Below is a transcript of my tutorial video: MCAT Math Video 3 – Multiplying/Dividing Complex Numbers Using Factor of 10 Trick
Leah here from leah4sci.com/MCAT and in this video I will show you how to do more advanced Multiplication and Division when you have multiple digits or multiple decimal spaces in your equation.
In my last video I showed you a simple trick on how to multiply and divide numbers that have a factor of 10 by moving the decimal to the right or the left. This trick can be taken a step further for more advanced questions such as this one.
“An elephant bumps a stone weighing 89 grams giving it an initial acceleration of 200 meters per second squared 200 m/s^2). With how much force did the elephant hit the stone?”
At first glance it seems to be an easy question if only you had a calculator. I’ll show you how to solve the question then we’ll break down Math trick. With any question like this, you have to look beyond the story and find the numbers that you’re going to calculate. In this case we are given 89 grams and 213 meters per second squared. In my physics videos I’ll break down the kinematic equations showing you how exactly when to use each formula.
For now, let’s just focus on the Math. We’re looking for Force is equals to Mass and Acceleration (F=ma). So looks like a simple plugin but be careful. Force has units of kilograms meter second squared (km/s^2) and here we’re given grams so we first have to convert. I’m only writing up the conversion because I want you to see what I’m doing but try to get comfortable doing this in your head.
We have 89 grams and we want to convert to kilograms. We know that 1 kg is equals to 1000 g and we set it up as 1 kg/1000 g allowing me to cancel out grams. In my last video I showed you the trick, when you’re dividing a thousand that has 3 zeros, you simply take the decimal and move it back 3 spaces. This gives me a new and usable number 0.089kg.
Now I have the appropriate units, I can plug them in. 0.089 kg x 213 m/s^2, we know the units are good because Force is kilogram meter per second squared (kg m/s^2), now let’s do the Math; 0.089 x 213 looks complicated, so let’s start by simplifying. 89 is close to 90 so we’ll change this number to 0.090, 213 is very close to 200 so we’ll round that down, so we’ve simplified the question.
But we still have the issue of numbers that we can’t necessarily do in our head. So here’s what I’m doing to do, I’ll take the first number and multiply it by a hundred (0.090 x 100), I’ll take the second number divided by a hundred (200/100), this gives me 9×2 which I can certainly do on my head and the answer is it’s 18 N. Punching this in a calculator, I got the answer 18.957 which on the MCAT it’s just close enough.
So let’s break down this trick going through series of step by step exercises. You already know that any number times 10 means that you move the decimal one space to the right and any number divided by 10 means that you move the decimal one space to the left. There’s another trick at work here; the second trick is multiplying a number by 1. Ninety Seven times one is Ninety Seven (97×1=97). A hundred times one is a hundred (100×1=100). Multiplying the number by 1 doesn’t change the number.
But the trick is being able to multiply by 1 when you see the number differently. For example, one over one is one (1/1=1), two over two is not two but still one (2/2=1), thirty three over thirty three is one (33/33=1), a green duck divided by a green duck is still one. And the idea here is, that it doesn’t matter what form your number takes, as long as it’s the same above and below the divisor, the answer is still 1. So for example if I take twenty times one over one is twenty three (23 x 1/1 = 23). Twenty three times Four point Six Seven divided by Four point Six Seven is still 23 (23 x 4.67/4.67=23) because I am simply multiplying 23 by 1.
Okay, so let’s take this one step further. If I have the equation 4 x 6 x 1/1, the answer is just 4×6. We’re not actually solving that, I just wanna show you that the identity doesn’t change. Now if I take 4 x 6 and I multiply it by 10/10, it’s still just 4×6. When you’re multiplying and dividing, you can move the numbers around because the order of operations doesn’t matter within multiplication and division, so watch.
If I have 4 x 6 x 10/10 that’s still 4×6 because I have 10 divided by 10, that’s 10/10. I can also write it as 4 x 6 x 10 x 1/10, one more time I have times 10 divided by 10 and I can even write it as 4/10 x 6 x 10 . This is the key; notice that I took the two numbers that I initially started with and I took 1 and multiplied it by 10, the other to divide by 10, that still gives me 4×6. Now let’s try an application on this.
Let’s take a number that you can probably do in your head, let’s say I ask you to calculate 2 x 20. Two times twenty is just twenty and twenty which is forty (2×20=40). So let’s prove that using this trick still gives us 40. If I take the numbers Two times Twenty and then do times 10 and divided by 10 (2x10x20/10) I still should have the answer 40, let’s prove it. 2 x 10 means we move the decimal one space to the right giving me 20, 20 divided by 10 means we move the decimal to the right giving me 2. I have 20 x2 which is still 40.
Let’s try something even more complicated. Let’s take the number 0.05 x 400. This is something that you may come across and can necessarily do it from your head but what if now instead of times 10 divided by 10 we try times a hundred divided by a hundred. What I am trying to do and how we choose where to put the multiplication and where to put the division is take the big number and make it smaller and take the smaller number and make it bigger. Specifically, I’m trying to get rid of the decimal because let’s face it, decimals are confusing. So if I take the point zero five (0.05) and I multiply that by a hundred, to make that fair, we have to take the 400 and divided by 100 because this way I have times a hundred divided by a hundred and that means I’m not changing the total identity of the question. 0.05 times 100, 100 has two zeros that’s why I move the decimal two spaces to the right giving me 5.
Look at 400 divided by a hundred, 100 hundred has two zeros and because we’re dividing, we move the decimals two spaces to the left, that gives me 4. Now I don’t know about you but 0.05 x 400 is difficult, 5×4 is not, it’s just 20. So what we did here is instead of writing it up the long way, don’t even write this up.
On your MCAT just mentally move the decimal to the right for the small number, to the left for the big number by the same number of space, so watch me do this without even writing times a hundred divided by a hundred. Here’s how I would look at it if this was an MCAT question. 0.05 is too small, I move the decimal 2 spaces to the right. To make up for that, I move the big decimal 2 spaces to the left. 5×4 as we already said is 20.
Now let’s go back to the initial question that we look at. We had 0.090 x 200, look at what we did. We move the decimal from 0.090 two spaces to the right, giving me 9, we move the decimal 200 two spaces to the left, giving me 2 and without any long and detailed calculation we know that 9 times 2 is 18 and that’s how we quickly got our answer.
Be sure to join me in the next video where I show you how to apply a similar trick when it comes to negative and positive exponents.
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Watch The Video Here: Multiplying/Dividing Complex Numbers Using Factor of 10 Trick