Do more with less memory

Chapter 16 Improving Mental Arithmetic Ability Enhances Memory
Mental arithmetic ability is closely linked with memory.Learn the mental arithmetic shortcuts in this chapter and use them frequently in your daily life. Practicing them can strengthen your memory.After studying this chapter, you will have the following abilities:

(1) Shop with ease.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.

Can you complete the exercises below in 3 minutes?Paper and pencil can be used for rough drafts.

(1) 900÷25=
(2) 45x22=
(3) 40x7.9=
(4) 19x25=
(5) 630-485=
(6) 17.6÷0.4=
(7) 726x11=
(8) 62÷99=
(9) 170x10.75=
(10) 5825÷64=
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] Calculate the value of 6x14.

Step 14: Divide 7 into 2x[-].

Step 6: Now the problem becomes to find the value of 7x2x[-].

第三步：6×7=42；42×2=84。

So the answer is: 6×14=84.

[Example 2] Calculate the value of 90×1.4.

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=126.

Practice questions for Shortcut 1:

(1) 60×1.6=
(2) 7×120=
(3) 17×22=
(4) 15×320=
(5) 78×12=
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 ones 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] Calculate the value of 21×23.

第一步：将个位数相乘，个位数是1和3，所以1×3=3。

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）+（1x2）=6+2=8
8 is the tens digit.

Step [-]: Multiply the ten digits together.

2 × 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] Calculate the value of 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 [-]:

（3x3）+（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.

Practice questions for Shortcut 2:

(1) 31×24=
(2) 72×54=
(3) 67×89=
(4) 81×38=
(5) 43×16=
Here we will learn how to multiply two digit numbers by 11, or 1.1, 0.11, 110, etc.The method is also very simple. Add the two-digit tens digit and the ones digit, and place the conclusion in the two digits, which is the answer we want.

Take a look at the example below.

[Example 1] Calculate the value of 35×11.

Step [-]: Add two numbers with two digits, then it 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] Calculate the value of 5.4×11.

Step 54: Don't worry about the decimal point. What you think about is multiplying 11 by 54. Now the problem becomes 11×[-].

Step 5: Add two numbers with two digits, then 4+9=[-].

Step 9: Put 5 in the middle of 4 and [-].

Step 54: The result of 11×594 is [-].

Now it’s time to consider the decimal point. You actually ignored a decimal point in the above calculation. Here you have to take the decimal point into account, count one digit from right to left, and the decimal point should be placed between 9 and 4, so 5.4 The final result of ×11 is 59.4.

[Example 3] Calculate the value of 9.7×1.1.

Note that here, 9+7=16, the resulting number is a two-digit number, what should I do in this case?Keep the ones digit 6, place 6 between 9 and 7, and add the tens digit 1 and 9:

9 (16) 7
1067
So the conclusion is 1067. Now let's consider the position of the decimal point, which should also be two digits from right to left, so the final result of 9.7x1.1 should be 10.67.You can also use another method to quickly draw a conclusion, and quickly estimate that the result of 9.7×1.1 is about 10, and the result must be 10.67.

Practice questions for Shortcut 3:

(1) 45×11=
(2) 56×1.1=
(3) 65×110=
(4) 9.3x11=
(5) 4.7x110=
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 number that is difficult to calculate into two or more numbers that are easier to calculate.

Let's look at a few examples below:

[Example 1] Calculate the value of 13x12.

Divide 13 into 12+1, so we can calculate like this:

（12×12）十（12×1）=144+12=156
[Example 2] Calculate the value of 507x6.

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
Practice questions for Shortcut 4:

(1) 58×7=
(2) 74×9=
(3) 6×93=
(4) 34×70=
(5) 45×21=
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] Calculate the value of 9x28.

第一步：将28加上2变为30，9×30-270。

Step 270: Subtract 2x9 (=18) from [-].

The third step: 270-18=252.

So to conclude:
9 × 28-252
[Example 2] Calculate the value of 39×99.

第一步：将99加上1，变成100，39×100-3900。

Step 3900: Subtract 1x39 from 39, that is, subtract [-].

The third step: 3900-39=3861.

Practice questions for Shortcut 5:

(1) 79×5=
(2) 29×12=
(3) 14×48=
(4) 89x20=
(5) 17x25=
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 must be 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) If a number is divisible by 5, its mantissa must be O or 5, such as 225, etc.

(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.

【Exercise questions】

(1) Which of the following numbers is not divisible by 3?

111￥183￥166￥141
(2) Which of the following numbers is not divisible by 4?

348￥488￥834￥384
(3) Which of the following numbers is not divisible by 6?

282￥474￥390￥256
(4) Which of the following numbers is not divisible by 9?

239￥234￥918￥630
(5) Which of the following numbers is not divisible by 12?

156￥384￥468￥150
Divides a larger, more complex number into several smaller numbers and multiplies them.

It can get complicated if you divide a number by 24, where you divide 24 into 2 and 12, or 3 and 8, or 4 and 6.Take whichever method is more beneficial to your calculations.

[Example] Calculate the value of 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 calculation is to divide 24 into 3 and 8.

Step 4488: 8÷561-[-].

Step two: 561÷3-187.

This algorithm is also very simple.

【Exercise questions】

(1) 1300÷25=
(2) 390÷15=
(3) 168÷14=
(4) 252÷36=
(5) 5824÷64=
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.

(1) 192÷24=
(2) 496÷8=
(3) 198÷18=
(4) 322÷14=
(5) 228÷12=
Mental multiplication is easier than mental 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) If a number is to be divided by 15, first multiply by two and then divide by 30.

For example: To calculate 135÷15.

Calculate 135×2-270 first.

Then calculate 270÷30, the number is 9.

(3) To divide a number by 7.5, first multiply by 4 and then divide by 30.

For example, to calculate 390÷7.5.

First calculate 390×4=1560.

Then calculate 1560÷30-52.

(4) If a number is to be divided by 12.5, first multiply the number by 8 and then divide by 100.

For example, to calculate 175÷12.5.

The calculation method is 175×8=1400.

1400÷100=14.

(5) If a number is to be divided by 37.5, first multiply it by 8, then divide by 300.

For example, to calculate 675÷37.5.

先计算675×8-5400，然后除以300，等于18。

【Exercise questions】

(1) 795÷5=
(2) 195÷15=
(3) 105÷7.5=
(4) 162.5÷12.5=
(5) 300÷37.5=
If a number is divided by 999999, the result must be a repeated number. If the number you calculated 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) 7÷9=0.7777
(3) 8÷9=0.8888
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
【Exercise questions】

(1) 53÷99=
(2) 763÷99=
(3) 514÷9=
(4) 2000÷999=
(5) 760÷99=
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. Quickly calculate the amount of money that should be returned when shopping.

2. The tip due to the waiter can be calculated immediately after dinner.

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

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

5. Reasonable investment and financial management.

6. Be confident in your computing skills.

Above we covered various ways of calculating multiplication and division.Below are the answers to the practice questions.

1. Answers to self-test questions

(1) 36
(2) 990
(3) 316
(4) 475
(5) 145
(6) 44
(7) 7986
(8) 0.6262
(9) 1827.50
(10) 91
2. Answers to multiplication practice questions
1) Shortcut 1
(1) 96
(2) 840
(3) 374
(4) 4800
(5) 936
2) Shortcut 2
(1) 744
(2) 3888
(3) 5963
(4) 3078
(5) 688
3) Shortcut 3
(1) 495
(2) 61.6
(3) 7150
(4) 102.3
(5) 517
4) Shortcut 4
(1) 406
(2) 666
(3) 558
(4) 3150
(5) 945
5) Shortcut 5
(1) 395
(2) 348
(3) 672
(4) 1780
(5) 425
3. Answers to division practice questions
1) Shortcut 1
(1) 166
(2) 834
(3) 256
(4) 239
(5) 150
2) Shortcut 2
(1) 52
(2) 26
(3) 12
(4) 7
(5) 91
3) Shortcut 3
(1) 8
(2) 62
(3) 11
(4) 23
(5) 19
4) Shortcut 4
(1) 159
(2) 13
(3) 14
(4) 13
(5) 8
5) Shortcut 5
(1) 0.5354
(2) 7.7070
(3) 57.111
(4) 2.0020
(5) 7.6768
(End of this chapter)