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Chapter 39 Improving Mental Arithmetic Ability Enhances Memory
Mental arithmetic ability is closely related to memory.Learn the mental arithmetic shortcuts in this chapter and use them frequently in your daily life. Practicing them can strengthen your memory.Here's what you'll get by studying this chapter:

(1) Easy shopping.You can immediately know the amount of money to get back.

(2) Budget your return on investment.

(3) It will quickly calculate the percentage and calculate the proportion of your money spent in different places.

(4) Calculate the various taxes you need to pay in advance.

(5) Calculate the profit and loss ratio for your company, you will be very skilled and sensitive to numbers.

These mental math skills are not necessarily memory skills.But they can definitely help improve your memory and enhance your flexibility.Simply put, they allow you to memorize more efficiently.

Before you start working on this chapter, do some homework, keep in your head what we've talked about above, and have a pen and paper ready to let the answers pop into your head.After studying this chapter, you will find yourself progressing in mental arithmetic.

Test your basic mental arithmetic skills
Can you complete the exercises below in 3 minutes?Paper and pencil can be used for rough drafts.

(1)900÷25=
(2)45×22=
(3) 40×7=
(4)19×25=
(5) 630－485=
(6)17。6÷0。4=
(7)726×11=
(8)62÷99=
(9) 170×10=
(10)5825÷64=
Mental Multiplication Shortcut 1: Speed ​​Up Your Speed
Large numbers can be divided into sums of several decimals without changing them.The numbers are too large to be multiplied. You can divide the large numbers into decimals and multiply them separately.

Example 1:

6 × 14 =
Step 14: Divide 7 into 2×[-]
The second step; now the problem becomes 6×7×2=
第三步： 6×7=42 42×2=84
So the answer is: 6×14=84
Example 2:

90×1=
Step 90: Divide 9 into 10×[-]
Step 9: Now the problem becomes 10×1×4. [-]=
第三步：10×1。4=14  14×9=126
So the answer is 90×1=4
Practice questions for Shortcut 1:

(1) 60×1=
(2)7×120=
(3)17×22=
(4)15×320=
(5)78×12=
Mental Multiplication Shortcut 2: Double-digit Multiplication

We'll introduce you to the cross-multiply method and right-to-left calculations here.After you master this method, you can multiply two-digit numbers without paper and pencil.

The method is: first multiply the single digits, then cross-multiply, and finally multiply the tens digits.

This method is very simple and easy to master.Let us look at a few examples in detail.

Example 1:

21 × 23 =
第一步：将个位数相乘，个位数是1和3，所以1×3=3
The second step: cross multiplication.The method of cross multiplication is to multiply the tens digit of the first number by the ones digit of the second number, and then multiply the ones digit of the first number by the tens digit of the second number.Add up the obtained results.

（2×3）＋（1×2）=6＋2=8
8 is the tens digit.

Step 2: Multiply the ten digits together. 2×4=[-]
4 is the hundreds digit

Step 483: Get the result, [-].

Note, what to do if the result is greater than 9, see the example below.

Example 2:

34 × 23 =
Step 3: Multiply single digits, 4×12=[-]
12 is a number greater than 9, you keep the unit digit 2, and bring the tens digit 1 in the following calculations
Step 1: Cross multiply, then add [-]
（3×3）＋（4×2）=9＋8=17
Don't forget to add 1, 17+1=18
At this time, you keep the ones digit 8, and bring the tens digit 1 in the following calculations
Step 1: Multiply the tens digit and add [-].

3×2+1=7
Step 782: Draw conclusions: [-]
So the final conclusion is 34×23=782
Exercise for Shortcut 2:
(1)31×24=
(2)72×54=
(3)67×89=
(4)81×38=
(5)43×16=
Shortcut to mental multiplication 3: Learn to multiply two-digit numbers and 11

Here we will learn how to multiply two digits by 11, or 1, 1, 0, etc.The method is also very simple. Add the two-digit tens digit and the ones digit, and put the conclusion in the two digits, which is the answer we want.

Consider the following example:
Example 1:

35 × 11 =
Step [-]: Add two numbers with two digits, then that is,

3+5=8
Step 8: Put 3 in the middle of 5 and [-],
The third step: come to the conclusion 385, which is the answer of 35×11.

Example 2:

5. 4×11
Step 54: If you want to ignore the decimal point, all you have in mind is to multiply 11 by [-].

Now the problem becomes 54×11.

Step [-]: Add the two numbers with two digits together, then
5+4=9
Step 9: Put 5 in the middle of 4 and [-].

Step 54: The result of 11×594 is [-].

Now to consider the issue of the decimal point, you actually ignored two decimal points in the above calculation, here you have to take the decimal point into account.Counting two digits from right to left, the decimal point should be placed between 5 and 9, so the final result of 5×4 is 11.

Example 3:

9×7=
注意在这里，9＋7=16 得出的数字是一个二位数，这种情况下怎么办呢？保留个位数6，将6放在9和7之间，将十位数1和9相加：

9 (16) 7
1067
所以得出的结论是1067，现在我们来考虑小数点的位置，也应该是从右往左数两位，所以9。7×1。1的最后结果应该是10。67。你也可以通过另一种方法来很快得出结论，很快的估算一下，9。7×1。1得出的结果是10左右，结果就肯定是10。67。

Exercise for Shortcut 3:
(1)45×11=
(2) 56×1=
(3)65×110=
(4)9×3=
(5)4×7=
Shortcut to multiplication in mental arithmetic 4: use recombination method to calculate multiplication
The method we will introduce here is similar to the method introduced in shortcut 1, which is the method of dividing relatively large and complex numbers into several smaller numbers for calculation.Divide a difficult number into two or more easier numbers.

Let's look at a few examples below:

Example 1:

13 × 12 =
将13分为12＋1 所以我们可以这样来计算：（12×12）＋（12×1）=144＋12=156
Example 2:

507 × 6 =
In the same way, 507 can be divided into 500+7 so we can calculate it like this:

（500×6）＋（7×6）=3000＋42=3042
Exercises for Shortcut 4
(1) 58×7
(2)74×9=
(3)6×93=
(4)34×70=
(5)45×21=
Shortcut to multiplication in mental arithmetic 5: Use rounding method to calculate multiplication
Integers are always easy to compute.We can find a way to get the number close to the number of tens and hundreds.Then subtract the overcounted number and add the undercounted number.

Take a look at the example below.

Example 1:

9 × 28 =
第一步：将28加上2变为30，9×30=270
Step 270: Subtract 2×9=18 from [-]
The third step: 270－18=252
So to conclude:
9 28 × = 252
Example 2:

39 × 99 =
第一步：将99加上1 变成100。39×100=3900
Step 3900: Subtract 1×39 from 39, that is, subtract [-]
The third step: 3900－39=3861
Shortcut 5: Practice Questions for

(1)79×5=
(2)29×12=
(3)14×48=
(4)89×20=
(5)17×25=
Shortcut to mental division 1: How to judge whether a number is divisible

To determine whether a number is divisible by another number, here are some basic rules you can remember:

(1) If a number is divisible by 2, its mantissa is an even number, such as the number 1996
（2）如果一个数可以被3整除，它的各个数位上的数字和应该可以被3整除，如数字369，3＋6＋9=18，18可以被3整除。

(3) If a number is divisible by 4, its last two digits should be divisible by 4.Such as the number 384.

（4）如果一个数要被5整除，它的尾数必须是0或者5，如225等。

(5) If a number is divisible by 6, it must be divisible by both 2 and 3.

(6) If a number is divisible by 8, its last three digits must be divisible by 8, such as 1992.

(7) If a number is divisible by 9, the sum of its digits should be divisible by 9, such as 423.

(8) If a number is divisible by 10, its last digit must be 0, such as 230.

(9) If a number is divisible by 12, it must be divisible by both 3 and 4, such as 144.

The rules about 7 and 11 are too difficult for us to cover here.

Practice questions:
(1) Which number below is not divisible by 3?

111 183 166 141
(2) Which number below is not divisible by 4?

348 488834 384
(3) Which number below is not divisible by 6?

282 474 390 256
(4) Which number below is not divisible by 9?

239 234 918 630
(5) Which number below is not divisible by 12?

156 384 468 150
Shortcut to Mental Division 2: Simplified Division

将一个较大较复杂的数分成几个较小的数字相乘。如果你要算一个数除以24，可能就会很复杂，你可以将24分成2和12，或是3和8或是4和6。哪个更利于你的计算，就采取哪种方法。

Example:
4488 ÷ 24 =
Step 24: Divide 4 into 6 and multiply by [-]
Step 4488: Divide 4 by 1122 and the number you get is [-].

Step 1122: Divide 6 by 187 to get [-]
That's easy to figure out.

Yet another way of calculating is to divide 24 into 3 and 8
The first step: 4488÷8=561
The second step: 561÷3=187
This algorithm is also very simple.

Practice questions:
(1)1300÷25=
(2)390÷15=
(3)168÷14=
(4)252÷36=
(5)5824÷64=
Mental division shortcut 3: how to deal with division of two even numbers
Let's give an example to illustrate the division of two even numbers. For example, if you want to divide 136 by 8, it is equivalent to calculating 68 divided by 4, which is equal to 34 divided by 2, which is equal to 17.This method is not very simple.

Practice questions:
(1)192÷24=
(2)496÷8=
(3)198÷18=
(4)322÷14=
(5)228÷12=
Shortcuts to mental division 4: How to deal with division by 5

Mental multiplication is easier than division.We'll cover here how to do division by 5 through multiplication.Take a look at the example below.

(1) If a number is to be divided by 5, then you should first multiply it by 2 and then divide by 10.

For example: 725÷5
我们可以先来计算725×2=1450，然后将1450除以10，得数是145。

(2)一个数字要被15除的话，先乘以二，然后除以30。比如：要计算135÷15=
First calculate 135×2=270
Then calculate 270÷30, the number is 9.

(3) If a number is to be divided by 7, first multiply by four and then divide by 5.Examples are as follows;
390÷7=
First calculate 390×4=1560
Then calculate 1560÷30=52
(4)如果一个数要除以12。5的话。先将这个数字乘以8，然后除以100。例子如下：
175÷12=
The calculation method is 175×8=1400
1400 ÷ 100 = 14
(5)如果一个数要被37。5除，先将它乘以8，然后除以300。例子如下：
To calculate 675÷37. 5=
先计算675×8=5400，然后除以300，等与18。

Practice questions:
(1)795÷5=
(2)195÷15=
(3) 105÷7=
(4)162。5÷12。5=
(5) 300÷37=
Shortcut to mental arithmetic division 5: Calculate the title of dividing by 9

If a number is divided by 9, 99, 999, the result must be a repeated number. If the number you calculate is not characterized by repetition, then you must have made a mistake.

If the first number is smaller than the second number, then the first number will repeat itself, see the following example:
(1)5÷9=0。5555
(2)73÷9=0。737373
(3)18÷9=0。018018
Note that when the first number is greater than the second number, the resulting number will still repeat, but in this case, the number is repeated in another way.See the example below:
(1)40÷9=4。4444
(2)900÷99=9。090909
(3)2500÷999=2。502502
Practice questions:
(1)53÷99=
(2)763÷99=
(3)514÷9=
(4)2000÷999=
(5)760÷99=
You will be proud of your mental arithmetic ability

Have you been inspired by the mental math shortcuts you learned in this chapter?In fact, there are many rules in numbers. As long as you pay attention to observation and discover the rules, you can also sum up better and faster calculation methods than these methods.At the same time, your memory will also be exercised.Through the study of this chapter, your memory can be greatly improved, because mental arithmetic itself requires a lot of memory, and you will be proud of the following things you can do:

1.When buying something, quickly calculate the amount of money that should be returned.

2.Tips due to waiters can be calculated right after dinner.

3.Before you go to the counter to pay your bill, estimate the amount due.

4.Figure out the most reasonable method of borrowing and lending money.

5.Reasonable investment and financial management.

6.Have confidence in your computing abilities.

Answers to practice questions

Above we introduced five methods each of calculating multiplication and division.Below are the answers to the practice questions.

Answers to self-test questions

(1)36
(2)990
(3)316
(4)475
(5)145
(6)44
(7)7986
(8)6262
(9) 1827
(10)91
Answers to Multiplication Practice Problems
shortcut 1
(1)96
(2)840
(3)374
(4)4800
(5)936
shortcut 2
(1)744
(2)3888
(3)5963
(4)3078
(5)688
shortcut 3
(1)495
(2) 61
(3) 71
(4) 102
(5)517
shortcut 4
(1)406
(2)666
(3)558
(4)945
(5)517
shortcut 5
(1)395
(2)348
(3)672
(4)425
(5)2670
Answers to division practice questions
shortcut 1
(1)166
(2)834
(3)256
(4)239
(5)150
shortcut 2
(1)52
(2)26
(3)12
(4)7
(5)91
shortcut 3
(1)8
(2)62
(3)11
(4)23
(5)19
shortcut 4
(1)159
(2)13
(3)14
(4)13
(5)8
shortcut 5
(1) 0
(2) 7
(3) 57
(4) 2
(5) 7
(End of this chapter)