## What Is The Mental Math (MM) Bonus Package?

The Mental Math (MM) Bonus Package provides additional mental math strategies, serving as an extension to the materials in the Mental Math Unleash The Power eBook (referred to as “eBook” for the sake of discussion). However, it is not another revision but a standalone set of mental math techniques to be given FREE with the purchase of the eBook. Some materials were written in parallel with the eBook, while others towards the end of completing the eBook. The MM Bonus Package was scheduled to be done in the fall of 2016; it has been completed in November of 2106. Originally, the MM Bonus Package was supposed to be an additional part for the second edition of the eBook in the coming years, but instead of waiting years for it to be made available, the decision was to have it completed as quickly as possible, further expanding the mental math knowledge base. A revision of both materials (also combining them into one product) is likely in the future and will include additional techniques.

So with the eBook and Bonus Package combined, the total number of pages exceeds 2300, representing an incredible amount of mental math knowledge. Don’t worry about the size of both materials. Each person can learn at his or her own pace. Certain techniques could be learned first without having to move through the materials sequentially, though recommended. Be sure to read and be aware of any “RECOMMENDATION” anywhere in the materials.

The MM Bonus Packages includes only mental math materials for seven subjects and not the full 14 as in the eBook. More details of the techniques are provided further down this page. The Bonus Package even includes a few extra techniques at the end; these are very effective and efficient techniques for the types of numbers mentioned.

**ADDITION**

**SQUARES**

**SUBTRACTION**

**CUBES**

**MULTIPLICATION**

**FRACTIONS**

**DIVISION**

### New And Modified Format

Besides introducing new materials, several techniques (only a few) from the eBook are also re-written in the MM Bonus Package, mainly for two reasons: (i) illustrating the differences between the original style or format in the eBook versus the one introduced in the Bonus Package and (ii) comparing a technique from the eBook to new ones for performing the same types of computations.

Almost every topic in the Mental Math Unleash The Power eBook consists of three sections: (i) **NOTES TO CONSIDER**, (ii) **EXAMPLES**, and (iii) **EXERCISES**. These same three sections are also part of the MM Bonus Package but with some differences. Since the **EXERCISES** section has the least amount of change, we’ll start from there – the only change is in the heading.

In terms of the **NOTES TO CONSIDER** section, some subtle and minimal changes were made. As to the **EXAMPLES **section, two versions may be provided for a technique. The first one is based on the original format in the eBook with some minor changes, and the second is an entirely new style. Some techniques are shown in both **EXAMPLES **formats, while others are presented in either one or the other style. These arrangements give different vantage points of how the techniques appear stylistically in terms of the learning process. Furthermore, a particular format may be better suited for a certain type of technique in terms of the visuals and how the written elements are presented, but in any case, both are useful.

Notice the changes in the **NOTES TO CONSIDER** and EXAMPLES below as compared to the corresponding ones on the __ More On eBook__ page. The pages (images) under

**NOTES TO CONSIDER**and

**EXAMPLES**resemble the formats in the Bonus Package (some minimal changes) and the new style.

Here is another example of the identical three sections but this time the format for the **EXAMPLES** section is an entirely new style (see **EXAMPLESEXAMPLES** section is written under the **NOTES TO CONSIDER**.

Because the MM Bonus Package is not another eBook (“extended” material), a glossary, appendixes, and other features are not given. However, a table of content is provided for easier navigation. Moreover, exercises are also provided for all new techniques except for a few of those techniques that were repeated from the Mental Math Unleash The Power eBook.

### What Are The New Techniques In The MM Bonus Package?

The Mental math (MM) Bonus Package when fully completed will contain over 700+ pages, therefore, only a few of the new techniques listed. A few techniques from the Mental Math Unleash The Power eBook have been rewritten for the purposes mentioned earlier. The vast majority of techniques are new and others are extensions to existing ones from the eBook.

### ADDITION

##### Place Values

Using places values is extended from the eBook, and is applied more in depth along with combining it with other techniques. By doing so, greater efficiency is gained. Other variants of using places values are also introduced, and they increase the effectiveness of the applications.

##### Miscellaneous

A few miscellaneous addition techniques were introduced at the end of the eBook, and they are expanded in this bonus package to cover more depth, allowing for more ways to add numbers to each other. For example, using smaller manageable parts (SMP) to add 12479 to 34 involves simplifying 34 into 30 and 4, and then adding 12479 to 30 to get 12509. Finally, add 12509 to 4 for 12513 as the answer. However, with one of the miscellaneous techniques, we can basically add 12479 to 21 to get 12500, and add 12500 to 13 to get 12513.

##### Reference Number

This technique is used for adding several numbers to each other much like applying place values, and can add as many numbers as can be managed mentally. So there is not set rule as to the number of numbers that can be added. It applies a refer.

### SUBTRACTION

##### From The Subtrahend

Typically, when subtracting one number from another, the minuend is the number doing the subtracting, while the subtrahend is the number being subtracted. For example, if 15 is subtracted from 40, the minuend in this case would be 40, and 15 as the subtrahend. This technique, for the most part, depending on the numbers present, uses the subtrahend or a part of it to do the subtracting. In other instances, the minuend is also involved.

##### All From 9 And Last From 10

Subtracting 345 from 1000 is a typical math problem solved with this technique. Basically, every zero in 1000 except the last is replaced by 9 with the last being 10, and then subtraction is performed. However, this technique offers much more. What if we are subtracting 540 from 3400 or 756 from 2040? Variances of the technique are also available to handle such problems.

##### Miscellaneous

As with math addition, miscellaneous subtraction techniques are also extended here from the eBook. These techniques offer several methods to handle a wide variety of subtraction problems. For example, using smaller manageable parts (good solid method) for subtracting 69 from 7852, then the arithmetic usually consists of simplifying 69 into 60 and 9. Next, 60 is subtracted from 7852 to get 7792, and finally subtracting 9 from 7792 for 7783 (answer). However, with one of the miscellaneous techniques, we can just subtract 17 from 7800 to get 7783.

### MULTIPLICATION

##### Any Number By One

This technique is used for multiplying any whole numbers by single-digit multipliers such as in multiplying 3456 by 4. Yes, there is already a technique in the eBook used for the same types of tasks but this technique is different – the arithmetic processes are started from the left or the first digits of the numbers. For example, other techniques begin from the right of the number or in other forms. With this technique using the problem mentioned above, the 3 in 3456 is multiplied by 4 first with the arithmetic process continuing to the right.

##### By 9

Multiplying any whole number by 9 is made easy and quick with this technique, especially when multiplying two and three-digit numbers by 9. The technique is also capable of multiplying large numbers by 9 like four digits or greater. In addition to this technique, other techniques are introduced for multiplying whole numbers by single-digit multipliers.

##### Two By Two

Four more techniques are introduced for multiplying two-digit numbers to each other. Each technique utilizes different types of strategies to accomplish the same tasks. The advantages of having more than one technique come from exposure to additional ways of doing things.

##### Three By Two

Additional techniques are introduced for multiplying three-digit numbers by two-digit numbers. These techniques increase the repertoire available for tackling these types of numbers. Most of the techniques use bases as the main strategies.

##### Three By Three

More techniques are introduced for multiplying three-digit numbers to each other besides the vertically crosswise (called standard in the eBook). These additional techniques provide good foundations to accomplish the necessary tasks required. Understanding them allows for easier computing in solving problems like multiplying 345 by 256 and many others. Even techniques for multiplying larger numbers are provided up to eight digits – learn how to multiply up to eight-digit numbers to each other. It is already very good if we can multiply three to fourdigit numbers mentally, but if we can do more, that would be great.

##### BASE 10, 20, 30, 40, 60, 70, 80, or 90

These techniques are designed to multiply two and three-digit numbers together using different base amounts. These bases (and principles related to them) are also used in squaring two and three-digit numbers as well. For example, using base 10, we can multiply the same sets of numbers as with any of the other bases mentioned above in the title. The key here is learning how to use them effectively under the right situations.

### DIVISION

##### By 15, 35, 40

These techniques are strategies for dividing numbers by either 15, 35, and 40 including the family members of each divisors such 0.15, 1.5, 150 for 15, 0.035, 3.5, 3500 for 35, 0.045, 450, 4500 for 45, and many other members respective of each divisor. So this means there is a whole range of techniques for these groups, providing many methods.

A great feature about these techniques is that each family member uses the same “initial computation(s),” and the difference is in moving the decimal point. For example, if 472 is divided by 0.35, and 472 is also divided by 350, both apply identical “initial computation(s).” Therefore, learning a new set of rules for the different divisors in the same family is not required. Furthermore, additional variances of these techniques in each family are given in the MM Bonus Package, providing even more ways of how to use these divisors.

##### Common Fractions

These techniques use common fractions to decimal equivalents for dividing one number from another. They are the same types of values featured in the eBook but this time they are used as division methods. The materials in the MM Bonus Package show how to use these divisors effectively without having to use math division throughout the whole arithmetic processes. Because decimal values are involved, keep in mind the answers are affected by the number of digits after the decimal point and rounding.

The key is knowing the common fractions to decimal equivalents. Review them in the Mental math Unleash The Power eBook if necessary. If they are already known, dividing 6 by 14 is simple; the answer is 0.4285714 and can be computed rather quickly mentally with ease. So there should not be any problems dividing 0.38 by 0.016 and many other math problems, including others involving decimal numbers. Their applications are straight forward, and only a short amount of time is needed to learn them.

##### Three To Eight-Digit Dividends By 99

Two versions of the techniques are given with both capable of dividing three to eight-digit numbers (or even larger numbers) by 99. The best feature of these two techniques is that both use simple math addition and little division, when required, to do the jobs. For example, dividing 34521 by 99 involves only math addition. Here is one short way in producing the answer. Add 345 to 3 to get a quotient of 348, and the remainder is the sum between 48 and 21. Therefore, the answer is a quotient of 348 with a remainder of 69.

These two techniques are also extended to dividing by 999, which incorporates similar principles with some modifications because 999 is a three-digit value.

##### Find Remainder

What is the remainder if 4511109 is divided by 3? These are the types of math problems solved under several techniques. Depending on the divisor, some problems are so easy to solve, using it one time is all that is required to master it. Others are more “complex” but not in the sense that they are harder. The logical requires more time to explain. For example, finding the remainder when 91340 is divided by 6 is not difficult at all but in explaining the details, it requires more input than when describing the arithmetic process of another divisor.

### SQUARES

##### Close To Between Two Full Base Amounts

This technique is used for squaring a number, which is between two full base amounts. A full base amount in the context of the bonus package and eBook is a number with all zero/zeroes, except the first digit such as in 100, 200, 400, 1000, 2000, and so forth. A number like 1020 is not considered a full base amount value. Squaring a number such as 252 is made easy with this technique. Notice 252 is basically between the two full base amounts of 200 and 300.

##### Two-Digit Numbers: Base 20, 25, 30, 40, 50, 60,70, 75, 80, 90, or 100

Several techniques dependent on bases (see list above) are available for squaring two-digit numbers. This means base 20 can be used to square 45 as well as base 40. Therefore, there is flexibility when these bases are applied, is especially efficient in the right situations. An important component is understanding the similarities and differences between the bases, so they can be used effectively.

##### Three-Digit Numbers: Base 20, 25, 30, 40, 50,60, 70, 75, 80, 90, or 100

The same bases above are also used in squaring three-digit numbers. The methodologies between squaring two and three-digit numbers are closely related with some twists due to the difference in the two types of numbers. As with squaring two-digit numbers, a base such as 50 can be used to square a number like 165 and others. At the same time, a base of 75 and the rest of the other bases can square the same numbers being squared by base 50.

##### Sum Of Squares

Several techniques are provided for computing the sum of squares such as adding 15 squared to 16 squared. One of the techniques is capable of finding the sum between the squares of these two numbers (38 and 24) and others meeting the conditions of the technique in an easy manner. For example, the sum between the squares of 38 and 24 is produced from multiplying 20 by 101 to get 2020. This means squaring one number at a time and then adding the results to each other is not needed.

### CUBES

##### Two-Digit Number: Base Amounts

This is a technique used for cubing two-digit numbers by applying base amounts. The application of bases where possible allows for easier computing since numbers ending in a zero are used in the arithmetic processes.

##### Close To 50

Here is a specific technique for squaring numbers close to 50. Generally, a specific driven technique does a better job for certain types of numbers due to relationships that may exist between those numbers and some reference value. This technique is more efficient than a general-purpose one if the numbers are close to 50 such as 44, 45, 47, 52, 56, 62, and so forth.

##### Three-Digit Number

Cubing three-digit can be challenging when done mentally. Two methods are provided in the bonus package where the arithmetic processes are broken down into smaller parts. By computing a part at a time, the answers can be determined mentally without too much difficulty. Being proficient in adding large numbers, multiplying two and three-digit numbers, and squaring and cubing two-digit numbers is recommended before cubing three-digit numbers.

### FRACTIONS

##### Sum Of Fraction Reverses

Many techniques involving fractions are in the eBook, and the MM Bonus Package offers several new ones. One of these techniques is for adding fractions, which are reverses of each other. For example, with this technique, we can add fractions like 5/6 to 6/5 or between others without using the lowest common denominator (LCD). The answers can be in two formats (mixed and improper fractions), and depending on the format required, applying different methodologies are required but involving only simple computations.

##### Add Consecutive Fractions

Adding fractions whose values are in consecutive order from one fraction to the next involves another specific type of technique. The set of fractions must have all four values in consecutive order starting with the numerator from either fraction. Here are some examples of the sets of fractions meeting this requirement: 1/2 + 34, 5/6 + 7/8, 11/12 + 9/10, 21/22 + 19/20, and others. Once again, the LCD is not needed, increasing efficiency, and the technique makes it simple to do the job.

##### Add Fractions: Factors Or Multiples Of Denominators

Yet, here is another specific driven technique. This time it is used for adding fractions whose denominators are multiples of one another. What is a value that is a multiple of another value? If we have 2, then a multiple of 2 is 4. Therefore, numbers such as 6, 8, 10, 12, and so forth are multiples of 2. In terms of fractions, the multiples are defined by the denominators. The technique applies simple multiplication and addition processes to create the answers without using LCDs and other math related tasks from the usage of LCDs. For example, finding the sum between 1/3 and 5/6 involves adding 2 to 5 for the numerator while keeping 6 as the denominator; the answer is 7/6 in improper format or 1 1/6 in mixed fraction.