This content originally appeared on DEV Community and was authored by Jose Maria Iriarte
In a world overflowing with digital tools, it’s easy to dismiss mental math as outdated — a quaint relic from the pre-calculator age. But there’s something uniquely powerful about sharpening your ability to think numerically without external support. Like optimizing code for runtime efficiency, refining your mental math streamlines your internal thought processes and reduces your cognitive load.
This article is based on a short guide I wrote called Multiplying and Dividing Fast - Essential Tactics to Speed Up Mental Calculations with Whole Numbers, the second in a three-part series I began in 2018. The project started as a personal challenge: to improve how I processed numbers mentally. I wanted to go beyond fuzzy estimations and reclaim the skill of clear, structured calculation — the kind that feels more like flow than friction.
What began as scattered notes turned into three compact books, each focused on two core operations: addition & subtraction, multiplication & division, powers & roots. This article distills the key strategies from the second of those books.
These are not “tricks” in the superficial sense — they’re practical, reusable mental algorithms. Some are intuitive, others might feel surprising at first, but all are meant to reduce latency between seeing a problem and arriving at a confident answer.
Whether you’re optimizing memory usage or tallying totals in your head, these techniques will train you to think faster, reduce errors, and — yes — enjoy math a little more.
Note that the original guide contains detailed explanations on these and other techniques, impossible to reproduce in an article of this nature or length. If you are serious about incorporating these techniques, I strongly recommend you read the full guide, linked at the end of this article. It's completely free to download and read.
📃 Core Multiplication Tactics
🔄 Doubling
Use for multiplying by 2, 4, 8, etc. (powers of two).
Example:
354 × 2 = 708
🔄 Doubling Twice or Thrice
Used for ×4, ×8, ×16, etc.
Example:
42 × 16 = 672
84, 168, 336, 672
🔄 Multiples of 10
Multiply as if zeroes weren’t there, then append them.
Example:
40 × 45,000 = 1,800,000
💵 Multiplying by 5 (Halving)
Multiply by 10, then divide by 2.
Example:
5 × 74 = 370
740 / 2 = 370
🔢 Squares Ending in 5
Multiply tens digit by (n+1) and append 25.
Example:
95 × 95 = 9025
9 x 10 = 90
🔑 Multiplying by 11 (2-digit)
Add digits and sandwich between.
Example:
11 × 72 = 792
7 + 2 = 9
7 9 2
🔑 Multiplying by 11 (3+ digits)
Pairwise addition from right to left, carry included.
Example:
11 × 3,480 = 38,280
8 + 0 = 8
4 + 8 = 12
3 + 4 = 7
30000
07000
01200
00080
00000
📃 Multiplying by 15
Decompose into (10 + 5) then apply halving.
Example:
15 × 382 = 5,730
10 x 382 = 3820
3820 / 2 = 1910
3820 + 1910
📈 Two-Digit Products (Lattice-like)
Apply place-value cross multiplication.
Example:
34 × 65 = 2,210
3 x 6 = 18
4 x 5 = 20
3 x 5 + 4 x 6 = 39
1800
0390
0020
📈 Three-Digit Multiplication
Break into hierarchical place-value multiplications.
Example:
123 × 432 = 53,136
4 x 1
1 x 3 + 4 x 2
2 x 1 + 2 x 3 + 3 x 4
3 x 3 + 2 x 2
3 x 2
40000
11000
02000
00130
00006
📈 Two-Digit by Three-Digit
Use the same principle, append 0s where needed.
Example:
44 × 452 = 19,888
💸 Balancing for Multiplication
Shift factors to easier equivalents.
Example:
144 × 24 = (288 × 12) = 3,456
(144 x 2) / (24 / 2) =
288 x 12 =
288 x 10 + 288 x 2 =
2880 + 576
🌐 Rounding and Compensation
Round one number to an easier figure, then subtract excess.
Example:
16 × 191 = (16 × 200) - (16 × 9) = 3,056
3,200 - 144
🔹 Associative Property
Break one factor into primes or easier groups.
Example:
22 × 267 = 11 × 534 = 5,874
11 x 2 x 267
267 doubled, 534
534 x 11
3 + 4 = 7
5 + 3 = 8
📃 Distributive Property
Use expanded form: a(b + c)
Example:
51 × 295 = (50 × 295) + 295 = 15,045
📃 Products Near Powers of 10 (One Away)
Use complements to subtract from powers of 10.
Example:
387 × 999 = 386,613
387000 - 387
📃 Products Near Powers of 10 (Both Close)
Use cross-adjustment and product of differences.
Example:
997 x 998 = 995,006
997 - 1,000 = -3, 998 - 1000 = -2
997 + (-2) = 995
-3 x (-2) = 6
995000
000006
📃 Core Division Tactics
📃 Division as Inverse Multiplication
Think "what times X gives me Y?"
Example:
72 ÷ 6 = 12
6 x 12 = 72
🧰 Advanced Halving
Successive halving for dividing by 4, 8, 16, etc.
Example:
3,760 ÷ 16 = 235
1880, 940, 470, 235
📉 Divisibility Rules (3, 5, 9, 10, etc.)
Use digital sums or ending digits.
Detailed explanations of this family of subtechniques can be found in the ebook linked below.
Example:
495
Digital sum = 4 + 9 + 5= 18
Digital sum = 1 + 8 = 9
Divisible by both 3 and 9
📃 Rounding & Compensation (Division)
Round dividend/divisor then adjust the result.
Example:
684 ÷ 3 = (690 - 6) ÷ 3 = 228
230 - 2
📈 Balancing in Division
Shift dividend/divisor by a common factor.
Example:
77 ÷ 11 = (770 ÷ 110) = 7
🔹 Associative in Division
Break divisor into friendly parts.
Example:
264 ÷ 12 = (264 ÷ 3) ÷ 4 = 22
📃 Distributive in Division
Break up dividend to divide in parts.
Example:
(840 + 60) ÷ 30 = 28 + 2 = 30
🧰 Practice & Progress
Like any skill, speed and confidence come with practice. Try:
Estimating bills or invoices in your head
Practicing products while walking
Playing games that push number recognition and recall
The goal isn’t to become a human calculator, but to develop a mental agility that helps in work, school, and everyday thinking.
📘 Download the Full Guide
This article is a condensed version of my guide:
👉 Download the PDF: Multiplying and Dividing Fast
Multiplying and Dividing Fast (PDF)
This guide contains much more detailed information about how to profit from the various tactics including various reference multiplication and division tables and expanded explanations, in particular, for the sections regarding divisibility rules. If you are serious about these techniques I strongly recommend you read the guide.
🚀 Help Others and Keep the Conversation Going!
If you found this article helpful, here’s how you can contribute to spreading the knowledge:
👍 Like this article to show your support and help others discover it.
📁 Bookmark it so you can easily come back to these techniques anytime you need them.
🔄 Share it with friends or colleagues who might benefit from faster mental math strategies.
💬 Comment below with your favorite tip, a trick you’ve used, or any questions you might have — let’s keep learning together!
By liking, sharing, or commenting, you're not just helping me reach more readers — you're contributing to a community that thrives on growth and knowledge. Sharing this article helps others enhance their mental math skills, reducing cognitive load and boosting productivity. Your engagement could make someone’s day easier or help them see math in a new, empowering way!
Thank you for being part of this learning journey! 🙌
This content originally appeared on DEV Community and was authored by Jose Maria Iriarte
Jose Maria Iriarte | Sciencx (2025-05-17T03:00:00+00:00) Optimizing Mental Math: Fast Multiplications and Divisions for Software Engineers. Retrieved from https://www.scien.cx/2025/05/17/optimizing-mental-math-fast-multiplications-and-divisions-for-software-engineers/
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