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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 is eight times two. We could take that factor of two away from the eight and give it to the 35. So, 16 times 35, and of course that’s eight times two times 35 and imagine we group it in a different way, of course, we can group multiplication in any order we want, that effectively known as the associate of property.
So, we can group that as eight times two times 35, well, two times 35 is 70. Well, all right, well now, this is something I can do in my head. So, you should be very comfortable with your one-digit multiplication table so eight times seven is 56. You should be comfortable with practicing your one digit multiplication table so that much is obvious.
Well, if eight times seven is 56 we just tack an extra zero on it that would be 560. So this entire thing I can do in my head very easily because I stole a factor of two away from 16, and gave it to 35. Think about what happened there. One factor lost a factor of two, and the other factor gained a factor of two.
In other words, one factor was halved, divided by two, and the other was doubled multiplied by two, but the product remains the same. And that kinda makes sense, that if we divide one factor by two, and multiply the other by two, that those will cancel, and you’ll still have the same product. In any multiplication, we can always double one factor and find half 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 five, or it ends in 50, doubling it would produce a round number, produce a nice multiple of 10 or a multiple of 100. Conceivably you could even have something ending in 500 and then if you doubled it it would be ending in, it would be a multiple of 1,000. So that’s what you’re looking for, especially when one of the factors ends in five or ends in 50.
As long as the other number is even, then we can take half of it and we make it smaller, then the number that ended in five or 50 will become a round number. So then we’re multiplying a smaller number by a round number, and it’s always considerably easier. So, for example, if we have to do 84 times 50, well, doing that straight multiplication that would be relatively challenging.
Instead what’s half of 84? Obviously half of 84 is 42. So, 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 times 15. Well the 15, I’d like to double that. So, 260, half of the would be 130, and 15 doubled would be 30, so I get 130 times 30.
And I’m just gonna separate this out to 13 times three times ten times ten and then, the 13 times three. Again, this is the kind of simple multiple, simple mental math, you know, one digit number times a low two digit number. That’s something you should be able to do in your head, 13 times three, that’s 39 and then, just tack on two zeroes and we get 3,900, that’s the answer.
56, 25, well, we’d like to double that 25, so divide 56 by two, that’s 28, times 50. Well, now we have something ending in 50, so we’ll do mental math again, we’ll, we’ll double one-half again. Half of 28 is 14, double of 50 is 100, 14 times 100 is 1400. Very easy to do in your head. 24 times 75, same thing.
Half of 24, double 75, I get 12 times 150. Let’s do it again. Half of 12 is six, double of 150 is 300 and then six time 300, well six times three 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 am 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 resultant product. When doubling one number, on, when doubling one number will make it a round number a multiple of ten or 100, then doubling that number and having 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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