**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!

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