* Note: The linear trend is that exponential series generally develops in one direction, that is, the numerical value is getting bigger and bigger, or getting smaller and smaller, and the change of numerical value is directly related to the number of terms (don't feel too mysterious, in fact, you can have this intuition after doing some questions).
Step 2: Idea A: Analyze the trend
1, increase (including decrease) is generally addition and subtraction.
The basic method is to do it almost, but if you can't find the rule by doing it three levels worse, you should change your mind immediately, because arithmetic progression and his variants above level three failed the public examination.
Examples 1:-8, 15, 39, 65, 94, 128, 170, ()
180
Solution: the observation is linear, and the value increases gradually, and the increase is general. Considering the difference, you have to take 23, 24, 26, 29, 34, 42 to form a linear sequence of small increments again, and then do the difference to get 1, 2, 3, 5, 8, an obvious and recursive sequence, and the next item is 5+8.
Summary: the difference will not exceed level 3; Some typical series should be memorized.
2, do the multiplication and division method with large increase.
Example 2: 0.25, 0.25, 0.5, 2 16, ()
64 C. 128 D.256
Solution: the observation shows a linear law, increasing from 0.25 to 16, considering multiplication and division. The latter item is divided by the former item to get 1, 2, 4, 8. Typical geometric series, the next term of the second-order series is 8*2= 16, so the next term of the original series is 16 * 655.
Summary: Doing business will not exceed level 3.
3, the increase is very large. Consider power series.
Example 3: 2, 5, 28, 257, ()
2006. 1342 C .3503 D .3 126
Solution: the observation is linear and the increase is very large. Considering power series, the law of maximum number is obviously the breakthrough of this problem. Note that there are powers of 256 near 257, 4 and 8 near 27, 25 and 28, and 1 and 4 near 2. And each term of a series must be related to the number of terms, so the power series related to the original series should be 1, 4,27,256 (each term of the original series is added with 1, that is,1,22,3,4), and the next term should be 5.
Summary: Familiar with power numbers.
Step 2 Idea B: Find the visual impact point *
* Note: Visual impact point is a special and unusual phenomenon in exponential series, and it is often the direction of solving problems.
Visual impact point 1: a long sequence of more than 6 items. The basic idea to solve the problem is to group or separate the projects.
Example 4: 1, 2, 7, 13, 49, 24, 343, ()
A.35 B .69 degrees Celsius. 1 14 D .238
Solution: The first six items are relatively small, and the seventh item suddenly becomes larger, which is not a linear law. Consider idea B. For long series, consider grouping or separation, and try two separation series: 1, 7, 49, 343; 2, 13,24,()。 Obviously, the first branch series is a geometric series, and the second branch series is a arithmetic progression with a tolerance of 1 1, and the answer A is quickly obtained.
Conclusion: Interval mixing of arithmetic and geometric progression is a common test.
Visual impact point 2: swing series, numerical fluctuation, showing a swing shape. The basic idea of solving problems is sub-topic.
20 5
Example 5: 64, 24, 44, 34, 39, ()
10
A.20 B .32 degrees Celsius 36.5 degrees. 19
Solution: the observed values are small and large, and every other item is observed immediately. If the difference is as above, it is found that the difference becomes a geometric series, and the next difference should be 5/2=2.5, and the easy answer is 36.5.
Summary: The number of items is not necessarily regular, but it is also possible to form a comprehensive law like this problem.
Visual impact point 3: double brackets. There must be a regular pattern!
Example 6: 1, 3, 3, 5, 7, 9, 13, 15, (), ()
A.19,21b.19,23 degrees celsius.21.23d.27,30
Solution: Find the rules directly when you see the double brackets, and use 1, 3,7, 13, (); 3, 5, 9, 15, (), which is obviously a two-level arithmetic progression with a tolerance of 2, easy to answer 2 1 23, and choose C.
Example 7: 0, 9, 5, 29, 8, 67, 17, (), ()
A. 125, 3 B. 129, 24℃ .84, 24 days. 172,83
Solution: I noticed that this is a swing sequence with double brackets, and I did not hesitate to look for a pattern every other item! 0, 5, 8, 17, (); 9,29,67,()。 There are many branch series 2, and the law is obvious. Noted a big increase. Considering multiplication and division or power series, I flashed 8, 27 and 64 in my mind, and found that branch series 2 is a variant of 2 3+1,3 3+2 and 4 3+3, and the next term should be 5 3+4 = 65438. Choose b directly. Looking back, we will find that the first branching sequence can be simplified to 1- 1, 4+ 1, 9- 1, 16+ 1, 25- 1.
Summary: Generally, the rule of parentheses can be found only by determining one branch series, and the other branch series can be ignored to save time.
Visual impact point 4: score.
Type (1): Mix and match integers and fractions, prompting multiplication and division.
Example 8: 1200, 200, 40, (), 10/3
10 .20 degrees Celsius. Thirty days. Five
Solution: Mix and match integers and fractions, and immediately associate them with quotient, so it is easy to get the answer of 10.
Type (2): Full score. The solution to the problem is: the first point that can be reduced; If it can be unified, it will be unified first; The breakthrough is the score that should not be changed, which is called the benchmark number; The numerator or denominator must be related to the number of terms.
Example 9: 3/ 15, 1/3, 3/7, 1/2, ()
A.5/8 B. April 9 C. 15/27 D .-3
Solution: The first reducible fraction is 3/15 =1/5; The common multiple of denominator is relatively large, which is not suitable for unification; The breakthrough point is 3/7, because the denominator is large, so it is not suitable for multiplication, so as a reference number, other scores change around it; Looking for the relationship between terms, 3/7 molecules are exactly its terms, and 1/5 molecules are also exactly its terms, so it is quickly found that the fractional column can be transformed into 1/5, 2/6, 3/7, 4/8, and the next term is 5/9, that is,/kloc-0.
For example, 10:-4/9, 10/9, 4/3, 7/9, 1/9.
B 10/9 C -5/ 18 D -2
Solution: there is no irreducibility; But the denominator can be unified, and the molecular sequence is -4, 10, 12, 7, 1, and the former term is subtracted from the latter term.
14,2, -5, -6, (-3.5), (-0.5) Compared with the molecular sequence, the next term should be 7/(-2)=-3.5, so the next term of the molecular sequence is 1+(-3.5)= -2.5. Therefore, (-2.5)/9= -5/ 18.
Visual impact point 5: positive and negative overlap. The basic idea is doing business.
For example: 1 1: 8/9, -2/3, 1/2, -3/8, ()
A 9/32 B 5/72 C 8/32 D 9/23
Solution: positive and negative overlap, do business immediately, and find that it is a geometric series, and it is easy to get an A.
Visual impact point 6: radical.
Roots and integers are mixed in the type (1) series. The basic idea is to turn integers into roots and move numbers outside the roots into roots.
Example12: 0316 √ 212 () () 2 48
A.√3 24 b√3 36 c . 2 24d . 2 36
Solution: There are 0, 1, √2, (), 2 before the double brackets; 3, 6, 12, (), 48. The first branch number is a mixture of a root number and an integer, with √2 as the base number, and other numbers are deformed around it, and the integer is √0 √ 1 √2 ()√4, which is easy to know. The second branch series is a geometric series with obvious common ratio of 2, so the answer is a.
The basic idea of the addition and subtraction formula of formula (2) is to use the square difference formula: A 2-B 2 = (A+B) (A-B).
Example: 13: √ 2- 1,1(√ 3+1), 1/3, ()
a(√5- 1)/4 B 2 C 1/(√5- 1)D√3
Solution: Form 1: √ 2-1= (√ 2-1) (√ 2+1) = (2-1)/(√ 2+65438) Meanwhile,1.
Visual impact point 7: The first or first two items are small and close, and the second or third item suddenly becomes larger. The basic idea is group recursion, and the next number is obtained by performing five operations (including power) with the first term or the first two terms.
Example 14: 2, 3, 13, 175, ()
30625 B .3065 1 degree celsius.30759d.30952.30959959896
Solution: It is observed that 2 and 3 are very close, and 13 suddenly becomes larger. Consider using 2,3 to calculate that 13 has 2*5+3=3, 3 2+2 * 2 = 13 and so on. To make 3 1365438.
Summary: Sometimes recursive operation rules are hard to find, but don't waver. This is the general rule of this kind of topic.
Visual impact point 8: Pure decimal series, that is, all items in the series are decimals. The basic idea is to consider the integer part and decimal part separately, or form a separate series, or form a common law.
For example,15:1.01.02, 2.03, 3.05, 5.08, ()
A.8. 13 B .8.0 13 degrees celsius.7.12
Solution: The integer part is extracted by 1, 1, 2, 3, 5, (), which is an obvious and recursive sequence, and the next item is 8, excluding C and D; Extract the decimal part, including 1, 2, 3, 5, 8, () and recursive sequence, and the next item is 13, so choose A.
Summary: This question belongs to the independent law of integer and decimal parts.
Example: 16: 0. 1, 1.2, 3.5, 8. 13, ()
a 2 1.34 B 2 1. 17 C 1 1.34D 1 1. 17
Solution: The integer part and the decimal part are still considered separately, but when observing the overall characteristics of the sequence, it is found that the number is very similar to a typical and recursive sequence, so the integer part and the small tree part are considered together, and a new sequence 0, 1, 1, 2, 3, 5, 8, 13, (), is found.
Summary: This question belongs to the law of integer and decimal parts.
Visual impact point 9: Much like a continuous natural series but not coherent. Consider prime numbers or compound sequences.
Example: 17: 1, 5,1,1,9, 28, (), 50.
A.29 B .38 degrees Celsius.47 days.49 pieces
Solution: the observed value gradually increases linearly, and the increase is average. Consider that the difference is 4, 6, 8, 9, ..., which is very similar to the continuous natural sequence but lacks 5, 7, association and sequence. Next, it should be 10, 12, which is substituted to verify 28+10 = 38,38+.
Visual impact point 10: natural number, and natural numbers with more than 3 digits appear in the sequence. Because the operation intensity of the sequence problem is not strong, it is impossible to use natural numbers to do operations, so this kind of problems generally investigate the microscopic number structure.
Example 18: 76395 1, 59367, 7695, 967, ()
69 degrees Celsius in 5936 AD. 769.76 AD.
Solution: It is not practical to calculate natural numbers. Microscopically, it is found that the last item is one digit less than the last item, at least 1, 3,5, and the next default number should be 7; In addition, after one digit is defaulted, the numerical order is also reversed, so it is reversed after 967 is divided by 7. It should be 69, and I chose B.
Example 19: 1807, 27 16, 3625, ()
5 149 .4534 degrees celsius.4231d.5847.
Solution: four natural numbers, directly look at the relationship between the numbers with a microscope, and find that the sum of the first two digits of each four-digit number is 9, and the sum of the last two digits is 7. Observe the options and get option B quickly.
Step 3: Find another way.
Generally speaking, it is the last two steps, and most of the questions can find ideas, but it does not rule out that some laws are not easy to find directly. At this time, if the original series is slightly changed, it may be easier to see the law.
Variant 1: common factor decreases. The number of series is large and there is a common divisor. You can remove the common divisor first, transform it into a new series, and then recover it after finding the law.
Example 20: 0, 6, 24, 60, 120, ()
A. 186 .2 10℃ for 220 days.226
Solution: Because each value of this series is very large, we are not sure whether the increase is large or small, but we find that there is a common divisor of 6, which is rounded to get 0, 1, 4, 10, 20. It is easy to find that the increase is average. Considering addition and subtraction, we can easily find that it is a second-order arithmetic progression, and the next term should be 20+ 10+.
Variant 2: Factorization. There is no common divisor between items in the sequence, but adjacent items have common divisors. At this time, factorizing the numbers in the original sequence is helpful to find the law.
Example 2 1: 2, 12, 36, 80, ()
100 . 125 C 150 D . 175
Solution: Factorization includes 1*2, 2*2*3, 2 * 2 * 2 * 5, and it is easy to get 1* 1*2, 2*2*3, 3 * with a little modification.
Variant 3: General scoring method. There are not many least common multiples of the denominator applicable to each item in the fractional column.
Example 22: 1/6, 2/3, 3/2, 8/3, ()
10/3 B.25/6 C.5 D.35/6
Solution: It is found that the denominator is easy to divide, and the single molecular sequence 1, 4,9, 16, () can be obtained by dividing the denominator immediately. The increase is general, with a difference of 3, 5 and 7, and the next item should be 16+9=25. The score with reducing component 6 is b.
Step 4: The guessing method is not the right way.
Some topics are puzzling, and sometimes there are only one or two minutes left. Should we give up? Of course not! A penny for a penny, targeted speculation can often save the emergency, and the correct rate is not low. Here are a few guesses that I have worked out myself.
First Mongolia: There are integers and decimals in the options, and decimals are mostly answers.
See examples 5: 64, 24, 44, 34, 39, ()
A.20 B .32 degrees Celsius 36.5 degrees. 19
Just guess c!
Example 23: 2, 2, 6, 12, 27, ()
A.42 B 50 C 58.5 D 63.5
Guess: If you find that the options include integers and decimals, you can choose directly from C and D. The decimal ".5" indicates that there may be multiplication and division in the operation. If it is observed that the last item in the series is divided by the previous item no more than 3 times, guess C.
Positive solution: the difference is 0,4,6, 15. (0+4) * 1.5 = 6 (2+6) * 1.5 = 12 (4+6) * 1.5 = 15 (6+ 15) * 65443.
The second Mongolia: the series has negative numbers, and the options have negative numbers. Negative numbers are mostly answers.
Example 24:-4/9, 10/9, 4/3, 7/9, 1/9, ()
B. 10/9 C -5/ 18 D.-2
Guess: There are negative numbers in the series and negative numbers in the options. Guess in C/D, while observing the original series, the denominator should be related to 9. Guess C.
The third Mongolia: guess the closest value. Sometimes it seems that a rule has been found, but the calculated answer is not in the option, but it is very close to an option. Don't waste time looking for another rule, just guess the closest one and it will be close to ten!
Example 25: 1, 2, 6, 16, 44, ()
A.66 B .84 ℃.88d.120
Guess: the increase is average, and the subconscious difference is 1, 4, 10, 28. If the difference is 3,6, 18, the next term may be (6+ 18)*2=42 or 6* 18= 108. Either way, the next item in the original series is greater than 100, so guess d directly.
Example 26: 0. , 0, 1,5,23, ()
A. 1 19 B .79 C 63 D 47
Guess: The first two items are the same, which is obviously a recursive sequence. It needs multiplication to push from 1 5 to 25, and 5*23= 1 15. Guess the closest option 1 19.
The fourth mask: use the relationship between options to mask.
Example 27: 0, 9, 5, 29, 8, 67, 17, (), ()
24 C 84，24 D 172 83
Guess: First of all, I noticed that there is a common value of 24 in option B and option C, and I immediately laughed, knowing that this is an obstacle deliberately set by the insidious questioner, which is only a clue to us. The second bracket must be 24! According to the law summarized before, every other item of double brackets must be regular. We found that the even-numbered items 9, 29, 67 and () are all about twice as many as the previous item, so we guessed 129 and chose B.
Example 28: 0, 3, 1, 6, √ 2, 12, (), (), 2, 48.
A.√3.24 lbs. √3.36 C 2.24D√2.36
Guess: As above, the first bracket must be √3! But the double brackets are regular, 3,6, 12. It is easy to know that the second bracket is 24, and A will be chosen soon.
Well, I hope everyone can understand and skillfully use these methods to speed up the problem solving and improve the correct rate! Come on! ! !
Of course, it is impossible to cover all the methods, because there are endless problems. Welcome to share more good methods ~
PS: I found it on the Internet: Top Ten Quick Calculation Skills.
★ Fast calculation skills 1: estimation method
Key points:
There is no doubt that "estimation method" is the first fast calculation method in data analysis, and whether it can be estimated first must be considered before all calculations are made. The so-called estimation is a fast calculation method of rough estimation under the condition that the accuracy requirement is not too high, which is generally used in the case of large difference in options or comparison data. There are many ways to estimate, which require candidates to train and master more in actual combat.
The premise of estimation is that the difference between the options or the numbers to be compared must be relatively large, and the size of this difference determines the accuracy requirements for "estimation".
★ Quick calculation skill 2: Direct division
Key points:
"Direct division" refers to the method of quickly obtaining the first place (the first place or the first two places) of the quotient through "direct division" when comparing or calculating complicated fractions, so as to get the correct answer. "Direct division" is widely used in the rapid calculation of data analysis, because it is "simple" and "easy to operate".
"Direct division" generally includes two forms in terms of questions:
First, when comparing multiple scores, the maximum/minimum number in the first place is the maximum/decimal number in the case of equal quantity;
Second, when calculating the score, you can choose the correct answer by calculating the first place under different options.
"Direct score" is generally divided into three gradients according to the difficulty:
First, you can simply and directly see the first business place;
Second, you can see the first place of quotient through hands-on calculation;
Third, some complex scores need to be calculated by the first place of the "reciprocal" of the score to determine the answer.
★ Quick calculation skill 3: truncation method
Key points:
The so-called "truncation method" refers to a fast calculation method of "truncating the numbers in the calculation process within the precision range (that is, only looking at or taking the first few digits, so as to get the calculation results with sufficient accuracy".
When using "truncation method" in addition and subtraction, add and subtract directly from the upper left bit (and pay attention to whether the next bit needs carry and borrow) until you get the answer with the required accuracy.
When using truncation method in multiplication or division, we need to pay attention to the direction of truncation approximation in order to make the result as accurate as possible:
First, to expand (or shrink) a multiplier, it is necessary to shrink (or expand) another multiplier;
Second, to expand (or reduce) the dividend, you need to expand (or reduce) the divisor. If "the sum or difference of two products (that is, A× B C× D)" is required, it should be noted that: 3. To expand (or shrink) one side of the plus sign, we need to shrink (or expand) the other side of the plus sign;
Fourth, to expand (or shrink) one side of the minus sign, you need to expand (or shrink) the other side of the minus sign.
Which approximate direction to take depends on the similarity and the calculation difficulty after truncation.
Generally speaking, when "truncation method" is used in multiplication or division, if the answer needs to have N-bit precision, the data in the calculation process needs to have N+ 1 bit precision, but the specific situation depends on the size of truncation error and the cancellation of error; In the case of small error, the data in the calculation process may not even meet the requirements of the above truncation direction. Therefore, when applying this method, candidates need to be more familiar with and master the training errors. When the answer can be obtained in other ways and the truncation error may be large, try to avoid using multiplication and division truncation method.
★ Quick calculation tip 4: The same method
Key points:
The so-called "similarity method" refers to a fast calculation method that "when comparing the sizes of two fractions, the numerator or denominator of these two fractions are the same or similar to simplify the calculation". Generally includes three levels:
First, make the numerator (or denominator) exactly the same, so you only need to look at the denominator (or numerator) again;
Second, after the numerator (or denominator) is similar, there will be a situation that "the denominator of a certain fraction is larger and the numerator is smaller" or "the denominator of a certain fraction is smaller and the numerator is larger", so the size of two fractions can be directly judged.
Third, make the numerator (or denominator) very close, and then use other quick calculation skills to make simple judgments.
In fact, it is generally impossible to make the numerator (or denominator) the same in data analysis, so the same method is more about "making it similar" than "making it the same".
★ Fast calculation skill 5: Difference method
Key points:
"Difference method" is a quick calculation method when comparing the sizes of two fractions, and it is difficult to solve it by other quick calculation methods such as "direct division" or "identical method".
Applicable form:
When comparing two fractions, if the numerator and denominator of one fraction are only a little bigger than the other fraction, it is often difficult to compare the relationship between size by "direct division" and "identity method", and the problem can be well solved by "difference method".
Basic definition:
Among the two fractions that meet the "applicable form", we define the fraction with large numerator denominator as "big fraction" and the fraction with small numerator denominator as "small fraction", and we define the new fraction obtained by the difference between the numerator denominator of these two fractions as "difference fraction". For example, 324/53. 1 is compared with 3 13/5 1.7, where 324/53. 1 is "big score" and 313/5/kloc-0.
Basic criteria for using the "difference method"
Use "small fraction" to compare "difference" instead of "big fraction";
1, if the difference is greater than the fraction, the fraction is greater than the fraction;
2. If the difference value is less than the small value, the large value is less than the small value;
3. Difference is equal to small points, and big points are equal to small points.
For example, "11/kloc-0 instead of 324/53. 1 compare with 3 13/5 1.7" because1/kloc-0. 3 13/5 1.7 (which can be simply obtained by "direct division" or "same method"), so 324/53.6438+0 > 313/51.7.
Please pay special attention to:
First, the "difference method" itself is an "accurate algorithm" rather than an "estimation method", and the obtained dimensional relationship is accurate rather than rough;
Second, "difference method" and "similarity method" are often used together, and "difference method after similarity method" and "difference method after similarity method" are two situations that are often encountered in data analysis and rapid calculation.
Thirdly, direct division is often needed when comparing the "difference fraction" obtained by the "difference method" with the "small fraction".
Fourth, if the two scores are very close, we even need to use the "difference method" twice, which is relatively complicated, but if it is skillfully used, it can greatly simplify the calculation.
★ Fast calculation skill 6: interpolation method
Key points:
"Interpolation" refers to a fast calculation method of "reference comparison" with an intermediate value when calculating numerical values or comparing large and small numbers. Generally speaking, it includes two basic forms:
First, it is relatively difficult to compare the sizes of two numbers directly, but there is obviously a number between the two numbers that can be compared and easily calculated, so the size relationship between the two numbers can be quickly obtained from the intermediate number. For example, the comparison between a and b, if a number c can be found, and A >；; C, and b < C, that is, you can judge A> B.
Secondly, when calculating a numerical value F, it is difficult to judge two close numbers A and B, but we can easily find a number C between A and B, such as A.
★ Quick calculation skill 7: Rounding method
Key points:
"Rounding method" refers to a quick calculation method that rounds the intermediate result to an "integer" (whole hundred, whole thousand and other convenient calculation forms) in the calculation process, thus simplifying the calculation. "Rounding method" includes addition and subtraction rounding and multiplication/division rounding.
In the calculation of data analysis, it is basically impossible to make up a real "integer", but because data analysis does not require absolute accuracy, making up a number close to "integer" is the main content of data analysis "rounding method".
★ Fast Calculation Skill 8: Scaling Method
Key points:
"Scaling method" refers to a fast calculation method that can boldly "scale" (expand) or "shrink" (shrink) the intermediate results in the comparison and calculation of numbers, so as to quickly get the size relationship of the numbers to be compared.
Key points:
If A & gtB& gt；; 0, while C>D>0 has:
1)A+C & gt； B+D
2)A-D & gt； B-C
3)a×C & gt； B×D
4) analog-to-digital conversion B/C
These four relations, that is, the four mathematical inequality relations described in the above four examples, are very simple and basic inequality relations that we often need to use in doing problems, but they are mathematical relations that candidates easily ignore or miss in the examination room, and their essence can be explained by "scale method".
★ Fast calculation skill 9: growth rate correlation speed algorithm
Key points:
Calculating the data related to the growth rate is a common problem in data analysis, and there are some common quick calculation skills in this kind of calculation. Mastering these quick calculation skills is very important for solving data analysis problems quickly.
Two-year mixed growth rate formula:
If the growth rates of the second and third phases are r 1 and r2, respectively, the growth rate of the third phase relative to the first phase is:
r 1+r2+r 1× r2
Approximate formula of growth rate divided by multiplication:
If the value of the second period is a and the growth rate is r, the value of the first period is a':
A'= A/( 1+r)≈A×( 1-r)
(In fact, the left formula is slightly larger than the right formula, and the smaller R, the smaller the error, and the error order is R 2).
Approximate formula of average growth rate:
If the growth rate in n years is r 1, r2, R3...Rn, then the average growth rate: r≈ the arithmetic average of the above numbers.
(In fact, the formula on the left is slightly smaller than the formula on the right, and the closer the growth rate is, the smaller the error is. )
When seeking the average growth rate, we should pay special attention to the expression of problems, such as:
1, "average growth rate from 2004 to 2007" generally refers to the growth rate excluding 2004;
2. "Average growth rate in 2004, 2005, 2006 and 2007" generally refers to the growth rate in 2004.
Determination of the changing trend of "numerator and denominator simultaneously expand/reduce fractions";
In 1 and A/B, if a and b expand at the same time, then ① if the growth rate of a is large, A/B will expand; (2) If the growth rate of B is large, A/B will shrink; In A/B, if A and B contract at the same time, then ① if A decreases rapidly, A/B contracts ② if B decreases rapidly, A/B expands.
2. In A/(A+B), if A and B expand at the same time, then ① If the growth rate of A is large, A/(A+B) will expand ② If the growth rate of B is large, A/(A+B) will shrink; In A/(A+B), if A and B contract at the same time, ① if A decreases rapidly, A/(A+B) contracts ② if B decreases rapidly, A/(A+B) expands.
Multi-part average growth rate:
If quantity A and quantity B constitute the sum of "A+B", the growth rate of quantity A is a, the growth rate of quantity B is b, and the growth rate of quantity "A+B" is r, then A/B=(r-b)/(a-r), which is generally simply calculated by "cross method".
Pay attention to several issues:
1, r must be somewhere between A and B. When you subtract "cross", one r comes before and the other r comes after.
2. The calculated ratio is the ratio before growth. If you want to calculate the proportion after growth, you must multiply this proportion by their respective growth rates.
Equal-rate growth conclusion:
If a quantity grows at a fixed speed, its growth will be bigger and bigger, and the value of this quantity will become a "geometric series", and the square of the middle term is equal to the product of the two terms on both sides.
★ Quick calculation skills 10: comprehensive speed algorithm
Key points:
"Comprehensive speed algorithm" contains many quick calculation methods. Although it is not as good as the first nine quick calculation skill systems in our data analysis problem, these quick calculation methods are still effective means to improve the calculation speed.
Fast calculation of square number:
Remember the common squares, especially the squares of numbers within 1 1-30, which can improve the calculation speed:
12 1、 144、 169、 196、225、256、289、324、36 1、400
44 1、484、529、576、625、676、729、784、84 1、900
Fast calculation by mantissa method;
Because the data involved in the data analysis test questions are almost approximate results, we usually emphasize the first estimate when calculating, and the mantissa can often be ignored. Therefore, the mantissa method in data analysis is only applicable to the calculation without approximation or approximation. Historical data prove that mantissa method can not be used in the analysis of national examination data, but it can still effectively simplify the calculation in the analysis of local examination data.
Dislocation addition/subtraction:
A×9 fast calculation skills: A× 9 = A× 9 = A×10-a; ; Such as: 743×9=7430-743=6687.
A×9.9 Fast calculation skills: A× 9.9 = A×10+A ÷10; Such as: 743×9.9=7430-74.3=7355.7.
A× 1 1 Fast calculation skills: a×11= a×10+a; Such as: 743×11= 7430+743 = 8173.
A× 10 1 type quick calculation skills: a×101= a×100+a; Such as: 743×101= 74300+743 = 75043.
Fast calculation skills of multiplication and division method 5,25, 125;
A× 5 fast calculation skills: A× 5 =10a ÷ 2; A÷ 5 fast calculation skills: A÷5= 0. 1A×2.
For example, 8739.45×5=87394.5÷2=43697.25.
36.843÷5=3.6843×2=7.3686
A× 25 fast calculation skills: A× A× 25 =100a ÷ 4; ÷ 4; A÷ 25 Fast calculation skills: A÷25= 0.0 1A×4.
Example 7234×25=723400÷4= 180850.
37 14÷25=37. 14×4= 148.56
A× 125 quick calculation skills: A×125 =1000a ÷ 8; A÷ 125 Fast calculation skills: A ÷ a ÷125 = 0.001a× 8+0a× 8.
Example 8736×125 = 8736000 ÷ 8 =1092000.
4 1 15÷ 125=4. 1 15×8=32.92
Add half:
A× 1.5 fast calculation skills: a×1.5 = a+a ÷ 2;
Example 3406×1.5 = 3406+3406 ÷ 2 = 3406+1703 = 5109.
Fast calculation skills of the product of two numbers with the same initial number and complementary mantissa;
Product header = header× (header+1); Tail of product = Tail× Tail