Some Practical Multiplication Shortcuts (A Preview of “Master the Language of the Universe”)

By Matthew Canning, Become Better at Everything Founder

In just a few short weeks, my next book will be hitting shelves. Master the Language of the Universe takes you on a journey through practical math, removes the anxiety that often accompanies math, and shows you how to perform even complex calculations using only your head.

To celebrate the coming release, here’s a preview. While the book dives into topics like probability, sorting, and estimation, it also features some fun, easy, and practical shortcuts. You’ll find a few below that can be applied to specific types of small, common numbers. Enjoy!

The Teeny-Teeny

Numbers in the “teens” (11 through 19) are common in daily life. Luckily, there’s a trick that will allow you to multiply two teen numbers in just a few seconds.

The process:

  1. Take the ones place (rightmost digit) from the smaller number, and add it to the larger number.
  2. Add a 0 to the end of this sum.
  3. Hold this combined number in your mind.
  4. Multiply the two ones place (rightmost) digits of the two original numbers.
  5. Add the product of step 4 to the product of step 3.

That can seem daunting at first, so let’s go through the steps again with an example in mind: 15 × 13. Both numbers are “teens,” so they qualify for this method.

  1. Take the one’s place (rightmost digit) from the smaller number (the 3 from the 13), and add it to the bigger number (3 + 15 = 18).
  2. Add a 0 to the end of this sum: 180.
  3. Hold the 180 in your mind.
  4. Take the two rightmost (ones place) digits of the two original numbers (15 and 13), and multiply them (5 × 3 = 15).
  5. Add the outcome of step 4 (15) to the number you’re holding in your head (180). 180 + 15 = 195.

That’s not so bad, is it? In fact, you can do this in your head, with no tools or pencil and paper. Try to multiply 12 × 19 using only your head.

  1. 2 + 19 = 21
  2. Add the 0: 210
  3. Hang on to the 210.
  4. 2 × 9 = 18
  5. 210 + 18 = 228

Answer: 228.

Congratulations! This is your maiden voyage into the world of mental calculation.

With practice, multiplying using the Teeny-Teeny will become a quick and painless process.

Perform the following calculations; the answers are below. Don’t cheat yourself; figure out and write down the answers, then check them. Only write down the answer digits, never the problem steps or “work.” If you’ve made a mistake, try to figure out where you went wrong.

  1. 15 × 15
  2. 16 × 13
  3. 12 × 19
  4. 18 × 17
  5. 19 × 11

Answers:

  1. 225
  2. 208
  3. 228
  4. 306
  5. 209

The Splitsy-Doubly

If you need to multiply two numbers and at least one of them is a power of two (2, 4, 8, 16, 32, 64, 128, 256, 512…), you can work through the problem by repeatedly halving the “power of two” number while doubling the other. Repeat this “split” and “double” step until the problem is simple enough to solve. To perform this shortcut properly, you need to feel comfortable splitting numbers in half (that is, dividing them by two). This skill is covered in the chapter, “Fundamental Mathematical Concepts.” Let’s look at an example: 16 × 23.

16 is a power of 2, so you’re going to repeatedly split this while doubling its counterpart.

(16/2 = 8) × (23 × 2 = 46)

(8/2 = 4) × (46 × 2 = 92)

(4/2 = 2) × (92 × 2 = 184)

2 × 184 = 368

Answer: 368.

With practice, you will learn to recognize powers of two immediately. You can then decide whether this method would be the best choice given the circumstances. Eventually, when you recognize a power of two, you will also be able to determine which power of two it is. You can therefore simply double the other number this many times. For example, when faced with 16 × 23, you will be able to say, “Oh, 16; that’s a power of 2. In fact, 16 is 2 to the 4th, so I have to multiply 23 by 2 exactly 4 times.” This allows you to skip the “splitting” aspect and concentrate on “doubling.”

When starting out, it can be useful to use your fingers to keep track of which power of two you’re working with; with the example of 16, you’d begin with four fingers extended, then three, then two, etc. Let’s try a more difficult example: 64 × 31.

(64/2 = 32) × (31 × 2 = 62)

(32/2 = 16) × (62 × 2 = 124)

(16/2 = 8) × (124 × 2 = 248)

(8/2 = 4) × (248 × 2 = 496)

(4/2 = 2) × (496 × 2 = 992)

2 × 992 = 1,984

Answer: 1,984.

Technically, this method works when the problem contains any even number; it’s not necessarily reserved for powers of two. However, you can end up running into issues with non-power-of-two problems, as you may have to split an odd number along the way. In these cases, the Splitsy-Doubly can still be used to turn one of the two numbers into a smaller, more manageable number, transforming a daunting problem into an easier one. Let’s look at an example: 14 × 22.

 (14/2 = 7) × (22 × 2 = 44)

You’re stuck with an odd number, but at the same time, the problem is now arguably easier. You will soon know how to solve this problem using Trachtenberg’s method for speed multiplication by the number 7.

Answer: 308.

This only works some of the time. Let’s next look at a non-“factor of two” example where using this method will not help: 38 × 21.

(38/2 = 19) × (21 × 2 = 42)

You now have to multiply 19 times 42. That’s still ugly, and no simpler than where you started off (38 × 21). Remain mindful that the Splitsy-Doubly is an available option, but it is only suggested as your primary method for any multiplication problems involving a power of two.

Perform the following calculations; the answers are below. Don’t cheat yourself; figure out and write down the answers, then check them. Only write down the answer digits, never the problem steps or “work.” If you’ve made a mistake, try to figure out where you went wrong.

  1. 16 × 3
  2. 64 × 9
  3. 32 × 12
  4. 54 × 16
  5. 128 × 22
  6. 256 × 14

Answers:

  1. 48
  2. 576
  3. 384
  4. 864
  5. 2,816
  6. 3,584

The Double-Double

To multiply any number by four, perform this simple two-step operation:

  1. Double the number, and
  2. Double it again.

This would be expressed algebraically as 4n = 2(2n).

It may seem obvious, but it’s easy to forget that you can think of “4” as “2 × 2”. This method is best suited for n × 4 problems where n is small enough to multiply by 2 somewhat easily.

Here’s an example of an optimal Double-Double method use case: 55 × 4.

  1. 55 × 2 = 110
  2. 110 × 2 = 220

Answer: 220.

See? For most, multiplying 55 by 4 is considerably more intimidating than either of the two individual steps (55 × 2 and 110 × 2). Let’s look at another example: 74 × 4.

  1. 74 × 2 = 148
  2. 148 × 2 = 296

Answer: 296.

If precision isn’t required, you can round before multiplying, turning 74 into 75, leveraging your familiar with quarters and currency.

  1. 75 × 2 = 150
  2. 150 × 2 = 300

Generally, the Double-Double method is most useful for multiplying 4 by numbers smaller than 999 (or familiar numbers larger than 999). An example of a large-but-familiar number would be 2,800 (2,800 × 2 = 5,600; 5,600 × 2 = 11,200). An example of a less simple larger number would be 1,731. When numbers are too large or complicated for this method, you can round them if appropriate, or use the Trachtenberg System long multiplication method, which you’ll learn shortly.

Perform the following calculations; the answers are below. Don’t cheat yourself; figure out and write down the answers, then check them. Only write down the answer digits, never the problem steps or “work.” If you’ve made a mistake, try to figure out where you went wrong.

  1. 71 × 4
  2. 19 × 4
  3. 114 × 4
  4. 53 × 4
  5. 516 × 4
  6. 890 × 4
  7. 2,100 × 4

Answers:

  1. 284
  2. 76
  3. 456
  4. 212
  5. 2,064
  6. 3,560
  7. 8,400

The Five-Ten Split

To multiply any number by five, perform this simple two-step operation:

  1. Cut the number in half, and
  2. Multiply by ten.

Depending on how your mind works, it may be easier to think of the second step as “moving the decimal one place to the right” instead of multiplying by ten. Either way, it’s the same principle. This would be expressed algebraically as 5n = 10(1/2 n).

Remember how I stressed the importance of learning to split numbers quickly a little while ago? The Five-Ten Split is a prime example of that skill’s usefulness.

Let’s try an example: 84 × 5.

  1. 84/2 = 42
  2. 42 × 10 = 420

Answer: 420.

Let’s look at an odd number problem: 75 × 5.

  1. 75/2 = 37.5
  2. 37.5 × 10 = 375

Answer: 375.

As you can see, this method handles odd number problems (especially those involving small numbers) somewhat gracefully despite the involvement of the decimal place; this introduces only a marginal increase in complexity.

Generally, the Five-Ten Split method is useful for multiplying 5 by any number less than 9,999. Numbers that split more gracefully (those ending in 0, for instance) tend to cooperate better. For example, 3,020 × 5 is arguably simpler to solve than 3,196 × 5. Let’s break them down.

3,020 × 5:

  1. 3,020/2 = 1,510
  2. 1,510 × 10 = 15,100

3,196 × 5:

  1. 3,196/2 = 1,598
  2. 1,598 × 10 = 15,980

Which was easier for you?

When a problem involves numbers that are harder to Split (such as 3,393), you can round or, for precise answers, rely on the Trachtenberg System method for multiplication by 5.

Perform the following calculations; the answers are below. Don’t cheat yourself; figure out and write down the answers, then check them. Only write down the answer digits, never the problem steps or “work.” If you’ve made a mistake, try to figure out where you went wrong.

  1. 26 × 5
  2. 124 × 5
  3. 72 × 5
  4. 55 × 5
  5. 388 × 5
  6. 1,540 × 5

Answers:

  1. 130
  2. 620
  3. 360
  4. 275
  5. 1,940
  6. 7,700