Mystery Skill 2: 72 Times

Published on Sunday, August 11, 2013 in , , , , , ,

SPUI's 72 highway shield imageHopefully, you've been practicing your skills at multiplying by 63, and are ready to move on to the next level.

In this post, you'll learn how to quickly multiply by 72. This builds on your skills at multiplying by 63, so this post will actually combine both of them.

BASIC STEPS: For multiplying by 63, I listed 5 steps in the last post. By the time you've completed step 3, you've tripled and doubled the original given number. In other words, you've multiplied the given number by 6 at this point. For multiplying by 72, 6 happens to be the starting point, so this list will continue from that earlier list:

6) Recall the number calculated in step 3.
7) Double the number from step 6. Do not forget the number from step 6, however.
8) Write down the ones digit of the result you calculated in step 7.
9) To the hundreds and tens digits ONLY of the number you calculated in step 7, add the entire number from step 6.
10) Write this new total to the immediate left of the digit written in step 8.

As you can see, this is nearly identical to the steps for 63, with some minor rearrangement. Let's turn to some practical examples to help clear up any confusion.

WHOLE NUMBERS: For our first example, let's start with 13. First, we'll multiply it by 63, and then multiply by 72 after that. We'll use this form to write our answers:


13 × 3 = 39, and we write down the 9 next to the 63 (leaving room to place numbers to its left):

63: 9

Now, we double 39 to get 78, trying to keep both 39 and 78 in mind. We add 78 to the tens digit of the first answer, which is 3. 78 + 3 = 81, so we write 81 down to the left of the 6, and we're done with the first part:

63: 819

That's all for 63, now how about 72? The number we calculated in step 3 was 78, so we'll start from there. We double this number to get 156. As instructed in step 8, we write down the ones digit of 156, which is 6:

63: 819
72: 6

Now, we add 78 to just the hundreds and tens digit of 156. What does that mean? The hundreds digit of 156 is 1 and the tens digit is 5, so we're going to add 78 + 15 to get 93. Finally, as instructed in step 10, we write this total down to the immediate left of the number we wrote previously:

63: 819
72: 936

There we go! We've calculated 13 times both 63 and 72 fairly quickly. Notice how the last doubled number from the calculation for 63 becomes the first step for 72? Once you practice calculating in this way, you'll probably develop an apreciation for the efficiency of this approach.

I'll run through another number to help lock in the idea. This time, 19 will be the given number. 3 × 19 = 57, so we write down the 7:

63: 7

57 doubled is 114, so we add 114 plus the ones digit, 5, to get 119:

63: 1197

Not forgetting the 114 at this point, we double that to get 228, and write down the ones digit, 8:

63: 1197
72: 8

Now we add 114 to 22 to get 136, and we write that down to the left of the 8:

63: 1197
72: 1368

Quicker than you may have thought possible, we've multiplied 19 by 63 and 72.

Just as before, practice your skills for multiplication of both 63 and 72 by random whole numbers up to 33. Don't forget to check your answers for both number by using this link, and setting x to the appropriate given number.

QUICK EXPLANATION: Note that the ones digit from only the first multiplication and last multiplication is written down. The ones digit of the second number you calculate is never used in this way. I'll quickly explain why.

When you start with a given number, which we'll dub N, you multiply it by 3 to get 3N, then by 2 to get 6N, and finally 2 again to get 12N. Now, 60N is just 6N × 10.

What you're really doing in this method is calculating 3N and 6N, then adding 6N to all the digits except the rightmost digit, which effectively treats it like 60N. Adding 60N + 3N will result in 63N, which is why the first trick works.

Still thinking of 6N, you double that to 12N. Once again, you treat 6N like 60N by adding 6N to all but the rightmost digit. Not surprisingly, 60N + 12N = 72N, which is how we get our second answer.

Let's go back to that 13 example above to get a better idea of how this works. 13 × 3 = 39. Next, 39 is double to get 78. As mentioned, this should really be 780, since we're trying to figure 60N + 3N. 780 + 39 = 780 + 30 + 9 = 810 + 9 = 819. See what happened there?

Because of the multiplication by 10, we know the last digit of 60N will end in a 0, so we know the last digit won't change. All that remains then, is to add 780 + 30. This is much easier to do if we don't attach any important to the zeroes, so we simply do 78 + 3 = 81, and put the 81 in its proper place.

The explanation for 72 is similar. We start with 78 (6N), and double that to 156 (12N). To work out 60N + 12N, we'd be doing 780 + 156 = 780 + 150 + 6 = 930 + 6 = 936. Again, the rightmost digit of 156 (12N) won't change, so we can just write it down immediately. By performing 78 + 15 instead of 780 + 150, the whole thing is simplified for mental math.

If you're worried about adding two digits to two digits, just add the tens digit, followed by the ones digit. To simplify 78 + 15, perform 78 + 10 to get 88 first, and then add 5 to get 93. To better understand the advantages of left-to-right addition in mental math, check out this video.

NUMBERS ENDING IN .5: Here's some good news. By the time you've calculated 6N, as discussed above, you're dealing with a whole number, so as far as 72 goes, there's little difference at this stage. To demonstrate, let's use a given number of 14.5.

14.5 × 3 is 43.5 (Remember how to get here quickly? It was discussed in the previous post!), so we write down 3.5:

63: 3.5

43.5 doubled is 87, so we add 87 + 4 (the tens digit of 43.5) to get 91, we write down 91 to the left of the previous numbers:

63: 913.5

Starting from 87, we double that to get 174, so we immediately write down that 4 (the rightmost digit of 174):

63: 913.5
72: 4

87 + 17 = 104, so now we put down that 104 to the left of the previous number, and now we have both complete answers:

63: 913.5
72: 1044

See? With a number ending in .5, the multiple of 63 will always end in .5, but the multiple of 72 will always be an even number.

To familiarize yourself with multiplying 63 and 72 by all possible numbers you'll need to handle, you can use this link, which will occasionally generate random numbers ending in .5. Again, don't forget to check your answers for both 63 and 72 by using this link, and setting x to the appropriate given number.

When you've mastered these calculations, you're almost ready to handle the full mystery feat. The nature of this feat will be revealed in Thursday's post!

Spread The Love, Share Our Article

Related Posts

Post Details

No Response to "Mystery Skill 2: 72 Times"