on. We can use (100-2), (50+ 1) and so on instead. , making the operation easier and faster. Generally, we call 100 "approximate strong number" of 98, and 2 "complement" of 98; 50 is called "approximate weak number" of 5 1, and 1 is called "complement" of 5 1. Write a number as the difference (sum) between its approximate strong (weak) number and its complement, and then perform an operation, for example:
( 1)387+99
=387+( 100- 1)
=387+ 100- 1
=486
(2) 1680-89
= 1680-( 100- 1 1)
= 1680- 100+ 1 1
= 1580+ 1 1
= 159 1
(3)69× 10 1
=69×( 100+ 1)
=6900+69
=6969
Second, the reduction method
According to the structure of the formula, reduction can make the calculation simpler. For example:
Third, the cardinal number method.
According to the characteristics of data, choose a number from many numbers as the basis of calculation, and calculate quickly by "cutting" and "supplementing" and adopting the method of "multiplying without adding". For example:
17+ 18+ 16+ 17+ 14+ 19+ 13+ 14
(When solving problems, you can choose 17 as the benchmark number, and the multiplication and addition method is as follows. )
= 17×8+ 1- 1-3+2-4-3
= 17×8-8
= 128
Fourth, the formula method
Arithmetic progression refers to a series in which the difference between every two adjacent numbers is equal. Arithmetic progression can be summed by the formula: sum = (the first term+the last term) × the number of terms ÷2. For example:
13+ 14+ 15+ 16+ 17+ 18+ 19+20+2 1+22
=( 13+22)× 10÷2
= 175
In addition, if the number of added terms is odd, the number arranged in the middle (middle term) can be directly multiplied by the number of terms to find the sum. For example:
3+5+7+9+ 1 1+ 13+ 15+ 17+ 19
= 1 1 (middle term) ×9 (number of terms)
=99
Verb (abbreviation of verb) deformation method
Identity deformation is an important thought and method, and also an important problem-solving skill. It uses what we have learned to carry out purposeful mathematical deformation, which can often solve the problem quickly. For example:
1, calculate 9999× 2222+3333× 3334.
Multiply this problem directly, and the number will be big and easy to make mistakes. If you change 9999 to 3333×3, the law will appear. ) 9999× 2222+3333× 3332. 199989999896
=3333×3×2222+3333×3334
=3333×6666+3333×3334
=3333×(6666+3334)
=3333× 10000
=33330000
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Partial deformation of molecules can make the operation simple. )
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