Mental Math walkthrough

feat of strength: 724,573×746,934 = 541,208,209,182

This document is a basic walkthrough to give context on some of the mental math problems I’ve done since October of 2021. It is not meant to convey the entirety of how to learn or gain this ability and is more so proof of work or of my ability to do mental math. There is a sufficient amount of work in this document, however, there is still much of the process and cognitive steps that are not present. It won’t grant insight into how to do mental math, but it can certainly help in learning how to do these sorts of problems without a calculator and show part of the processes I take in order to solve these problems in my head.

• All red numbers are either critical points in a problem or the answer to a presented problem.

(1.) Six-digit multiplication 

724,573×746,934 

First step is to multiply the highest placement of ten by the other of its kinds, which is 100,000. After doing so, multiply that given number by the raised value from the first core number (7) and then multiply that by the raised value from the second core number (7). This process will be repeated many many times.

100,000×100,000=10B

10B×7= 70B×7=490B

490B

10,000×100,000=1B

1B×2=2B×7=14B

14B

1,000×100,000=100M

100M×4=400M×7=2.8B

2.8B

100×100,000=10M

10M×5=50M×7=350M

350M

10×100,000=1M

1M×7=7M×7=49M

49M

3×100,000=300,000×7=2.1M

2.1M

Make note that the numbers in red would’ve been added as they were concluded in real time. I am using this format for the sake of simplicity. I have cut out a significant number of minor steps and will gladly provide a detailed walkthrough of the sort if requested.

Now we shall add all the red numbers together and acquire our first benchmark.

490B+14B+2.8B+350M+49M+2.1M= 507,201,100,000

Benchmark #1: 507,201,100,000

724,573×746,934 

There’s two ways to go about this, either we could build upon the first benchmark, adding as we go, or we could acquire the second benchmark and then do the addition. I normally take the former since it’s less of a cognitive strain, but the latter seems to be more desirable when putting it into text.

100,000×10,000=1B

1B×7=7B×4=28B

28B

10,000×10,000=100M

100M×2=200M×4=800M

800M

1,000×10,000=10M

10M×4=40M×4=160M

160M

100×10,000=1M

1M×5=5M×4=20M

20M

10×10,000=100,000

100,000×7=700,000×4=2.8M

2.8M

3×10,000=30,000×4=120,000

120,000

Benchmark#2: 28,982,920,000

724,573×746,934 

100,000×1,000=100M

100M×7=700M×6=4.2B

4.2B

10,000×1,000=10M

10M×2=20M×6=120M

120M

1,000×1,000=1M

1M×4=4M×6=24M

24M

100×1,000=100,000

100,000×5=500,000×6=3M

3M

10×1,000=10,000

10,000×7=70,000×6=420,000

420,000

3×1,000=3,000×6=18,000

18,000

Benchmark #3: 4,347,438,000

724,573×746,934 

100,000×100=10M

10M×7=70M×9=630M

630M

10,000×100=1M

1M×2=2M×9=18M

18M

1,000×100=100,000

100,000×4=400,000×9=3.6M

3.6M

100×100=10,000

10,000×5=50,000×9=450,000

450,000

10×100=1,000

1,000×7=7,000×9=63,000

63,000

3×100=300×9=2700

2,700

Benchmark #4: 652,115,700

724,573×746,934

100,000×10=1M

1M×7=7M×3=21M

21M

10,000×10=100,000

100,000×2=200,000×3=600,000

600,000

1,000×10=10,000

10,000×4=40,000×3=120,000

120,000

100×10=1,000

1,000×5=5,000×3=15,000

15,000

10×10=100

100×7=700×3=2,100

2,100

3×10=30×3=90

90

Benchmark#5: 21,737,190

724,573×746,934

100,000×4=400,000×7=2.8M

2.8M

10,000×4=40,000×2=80,000

80,000

1,000×4=4,000×4=16,000

16,000

100×4=400×5=2,000

2,000

10×4=40×7=280

280

3×4=12

12

Benchmark#6: 2,898,292

724,573×746,934 = 541,208,209,182 

Benchmark #1: 507,201,100,000

Benchmark#2: 28,982,920,000

Benchmark #3: 4,347,438,000

Benchmark #4: 652,115,700

Benchmark#5: 21,737,190

Benchmark#6: 2,898,292

Final number:  541,208,209,182 

(2.) 2-digit multiplication 

37×56=

2072

30×50=1500

7×56=392

30×6=180

1500+392+180=

2072

37×56=

2072

Details:

• First, we need to multiply the 10s and their raised value, so that is 30 and 50.

Reference: 37×56

30×50=

1500

Once we do that, we set that aside for later.

•  Next, we remove the first core number’s 10 placeholder and its raised value then multiply the one value, which is 7, against the entirety of the second core number which is 56.

Reference: 37×56

7×56=

392

Like before, we set that aside for later.

• Now we get the first core number, we only want its 10 value and raised number, which is 30 and this is to be multiplied against the second core number’s one value, which is 6.

Reference: 37×56

30×6=

180

• Once these three problems are done, it is to be added while actively solving or once the three problems are complete.

30×50=1500

7×56=392

30×6=180

1500+392+180=

2072

37×56=

2072

(3.) 4-digit multiplication

The following is the multiplication method used in the last example. I’ve only used this method on problems that are 4 digits or less. For a long while I only used this method on 2-digit problems, but recently found out it is applicable to 4 digits and I’ve yet to attempt 5 digits with this method. It splits the problem into four problems, which are all to be added as they are solved. 

1. Multiply the thousand and hundred placements against each other.
   1. To make this simpler, remove the zeros and later reintroduce them.

6,732×5,946=

40,028,472

6,700×5,900=

39,530,000

• Temporarily remove the zeros for simplicity. 
• It is to then be solved in the manner of the second example listed in this walkthrough.

67×59

60×50=3000

7×59=413

60×9=540

3000+413+540=3,953

• Reintroduce the four zeros and that’s the first benchmark.

Benchmark #1: 39,530,000

Reference: 6,732×5,946

32×5,946=

190,272

• Take the last two numbers from the first core number 6,732 which is (32) and multiply it against the entirety of the second core number 5,946. Distribute the ten placement and it’s raised value.

10×5,946=59,460×3=178,380

• Now we must distribute the one value which is two.

2×5,946=11,892

• Then we must add these two numbers in order to acquire our second benchmark.

178,380+11,892=

190,272

Benchmark #2: 190,272

Reference: 6,732×5,946

6,700×46=

308,200

• For the final benchmark we must multiply the thousandth and hundredth of the first core number: 6,700 by the last two digits of the second core number.

6,700×10=67,000×4=268,000

6,700×6=40,200

268,000+40,200=

308,200

Benchmark#3: 308,200

• Now the last order of business is to add these benchmarks together to get our answer.

39,530,000+190,272+308,200=40,028,472

6,732×5,946=40,028,472

(4.) Percentage calculation 

73.48×0.42=

30.8616

• First thing to do is remove all decimals and then multiply.

• Distribute the 40 into a 10 then a 4.

  7,348×42

  7,348×10=73,480

73,480×4=293,920

    7,347×2=14,696

293,920+14,694=

308,616

• Now the decimal point will be reintroduced and will be moved 4 times from right to left.

73.48×0.42=

30.8616

(5.) Percentage calculation #2

10,541,655,223×0.25 =

2,635,413,805.75

• First order of business is to remove the decimal and then multiply.

10,541,655,223×25

10,000,000,000×25=

250,000,000,000

Benchmark #1: 250,000,000,000

541,000,000×10=5,410,000,000

5,410,000,000×2=10,820,000,000

541,000,000×5=2,705,000,000

10,820,000,000+2,705,000,000=

13,525,000,000

Benchmark #2: 13,525,000,000

655,000×25

655,000×10=6,550,000×2=13,100,000

655,000×5=3,275,000

3,275,000+13,100,000=

16,375,000

Benchmark #3: 16,375,000

223×25

223×10=2230×2=4,460

223×5=1115

4,460+1,115=

5,575

Benchmark #4: 5,575

250,000,000,000+13,525,000,000+16,375,000+5,575=263,541,380,575

• Now we reintroduce the decimal, which is moved 2 times.
  • 263,541,380,575
  • 2,635,413,805.75

10,541,655,223×0.25 =

2,635,413,805.75

(6.) Chain multiplication 

67×43×89×56=

14,358,904

• For this example, we will be operating from left to right and incrementally.

67×43

• For the sake of simplicity, I’m not going to use the unorthodox method and instead distribute. 

67×10=670×4=2,680

67×3=201

2,680+201=

2,881

2,881×89

2,881×10=28,810×8=230,480

2,881×9= 25,929

25,929+230,480=

256,409

256,409×56

256,409×10=2,564,090×5=12,820,450

256,409×6=1,538,454

12,820,450+1,538,454=

14,358,904

67×43×89×56=

14,358,904

(7.) PEMDAS

67×45-73+84×53=

7,394

• First things first are we want to follow the rules of PEMDAS. Multiplication goes first, then addition, and lastly subtraction. 
• We will be using the 4-step multiplication method.

67×45

60×40=2,400

7×45=315

60×5=300

2,400+315+300=

3015

84×53

80×50=4,000

4×53=212

80×3=240

4,000+212+240=

4452

4,452+3,015=7,467-73=

7,394

67×45-73+84×53=

7,394

(8.) Improper fraction division & conversion 

This last example entails the division and conversion of an improper fraction into a decimal.

39/68

39÷68=0.573529411764

1. First order of business is to multiply the numerator by 10.
2. Second is to then inquire how many times the denominator can fully go into the numerator after it has been multiplied by 10, which is 5 times.
3. Multiplication is needed to determine the remainder, which allows us to get our next decimal point.
4. 5×68=340: the goal is to get as close to 390 as possible without surpassing it. Going over even by 1 is too much.
5. Now we must subtract 340 from 390 and that gives us 50. 50 is the remainder and becomes the new numerator. These exact steps are to be repeated for as long as it is desired or possible.
6. A heads up, let’s say the remainder is 4 and our denominator is 52. If we multiplied 4 by 10, which gives us 40. 52 cannot divide into 40 even once, don’t panic. The number that will be placed there is a 0. Now once we’ve made that decimal point a zero. We then multiply the numerator (4) by 100 instead of 10. This gives us 400, After doing so we then see how many times the denominator (52) can perfectly go into 400, which is 7. 7×52=364. Below is the example explained in steps 1-5 in numerical terms.

39×10=390

390÷68=

5 

5×68=340

340-390=50, remainder of 50

50/68

50×10=500÷68=

7

remainder of 24

24/68

24×10=240÷68=

3

remainder of 36

36/68

36×10=360÷68=

5

remainder of 20

20/68

20×10=200÷68=

2

remainder of 64

64/68

64×10=640÷68=

9

remainder of 28

28/68

28×10=280÷68=

4

remainder of 8

8/68

8×10=80÷68=

1

remainder of 12

12/68

12×10=120÷68=

1

remainder of 52

52/68

52×10=520÷68=

7

remainder of 44

44/68

44×10=440÷68=

6

remainder of 32

32/68

32×10=320÷68=

4

remainder of 48

39÷68=0.573529411764

Miscellaneous:

The following are two more problems I’ve done; however, I’ve been lazy on making a step-by-step walkthrough for them. I’m currently chewing through algebra 2 right now, little bit of calc, with only using mental math methods and I strongly desire to get into the more advanced material at some point. It’s not a huge priority for me at the moment, I have several other projects and interests that I’m working on. For me it’s a nice little cognitive exercise that I do almost every day.

0.00001282 x 0.001723 = 2.20888600E-8

1/55,000 = 0.0000181818

This last problem is a seven-digit multiplication problem I attempted about a month ago. I attempted multiplying seven-digits back in 2021 of December and was 1 millionth of a percent off, had the exact outcome with this 3rd attempt. The six-digit multiplication problem that is featured at the beginning of this document was my 2nd attempt at six digits and was in December of 2021. That six-digit problem took me 15 minutes, seven-digits so far has taken up an hour plus. Each time attempt I’ve had to deal with interruptions, which can cause a great deal of cognitive strain. It’s like trying to run a hefty computer program and it advises not to run other programs during that period. There is well over 200 steps I need to do for a seven-digit multiplication, having to put that on hold and process something else can cause problems for the mental math exercise. What I am coming to find is that I need to figure out a more ideal environment. At first, I would do my mental math exercises while I worked. I work on an assembly line and have headphone privileges, so early on this was perfect for the smaller problems, but once I got to 5 digits plus, interruptions became fatal. Being in a constant state of motion, while listening to music that triggers my binaural brainwaves. It’s like a deep meditative state and once I got to 5 digits, I’d go on a drive around 3:00am when no one would be on the road, and I’d do those hefty problems. Two things come to mind with the seven-digit problem. During my most recent attempt I was halfway through, breezing through it, there was very little cognitive strain, and I was on the path to successfully solving it…. then…. a dear decided to be all suicidal and I almost hit the damn bastard, so I had to put that hefty problem on hold and process almost hitting a deer…. Preferably though, I’d love to just have a more ideal and safer environment to test my limits lol. Anyhow, sorry for the tangent, this is my most recent seven-digit multiplication attempt.

3,539,284×7,492,935 = 26,519,624,958,540

my answer: 26,519,614,958,540

26,519,614,958,540 ÷ 26,519,624,958,540 = 0.9999996229

1 millionth of a percent off, one day, someday soon.

feat of strength: 724,573×746,934 = 541,208,209,182

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