Learn what the distributive property means in the real world and how it can help you perform lightning fast multiplication in your head.
In a past episode, we talked about the distributive property. In particular, we talked about how you can visualize what the distributive property means. Today, we’re continuing our discussion of the distributive property by looking at two real life applications: the first is helpful in the kitchen, and the second will help you perform lightning fast multiplication in your head.
Review: What is the Distributive Property?
As we talked about before, we can summarize the distributive property with the formula:
a x ( b + c + d ) = a x b + a x c + a x d
In words, that says that if we take the sum of some numbers (in this case b, c, and d) and then multiply this sum by some other number (in this case a), then the answer you get will be the same thing you’d get if you first individually multiplied each number in the sum (that is, b, c, and d) by a, and then added these all up. For example, in the problem 2 x (3 + 4 + 5), we can first add up 3, 4, and 5 to get 12, and then multiply this by 2 to get 24. Or, we can first multiply each of 3, 4, and 5 by 2 to get
2 x 3 = 6
2 x 4 = 8
2 x 5 = 10
If we now add up these results, 6 + 8 + 10 = 24, we see that we get the same answer as before. That’s the distributive property. But what good is it?
A Real World Example of the Distributive Property
Before we get to the really useful and practical application of the distributive property, let’s first think a bit about what it means in the real world. In particular, let’s think about how you go about doubling a recipe—in other words, taking a recipe that’s for, say, 4 servings and turning it into a recipe for 8 servings. Let’s say you’re making a cake that requires 3 ingredients, which we’ll unimaginatively call b, c, and d. That means that the finished recipe ready to go into the oven for baking will look like b + c + d (addition here means that you combine all the ingredients together and mix them up well).
Okay, but what if you need to double the recipe. How can you do it? Well, you’ve got 2 options. First, you could get a bigger bowl and combine together twice as much of each ingredient. That way, your final batter ready for baking would look like 2b + 2c + 2d. Alternatively, you could make two separate batches of your b + c + d recipe, and then combine these 2 finished batches together. The result will be a batter that looks like 2 x ( b + c + d ). But, of course, that batter will also look exactly like the 2b + 2c + 2d batter from before. And that means we’ve found that 2 x ( b + c + d ) = 2b + 2c + 2d—the distributive property! And notice that if instead of doubling the recipe, we wanted to increase it by an amount we call a, then the equation we talked about earlier, a x ( b + c + d ) = a x b + a x c + a x d precisely describes our baking situation!
How to Multiply Quickly with the Distributive Property
Okay, we’re now ready for our second real life application of the distributive property…and it’s a good one. How does the ability to perform super fast multiplication in your head sound? Pretty hard to resist, right? Well, let’s see how it works with a simple example. How about 8 x 47. Now, this multiplication problem isn’t too hard to do the old-fashioned way—that is, first multiplying the 7 from 47 by 8, putting a 6 from the result 56 in the ones column and carrying the 5 to the tens column; then multiplying the 4 from 47 by 8 to get 32…plus 5 from before is 37…but, of course, that’s in the tens column so it’s actually 370. Add this to the original 6 and we get grand total of 376…right? In truth, it’s not that hard, but it’s definitely a little cumbersome.
But the distributive property can be used to turn one large, difficult, and slow multiplication problem into several small, easy, and fast multiplication problems. And that turns out to be a pretty good tradeoff. How does it work? Well let’s go back to the problem we just looked at: 8 x 47. But let’s instead write the number 47 as 40 + 7. Why did we choose that? Well, the quick and dirty tip is that when you’re breaking numbers apart like this to add up, you want to make as many numbers as possible multiples of 10—in other words, for them to end with a zero. Using multiples of 10 like this is the thing that’s going to make multiplication the easiest since it’s really easy to multiply a number by 10, 100, and so on. So instead of the problem 8 x 47, we instead have the problem 8 x (40 + 7). And, according to the distributive property, that’s just equal to 8 x 40 plus 8 x 7—which are both pretty easy problems to solve in your head: 8 x 40 = 320 and 8 x 7 = 56. So the total is 320 + 56 = 376. There’s no messy carrying of numbers or keeping track of extraneous digits. It’s all nice and tidy.
[[AdMiddle]And, of course, it works for more complicated problems too. How about the problem 7 x 437? In this case, we need to break the number 437 into 400 + 30 + 7. See how we were able to make 2 of the 3 numbers, 400 and 30, multiples of 10? Now, if we multiply this sum of numbers times 7, then instead of the one large and somewhat difficult problem 7 x 437, we have the three easier problems (7 x 400) + (7 x 30) + (7 x 7). So, 7 x 400 = 2800, 7 x 30 = 210, and 7 x 7 = 49. Put them all together and you get 2800 + 210 is 3010, and then 3010 + 49 gives a grand total of 3059. It might take a little practice before you can do it all in your head, but with a little effort you’ll be performing lightning fast multiplication in no time.
Web Bonus: Multiply Two 2-Digit Numbers Too!
The following grid of 4 rectangles gives you a new way to picture the problem of multiplying two 2-digit numbers. Just split each number up into 2 simple-to-multiply parts, then do the 4 simple multiplication problems, and finally add up the result. The total area has to equal the sum of the 4 smaller areas. See, math really is there to lend a helping hand and make your life easier!
Okay, that’s all the math we have time for today. Please email your math questions and comments to . You can get updates about the Math Dude podcast, the “Video Extra!” episodes on YouTube, and all my other musings about math, science, and life in general by following me on Twitter. And don’t forget to join our great community of social networking math fans by becoming a fan of the Math Dude on Facebook.
Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!