Mathematical calculations There are no easy formulas

A simpler algorithm is to start with the large denomination, calculate the maximum number of denominations needed, and then count the smaller denominations in turn.

11 = 5 2 + 1 1, you can use 2 pieces of 5 yuan, 1 piece of 1 yuan;

11 = 5 1 + 2 3, 1 piece of 5 yuan can be used, 3 pieces of 2 yuan;

11 = 5 1 + 2 2 + 1 2, you can use 1 piece of 5 yuan, 2 pieces of 2 yuan, 2 pieces of 1 yuan;

11 = 5 1 + 2 1 + 1 4, you can use 1 piece for 5 yuan, 1 piece for 2 yuan, 4 pieces for 1 yuan;

11 = 5 1 + 1 6, you can use 1 piece for 5 yuan, 6 pieces for 1 yuan;

11 = 2 5 + 1 1, you can use 5 pieces of 2 yuan, 1 piece of 1 yuan;

11 = 2 4 + 1 3, you can use 4 pieces of 2 yuan, 3 pieces of 1 yuan;

11 = 2 3 + 1 5, you can use 3 pieces of 2 yuan, 5 pieces of 1 yuan;

11 = 2 2 + 1 7, you can use 2 pieces of 2 yuan, 7 pieces of 1 yuan;

11 = 2 1 + 1 9, you can use 1 piece of 2 yuan, 9 pieces of 1 yuan;

11 = 1 11, you can use 1 piece for 1 yuan.

There are 11 ways to do this.

10 kinds, composed of 11 yuan, there must be a separate 1 yuan, so as long as you find a composition of 10 yuan there are several cases. First look at the 5 yuan, 5 + 5, 5 + 2 + 2 + 1, 5 + 2 + 1 + 1 + 1 + 1, 5 + 1 + 1 + 1 + 1 a total of 4 kinds, there are no 5 yuan, there are 0 1 2 3 4 5 2 yuan, a total of 6 situations, plus 10 kinds.

First, the overall simple calculation. The entire equation can be calculated in a simple way, and this form is the most common. For example:

2. Partial simple calculation. It is not uncommon for a part of an equation to be easily calculated.

3. Simple calculation in the middle. It's easiest to overlook the fact that you can't start a simple calculation, but you can do it after a step or two. For example:

Fourth, repeat and simple calculation. A simple calculation is performed more than once in a problem, and this situation often does not pay attention to the latter simple calculation. For example:

8 The second time.

1. The basis of the simple calculation is a, the law of multiplication b, the law of addition c, the nature of subtraction, and the arithmetic nature of division.

2. The type of short-term calculation is a, direct short-cost calculation b, partial short-term calculation c, conversion short-cost calculation d, and process short-term calculation.

Three formulas for short-term calculation:

Addition: a+b+c=a+(b+c) (additive associative law).

Multiplication: a b c c = a c b (multiplicative commutative law) a b c = a (b c) (multiplicative associative property) (a+b) c=ac+bc or (a-b) c=ac-bc (multiplicative distributive property).

Subtraction: a-b-c=a-c-b (commutative law of subtraction) a-b-c=a-(b+c) (associative law of subtraction).

Division: a b c = a c b (division commutative law) a b c = a (b c) (division associative law) (a + b) c = a c + b c or (a - b) c = a c - b c (division distributive law).

Note: The division distribution rate can only be assigned if the dividend is the difference or sum of the two numbers.

The simple equations are:

1. Multiplication operation per copy number of copies total number of copies of the total number of copies of the total number of copies of the number of copies of the number of copies of the number of copies.

2. Multiples are calculated as 1 multiple, multiples, multi

3. Distance calculation speed time distance speed speed time distance speed time distance speed.

4. **Calculate unit price, quantity, total price, unit price, total price, quantity, unit price.

5. Efficiency: Calculate work efficiency, working hours, total work, total work, working hours, total working hours, working hours, work efficiency.

6. Addition calculates the addition of the addition of the sum and the addition of one addition and the addition of the other.

7. Subtraction calculates the subtracted difference subtracted difference subtracted difference subtracted subtracted number.

8. Multiplication Problem Factor Factor Product One Factor Another Factor.

Multiplicative distributive property.

The most common method used in simple calculations is the multiplicative distributive property. The multiplicative distributive property refers to ax(b+c) = axb+axc, where a, b, c are arbitrary real numbers. Conversely, AXB+AXC=AX(B+C) is called the inverse of the multiplicative distributive law (also known as extracting the common divisor), especially when a and b complement each other, this method is more useful.

There are also times when the associative property of addition is used, such as a+b+c, where b and c are complements to each other, so you can combine b and c and multiply it with a. If we change the above equation to x, we can easily calculate it by using the associative property of multiplication.

The simple equations are:

1. Multiplication operation per copy number of copies total number of copies of the total number of copies of the total number of copies of the number of copies of the number of copies of the number of copies.

2. Multiples are calculated as 1 multiple, multiples, multi

3. Distance calculation speed time distance speed speed time distance speed time distance speed.

4. **Calculate unit price, quantity, total price, unit price, total price, quantity, unit price.

5. Efficiency: Calculate work efficiency, working hours, total work, total work, working hours, total working hours, working hours, work efficiency.

6. Addition calculates the addition of the addition of the sum and the addition of one addition and the addition of the other.

7. Subtraction calculates the subtracted difference subtracted difference subtracted difference subtracted subtracted number.

8. Multiplication Problem Factor Factor Accumulation of Rapid Accumulation of a Judgment Factor Another Factor.

Multiplication is divided into acres and the law of the hood.

The most common method used in simple calculations is the multiplicative distributive property. The multiplicative distributive property refers to ax(b+c) = axb+axc, where a, b, c are arbitrary real numbers. Conversely, AXB+AXC=AX(B+C) is called the inverse of the multiplicative distributive law (also known as extracting the common divisor), especially when a and b complement each other, this method is more useful.

There are also times when the associative property of addition is used, such as a+b+c, where b and c are complements to each other, so you can combine b and c and multiply it with a. If we change the above equation to x, we can easily calculate it by using the associative property of multiplication.

Simple calculation is a special kind of calculation, which uses the laws of operation and the basic properties of numbers, so as to make the calculation easy, so that a very complex formula becomes easy to calculate the number.

The most common method used in simple calculations is the multiplicative distributive property. The multiplicative distributive property refers to ax(b+c) = axb+axc, where a, b, c are arbitrary real numbers. Conversely, AXB+AXC=AX(B+C) is called the inverse of the multiplicative distributive law (also known as extracting the common divisor), especially when a and b complement each other, this method is more useful.

There are also times when the associative property of addition is used, such as a+b+c, where b and c are complements to each other, so you can combine b and c and multiply it with a. If we change the above equation to x, we can easily calculate it by using the associative property of multiplication.

The associative law of multiplication.

The associative law of multiplication is also a method of doing simple operations, which is expressed by letters as (a b) c = a (b c), and its definition (method) is: multiply three numbers, first multiply the first two numbers, and then multiply the third number; Or multiply the last two numbers first, and then multiply them with the first number, and the product remains the same. It can change the order of operations in multiplication operations, and the multiplication law is not used very much in daily life, mainly in some more complex operations to play a simple role.

The commutative law of multiplication.

The multiplicative commutative law is used to swap the positions of individual numbers: a b = b a additive commutative law.

The additive commutative property is used to swap the positions of individual numbers: a+b=b+a, the additive associative law.

a+b）+c=a+（b+c）

There is an additive commutative law in addition, and a combination law of jujube source in addition. Multiplication has the law of multiplication commutation, the law of multiplication combination, and the law of multiplication of the rock belt. There is also the nature of subtraction and the nature of division.

1. Simple calculation using the multiplicative distributive law.

The most common method used in simple calculations is the multiplicative distributive property. The multiplicative distributive property refers to:

ax(b+c)=axb+axc

cx（a-b）=axc-bxc

Example 1: 38x101, how do we dismantle it? See who is closer to the whole hundred or the whole ten, of course 101 is better, then we can split 101 into 100+1.

38x101

38x（100+1）

38x100+38x1

Example 2: 47x98, how to disassemble it? To dismantle 98, bring it closer to 100.

47x9847x（100-2）

47x100-47x2

Second, the benchmark number method.

Find a compromised number in a series of numbers to represent all numbers, and remember that the selection of this number cannot deviate from this series of numbers. Example:

2062x5）+10-10-20+21

3. Addition combined with law.

The application of the additive associativity law (a b) c=a (b c) allows for easier operations by changing the position of the additive modulus plexus. Example:

Fourth, the splitting method.

As the name suggests, the splitting method is to split a number into several numbers for the convenience of calculation. This requires mastering some "good friends" such as: 2 and 5, 4 and 5, 2 and, 4 and, 8 and so on. Be careful not to change the size of the number! Example:

5. Extracting the common factor method.

This method actually uses the multiplicative distributive property to extract the same factors. Example:

Here are all decimal addition and subtraction, it is easy to observe that some decimal fractions are the same, the absolute subtraction can be offset, and some of the decimal grinding macro oak decimal blind part can just be rounded.