Consider the following scenario. The process of using a calculator is not nearly as fast as it is portrayed to be. First, you need to get it out and turn it on. Then, you need to enter in the numbers. Finally, it will display the answer for you. Think about it. How long does it really take to do all this? I claim that I can do this task faster in my head, and that I can teach you to do the same. I’ll teach you how to be lazy, save time, and avoid the hassle of getting out the calculator all at the same time!
There are two methods that I would like to share with you: Round And Adjust, and Multiply In Parts. Along with these two techniques, I’ll share with you a couple of tricks that I apply to simplify both of these methods.
Trick #1: Ignore the trailing zeros
What this means is that if you have a number such as 250, pretend it is just 25. Once you do the multiplication, you just add the zero back to the end of your answer. This works for any number of zeros. 250000, 250, and 25000000 should all be treated the same as 25. Just remember to add the respective amount of zeros to the end of your answer. However, you cannot simplify a problem when the zeros are in the middle. You can only use this for trailing zeros (zeros at the far right side). For example, 205 must be thought of as 205. This cannot be reduced to anything smaller. On the contrary, 2050 can be treated as 205, because there was one trailing zero. This helps quite a bit if you understand Difference of Squares.
Trick #2: Ignore the decimals
This trick is the exact same as the first one. If you have a decimal, forget about it. Pretend it isn’t there. It only has an effect on your answer, so you can completely ignore it while you’re multiplying. Doing work in your head requires you to simplify things as much as possible, or you can mess up really easily. So, pretend there is no decimal. 2.54 becomes 254, and 12.3 becomes 123. It is as simple as that. Now, how does this affect the answer? You need to count the number of digits that come after the decimal. 12.3 has 1 digit after the decimal, and 2.54 has 2 digits after the decimal. You are multiplying two numbers together, so you add the total. If you were multiplying 12.3 and 2.54, there are 3 total digits that come after the decimal. This should be reflected in your answer. If there is a total of 3 digits after the decimals, you will place the decimal in your answer so there are 3 digits after the decimal! It is as simple as that. For example, if you have a number that is XX.X multiplied by Y.YYY, the answer will be in the form ZZ.ZZZZ. I think that is enough said about this trick.
Now for the main course, my methods!
Round And Adjust
I’d like to explain this method by using an example. Let’s say you want to multiply 98 by 52, and try and do it quickly in your head. The answer is 5096, and I did that in around 8-10 seconds – much faster than getting out a calculator. So, what did I do? I rounded, did the multiplication, and then adjusted my result. I rounded 98 up to 100. Then, it is extremely easy to multiply 100 by 52: 5200. Then, I think about how much I rounded by. I rounded up by 2. 100 x 52 is 2 x 52 more than 98 x 52. This means, that 98 x 5equals 100 x 52 minus 2 x 52. Therefore, I find a quick answer by multiplying 100 by 52, then I subtracted 2 times 52 from that. 2 times 52 is 104. 5200 – 104 is 5096. Simple as that.
Now, what if you round down? What is 72 times 45? 3240. In my head, this was the process that I followed. 72 is close to 70, so let’s start with that. 70 times 45 is like 7 times 45. 7 times 45 is 315. Since I turned 70 into 7, I’ve got to add a zero to the end. 3150. I rounded down by 2, this means that I now have to add this 2 back in. To do so I multiply 2 by 45, and add this to my answer. 2 times 45 is 90, and 3150 + 90 is 3240. I know that took a little while to read, but it goes much faster in your head once you understand the method.
Why does this work?
98 x 52 = (100 – 2) x 52 = (100 x 52) – (2 x 52) = 5200 – 104 = 5096.
See? This is the algebra that allows for this method to be used. The same proof can be applied to rounding down. Basically, you look to round one of the numbers to a multiple of ten. It needs to be somewhat close to that number already, in order for this method to really be useful. Remember that you can only round ONE of the numbers. Do not round both numbers. Only one. Try it out =)
Multiply In Parts
What the heck does this one mean? Well, actually, it is very similar to my first method, however, it is a bit more universal. It works because of the same logic, but it does not require rounding or adjusting. It is simply breaking down a number piece by piece. What do I mean by that? I’ll show you an example, using algebra to solve it.
1234 x 32 = (1000 + 200 + 30 + 4) x 32 = (1000 x 32) + (200 x 32) + (30 x 32) + (4 x 32)
= 32000 + 6400 + 960 + 128 = 39488
Could you have done that in your head? I think you can =) You need to break up the larger number into its parts. For example, think of 345 as a 3, 4, and 5 separately. Multiply them individually, and then add the results. Remember that 3 is really 300, so you must add two zeros to that result, and add one zero to the result from the 4. This is the same way we do multiplication when we do it on paper. The difference is the perspective in which you are thinking about it. You are multiplying each number individually, and then adding the results. It’s not all that hard once you get used to it. 64 x 55. 6 x 55 is 330, and 4 times 55 is 220. The answer is simply 3300 + 220. The extra zero on the 3300 was inserted because the 6 is really a 60. The answer is just 3520!
What do you think? I promise that doing multiplication in your head is much faster than doing it on paper, and can even be faster than using a calculator. If you have a calculator out already, then, by all means, use it. However, if it is out of the way to go get it, you should try to do it in your head! I’ll do a couple of examples using all the technical math tricks I showed you above.
31 x 4.5?
31 x 4.5 is like 30 x 45, which is like 3 times 45. 135. 1350. When you add back in the 1 that we rounded down from, this becomes 1395. There was 1 digit to the right of the decimal. The final answer is 139.5. This is exactly what I thought in my head. I know it takes a while to read, however it is much faster when I’m thinking it. Try it out! It is really easy to make up an example for yourself.
2.40 x 314?
2.40 x 314 is like 24 x 314. 24 x 3 is 72. 24 x 1 is 24. 24 x 4 is 96. 7200 + 240 + 96 = 7536. When you add back the zero and decimal, you get 753.60.
To be honest, I don’t add all those large numbers in my head all at once. I keep a running tally. This helps when the numbers get much, much larger. What does that mean? In the example above, I say 24 x 3 is 72, which is 7200. I repeat that number in my head. 7200. 24 x 1 is 24, which is really 240. 7440. I then repeat that in my head, so that I remember it better. 7440. I continue this, until I get to the last number, 24 x 4 which is 96. 7536. Then I add in my zeros and decimal, and get the final result. Check also this real-life Math post.
The most common use for this is shopping. Do you ever go to a store and see that the price is 30% off? Is that really a good deal? What if it is 30% off, but then take an additional 20% off? Doesn’t that get a bit more complicated? Of course, cell phones can act as a calculator on the go, but is it worth the time to get it out and type in the numbers? I think not.
You go into a store and see a pair of shoes you’d like to purchase. The price for them is 49.99, and they are on sale for 30% off. However, they are having a special 1 day only sale, that allows you to take an additional 20% off the original sale price.
So, you need to take 30% off of the original price. Then, take 20% off of that.
First, sales need to be thought of in an opposite way of thinking. If something is 30% off, that means you are paying 70%. If you multiply the price by 0.7, this will give you the sale price. So, find 49.99 x 0.7.
49.99 x 0.7 is like 5 x 7, or 35. With the zeros, 35000. Subtract 1 x 7, from rounding. 34993. Now, add back in the decimal. 34.993. This is the new price! 34.993, or just 34.99! Now, you can take 20% off of that price. So, we will multiply by 0.8, because you are really paying 80% of the price when you take off 20%.
34.99 x 0.8 is like 35 x 8, 280. 28000. Subtract 1 x 8. 28992. The answer is 28.992, with the decimal. The final cost of the shoes will be 28.99! That is a great deal =)
How else could we have done this problem? Well, you are taking off 30%, and then another 20%. This is the same as paying 70%, and then paying 80% of that. We could just do 0.7 x 0.8 = 0.56. Multiply this by 49.99, and you will get the same answer. Just another way to think about it =)
I know I’m really geeky and all, but when I’m bored I multiply things in my head sometimes. It keeps my mind sharp, I believe. I also like to do some calculus in my head. I work a part-time job after school, and this helps pass the time when things are slow. I know it seems useless to do this sort of thing randomly, however, when the time comes when you need to use it, you can do it much, much faster. I hope this has been somewhat helpful!