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Mental Math: Doubling and Halving


In this video we'll talk about one of my favorite mental math tricks, doubling and halving. The logic of the dividing-by-5 trick talked about in the last video suggests a more general and more widely applicable trick. Suppose we need to multiply say 16 times 35. Now that would be a little bit challenging to do in your head without a calculator, 16 times 35.

But of course, we know that 16 = 8*2. We could take that factor 2 away from the 8 and give it to the 35. So 16*35, of course that's (8*2)*35. And imagine we group it in a different way. Of course, we can group multiplication in any order we want. That's technically known as the associative property.

So we can group that as 8*(2*35). Well, 2*35 is 70. Well, all right, now this is something I can do in my head. So you should be very comfortable with your one-digit multiplication table. So 8*7 is 56. You should be comfortable with practicing your one-digit multiplication table, so that much is obvious.

Well, if 8*7 is 56, we just tack an extra 0 on it, that would be 560. So this entire thing I can do in my head very easily because I stole a factor of 2 away from 16 and gave it to 35. Think about what happened there. One factor lost a factor of 2 and the other factor gained a factor of 2. In other words, one factor was halved, divided by 2, and the other was doubled, multiplied by 2, but the product remains the same.

And that kinda make senses that if we divide one factor by 2 and multiply the other factor by 2 that those will cancel and you'll still have the same product. In any multiplication, we can always double one factor and find half of the other, and the product will still be the same. When would it be advantageous to employ this trick?

Well, if one factor ends in 5 or ends in 50, doubling it would produce a round number. It would produce a nice multiple of 10 or a multiple of 100. It conceivably could even have something ending in 500. And then if you doubled it, it would be a multiple of 1000. So that's what you're looking for, especially when one of the factors ends in 5 or ends in 50.

As long as the other number is even, then we can take half of it and we make it smaller, and the number that ended in 5 or 50 will become a round number. So then we're just multiplying a smaller number times a round number, and it's always considerably easier. So for example, if we have to do 84*50, well, doing that straight multiplication, that would be relatively challenging.

Instead, what's half of 84? Obviously, half of 84 is 42. And then we'll double 50. 50 doubled is 100. Well, 42 times 100 I can do that in my head very easily. That's just 4200.

If you practice this trick, you can get quite quick with it. Here's some practice problems. Pause the video, and then we'll talk about this. Okay, that first one, 260*15. Well, the 15, I'd like to double that. So 260, half of that would be 130, and 15 doubled would be 30, so I get 130*30.

And I'm just gonna separate this out to 13*3*10*10. And then the 13*3, again this is the kind of simple multiple, simple mental math, one-digit number times a low two-digit number. That's something you should be able to do in your head. 13*3, that's 39, and then just tack on two 0s. And we get 3,900, that's the answer.

56*25, we'd like to double that 25. So divide 56 by 2, that's 28*50. Now we have something ending in 50. We'll do mental math again. We'll double and half again, half of 28 is 14, double of 50 is 100, 14*100 is 1,400, very easy to do in your head.

24*75, same thing, half of 24, double 75, I get 12*150. Let's do it again. Half of 12 is 6, double of 150 is 300. And then 6 times 300, well, 6*3 is 18, so this will be 1800. Once again, this trick might feel a little anti-intuitive when you are first using it, but the more you practice it, the quicker you will become with it.

When you can perform doubling and halving calculations quickly and efficiently on the test, seemingly difficult calculations will be done in seconds. As I'm sure you appreciate every second you can save on the Quant section is absolutely golden. Very important to have as many time saving strategies as possible on the Quant section of the test.

In summary, in any product we always have the option of finding half of one factor and doubling the other, this does not change the resulting product. When doubling one number will make it a round number, a multiple of 10 or 100, then doubling that number and halving the other can enormously simplify the calculation. And as we saw, sometimes we apply the procedure twice in succession, for example, when one factor is 25 or a multiple of 25.

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