**Learn one of the handiest tricks for math without a calculator! **

Without a calculator, what is

1. 35 x 12?

2. 150 x 36?

3. 125 x 84?

On the GMAT, you don’t get a calculator. With the **doubling and halving** trick, all of these become much easier.

## Thinking about multiplication

Every positive integer has a unique prime factorization — that is to say, there is a unique way to express each whole number as a product of prime factors. Therefore, whenever we multiply to positive integers, we can think of this the product of the prime factors of one times the product of the prime factors of the other — two big collections of factors being multiplied together. Furthermore, the associative law the commutative law tell us we can multiply in any order — we could even swap around factors from one number to the other, and the overall result of the multiplication would not change.

## Doubling and halving

Suppose one factor ends in 5, and suppose the other factor is even. In this case, we know the even factor must be divisible by 2, so we can easily remove a factor of 2 from that one (thereby “halving” it) and multiply the multiple of 5 by 2 (thereby “doubling” it), which will make that one a multiple of ten. In this process, both numbers become simpler, and the multiplication often becomes something you could easily do in your head.

For example, consider the multiplication 15 x 16. At first glance, that looks not-fun without a calculator. Now, we will perform “doubling and halving.” Remove a factor of 2 from 16, so 16 becomes 8 — it “halves.” Give that spare factor of 2 to 15 — multiply 15 by 2 to get 30. Therefore, 15 x 16 = 30 x 8 = 240. After using the doubling-and-halving trick, the problem just becomes one-digit multiplication, with an extra zero along for the ride.

In the case of 25 x 44, we can do doubling-and-halving once —- 44 becomes 22 and 25 becomes 50 —- to get 25 x 44 = 50 x 22. That’s better, but we can do doubling-and-halving *again* —- 22 becomes 11 and 50 becomes 100 —- so that 25 x 44 = 50 x 22 = 100 x 11 = 1100.

Take another look at the problems at the top, and see if you can simplify them with the doubling-and-halving trick before reading the solutions below.

## Practice question

Here’s a free GMAT practice question on which you can use this trick.

4) http://gmat.magoosh.com/questions/331

## Solutions

1) Halve 12, so it becomes 6. Double 35, so it becomes 70. 12 x 35 = 6 x 70 = **420**.

2) Halve 36, so it becomes 18. Double 150, so it becomes 300. 150 x 36 = 300 x 18 = **5400**.

3) Halve 84, so it comes 42. Double 125, so it becomes 250. 125 x 84 = 250 x 42. Now, perform the trick again. Halve 42, so it becomes 21. Double 250, so it becomes 500. 125 x 84 = 250 x 42 = 500 x 21.

Now, notice that 21 = 20 + 1, so 500*21 = 500*(20 + 1) = 500*20 + 500

Apply doubling & halving one more time, 500 to 1000, and 20 to 10:

500*21 = 500*20 + 500 = 1000*10 + 500 = 10000 + 500 = **10500**.

No calculator, no problem!

Very very useful technique.

Thanks!

Hey Mike,

Thanks so much for the tip, hard to believe how I made it all the way through high school without it! Do you have a set of questions (with answers) on this topic for us to practice? Thanks!

Kelly

Interesting take on this. Personally I’ve managed to get through school, highschool, and most of my life without ever learning the table of multiplication (am 33 years old and I still don’t know it). Instead, I’ve always used a dynamic shortcut system based on multiplication by 10 and division/multiplication by 2, or halving/doubling (very easy to do mentally), and it works exceptionally well once you get good at it. Especially if you’re the type of person that processes information visually instead of textually.

It’s hard to explain how it works, since it’s different every time, you just need to “optimize” the operation by decomposing it into simpler operations. It’s kinda like the reverse of using repeated division by 2 to “guess” numbers.

Here’s the system based on your examples, and an unfold of how I process it in my mind:

35 * 12 > 35 * (10 + 2) > 35 * 10 + 35 * 2 > 350 + 70 > 420

150 * 36 > (100 + 50) * 36 > 100 * 36 + 50 * 36 > (100 * 36) + (100 * 36) / 2 (no point calculating 50 * 36 since it’s much faster to take the first result and split it in half mentally) > 3600 + 3600 / 2 = 3600 + 1800 = 5400

125 x 84 = (100 + 25) * 84 > 100 * 84 + 25 * 84 > (again 25 is a quarter of 100, it’s much easier to mentally split 100 in half, and then split the result in half again) 8400 + 8400 / 4 > 8400 + 4200 / 2 > 8400 + 2100 > 10500

Pretty much everything can be disassembled into optimal sub-operations and once you get good at division and multiplication by 2 (doubling and halving) you can perform almost all of it visually without doing much math in your head at all.

But again, I’m not sure how well this works with people that think textually, my brain thinks in images, which makes doubling and halving almost instinctively thus this method works best for me, not sure about others.

Also, this method works for division too, not just multiplication. Anyway, hope this helps someone.

Thank you for amazing tricks

How quickly should we be able to solve problems like this in real-time after becoming proficient? I ask that as someone with very poor mathematical intuitions 🙂

That’s an excellent trick. 🙂

Dear Mike,

Thank you so much for the tricks, it helps us a lot 🙂

The solution which you discussed above for the question – shortcut method for 17 * 13? Does this method applies only when there is a difference of 4, I am just wondering by using the same method can we find out 17*19, please let me know your suggestions. thanks for your time.

Thanks,

Swaroop

Dear Swaroop,

I’m happy to respond. 🙂 First of all, see this blog, which discusses this perspective in more detail:

http://magoosh.com/gmat/2012/advanced-non-calculator-factoring-on-the-gmat/

Technically, you could use this to multiply two numbers that were separated by

any even number— e.g. 27*33 = (30 – 3)(30 + 3) = 30^2 – 3^2 = 900 – 9 = 891. Of course, this is only useful if you happen to know the square of the larger number.For 17*19, we can write this as (18 – 1)(18 + 1) = 18^2 – 1, but that’s helpful only if you happen to know what 18^2 is. If you know the perfect squares of all the numbers from 1 to 30, that would give you some more computational options, but for most folks, that’s too much to remember.

As a general rule, on the GMAT, if there is a product of two 2-digit numbers, it is usually better NOT to multiply they out, but leave the product unmultiplied, which will make it easier to cancel a factor in a later step. For example if, in a problem, we have to multiply 24*36, then later have to divide by 6, then by 12, there would be absolutely no point in finding the product of 24*36 — we would just leave it in that unmultiplied form, so it’s easier to cancel.

Does all this make sense?

Mike 🙂

Good stuff Mike. Thank you for sharing!

Nico,

You are quite welcome! Best of luck to you!

Mike 🙂

Amazing trick! Thank you soooo much! Was not aware a calculator was not provided. : )

Jennifer,

I am very glad you found this helpful! BTW, here are a couple more articles on this blog that provides strategies that are useful when you have no calculator:

http://magoosh.com/gmat/2012/the-power-of-estimation-for-gmat-quant/

http://magoosh.com/gmat/2012/gmat-divisibility-rules-and-shortcuts/

I hope all this helps! Best of luck to you in your GMAT prep!

Mike 🙂

I’m 48, why has no one ever taught me this! Thanks.

Dear Mac,

It’s a particular pleasure receiving kind words from someone else who remembers the 1970s. 🙂 I’m glad you found this helpful. Best of luck to you!

Mike 🙂

Is there any shortcut method for 17 * 13?

Dear Kzaman,

This is something the GMAT is highly unlikely to ask. Notice that you could find the units digit (it would be 1, the same as the units digit of 7*3). In the very rare instance that this appeared on the GMAT, probably you would have to take the few seconds to multiply it out on the pad.

IF you happen to notice that 17*13 = (15 + 2)*(15 – 2), and IF you happen to remember the Difference of Two Squares formula, so that this equals (15)^2 – (2)^2, and IF you happen to know off the top of your head that (15)^2 = 225, then it would be very easy to see that 17*13 = 225 – 4 = 221. Very easy, but that’s a lot of if’s!

Does all this make sense?

Mike 🙂

Hi Mike,

I think you misplaced a “6” instead of a “1” in the “Therefore, 65 x 16 = 30 x 8 = 240. After using the doubling-and-halving trick, the problem just becomes one-digit multiplication, with an extra zero along for the ride.”

I think 65 should be replaced by 15. Is that correct?

Regards

Leandro

Leandro — Good eye! That’s exactly what it should be! You are 100% correct! Thank you very much for spotting that typo — I just corrected it. Best of luck to you!

Mike 🙂

That’s fab! Thanks!

You are quite welcome.

Mike 🙂