Mixed problems with standard form

Key Stage 3

Maths

Indices and standard form

﻿Hello and welcome to today's lesson on the mix operations in standard form with me, Miss Oreyomi. For today's lesson, you will need a paper and a pen or something that you can write on and with. It would also help if you can minimise the distraction by putting your phone on silent and also trying to get into a space with less noise or distractions. If at any point during this video, you need to pause it or rewind it just so that you can go back and go over a point, then feel free to do so. Also, when I tell you to pause the video and attempt a task, please do so as that would help you understand further that particular point I am making. So should you now wish to pause the video and go get your equipment and put yourself in a space with less distraction, pause the video now and press play when you are ready to begin the lesson. Okay, let's think about our try this task. You've got five numbers on your screen. I want you to evaluate each of these. And then what do you notice? Once you get an answer for each one, what do you notice about your answers? So pause the video and attempt this, and then press resume when you are ready to carry on. Okay, hopefully what you notice is they all give you the same answer of 1,200,000. So hopefully that is what you notice. And even though we did it using different signs, so different operation, that this was multiplication, this was subtraction, this was addition, this was division, we're still able to get the same number. And today's lesson is just based on that. Being able to carry out calculations in standard form using different operations. So let's look at this question here. We have a table that is showing the population of five countries, and we have two questions, which of these countries have the largest population? And we want to calculate the difference between Nigeria here and Jamaica here. So let's attempt the first one, which country has the largest population? So we are thinking of the largest population. Well, in this case, we're just going to be looking at the power over here, isn't it? So the country where the largest population is Nigeria. So we're looking at, in this case, we're simply looking at the power to determine which have the largest power. Remember the first lesson we did on standard form, we said, we put large numbers in standard form as they help us to order, they help us to compare easily, they help us to see which is the biggest, which is the smallest and so on and so forth. So by looking at the power here, we can see that this number is rest of the power of eight, whereas the rest are raised to a smaller power. So Nigeria has to be the country with the largest population with this table. The second question, country calculate the difference between the population in Nigeria and the population of Jamaica. While I've gone ahead and written the population of Nigeria as an ordinary number. So I've written 1.87 to the power of eight, times 10 to of power eight is an ordinary number which is here. And then 3.5 times 10 to power seven as an ordinary number which is here. And I want to calculate the difference. So I'm going to be subtracting my numbers. So I'm just going to draw a line over here. So that would be zero, zero, zero, zero zero , seven take away five that is two, that is five and that is one. I'm not quite done because I was told to give my answer in standard form, remembering that the first part of my number, I'm just going to write my answer here. The first part of my number must be a number between one and 10. So that I'm going to say is 1.52. And from 1.52, to get to the end of my number here, I am going to times it by one, two, three, four, five, six, seven, eight. I'm going to multiply that number by 10 to the power of eight, to get back to my original number. What of question three, the population of China is 30 times the population of the UK. So I don't know what the population of China is. I want to calculate the population of China. If it's 30 times the population of the UK, what am I meant to do with those two numbers? I'm meant to multiply them together, ain't I? So I'm going to have 30 multiply by 1.4 times 10 to the six. I could take my numbers here and multiply them together first and then just add my 10 to the power of six. So if I've got 30 times 1.4, that would give me 42 times 10 to the power of six. Again, 42 is not a number between one and 10, so right I'm going to split or rather I'm going to divide this by 10. So that would be 4.2. And because I've divided by 10, I have to increase my power by 10. So it's going to be 4.2 times 10 to the seven. Let's look at another question. The table gives the surface area, in square kilometres of five seas. And we have three questions here. Write the number 4.95 times 10 to the seven as an ordinary number. And then we have question two and question three, and we're going to work through each one. So the first one, I want to write the number 4.95 times 10 to the power of seven as an ordinary number. Well, I'm going to start by writing 495. And then how many zeros would I need to add so that when I'm going back to standard form would give me 10 to the power of seven? Well, it's going to be five. So it's one, two, three, four, five. Think about it this way. If I put a decimal point here and I count to get to the end of my numbers, how many times would I count to get to the end of my number? It should be seven. So it's one, two, three, four, five, six, seven. So I know that I have put the correct number of zeros. So here is, yeah. So it's this one, 49,500,000. Number two, work out the total surface area for these five seas. So I've gone ahead again and written out my numbers from standard form to ordinary number. So we did that first and then 1.5 times 10 to the power of six. Well, if I put a decimal point there, I would count one, two, three, four, five, six times. So I know I've put the correct number of zeros. So I'm going to go ahead and add it up. So I've got zero plus zero plus zero plus zero plus zero, that is zero, zero, zero. That is three. That is again, another zero. That is 11, 15, 14. That is one. And that is five. And we were meant to give our answer in standard form. So if I want to give this answer in standard form, what should I write? I should have a number between one and 10. I'm just going to write my answer here. I should have a number between one and 10. So if I put a decimal point between the five and the one that will be 5.1403. What do I need to raise my number to, say that when I'm more to plate out, I would have this number here. And it's going to be, one, two, three, four, five, six, seven. So it's going to be times 10 to the power of seven. If I multiply this out, I would get an... If I write this rather, this standard form number as an ordinary number, I'll go back to this one there. Okay, let's think about this question. The surface area of the Mediterranean sea. Well, that is this information here, isn't it? So I'm going to write the surface area of the Mediterranean sea, which is 4.95 times 10 to the seven, is K times the surface area of the East China sea. What does 'is' mean in this question? Is means? It's equal, right? So it means this surface area here is equal to K times the surface area of the East China sea. What am I doing with K and what operation am I putting in between K and the East China sea? It's multiplication, isn't it? 'Cause it's K times the surface area of the East China sea. So it's K times the East China sea, which is 1.5 times 10 to the power of six. So whatever this value is, multiply by that value would give me the surface area of the Mediterranean sea. So if I want to find what K is, what am I doing to this side of my equation? Right, I am going to divide by 1.5 times 10 to of the power of six, wouldn't I? Whatever I do to one side I must do to the? I must do to the other side. Good. So I'm going to divide by 1.5 times 10 to the power of six. If I write that out more clearly, over here I've got 4.95 times 10 to the power of seven divided by 1.5 times 10 to the power of six, that is equal to K. How can I go about this then? Well, I could divide my base, can I? 4.95 divided by 1.5. I could make it into whole numbers. 495 divided by 15, that would give me 33. And if I put my decimal point back, it's going to be 3.3. 10 to the power of seven, take away 10 to the power of six, 'cause we're dividing. Well, that's going to be 10 to the power of one, isn't it? But I'm told to give my answer to the nearest whole number. What is the nearest whole number here? If I times it together, it's going to give me 33. 3.3 times 10 to the one is the same as 3.3 times 10. So just going to write it here. 3.3 times 10, which is the same as 33. So my answer is 33. So 33 times 1.5 times 10 to the power of six would give me the surface area of the Mediterranean sea, which is 4.95 times 10 to the seven. Your turn. Pause the video now. You have three questions on your screen, using the data given to you, pause the video and attempt the questions and then we'll go to have a chance to go over them again. So pause the screen now and have a go at these questions. Okay, let's look at question one. Which of these countries has the largest population. So we're looking at the country with the largest population. Well, I'm looking at the power value, isn't it? And the country where the largest power is Algeria. So I am going to write out Algeria here. Next question calculate the difference between the population of Malaysia and the population of Ethiopia. And you're told to give you answer in standard form. Right, the difference means I am subtracting, aren't `I? So I have written my answer as an ordinary number for the population of Ethiopia. Notice that I put in Ethiopia above Malaysia. 'Cause Ethiopia has got a bigger value. And when I'm subtracting, that really matters. So I've got zero, zero, zero, zero. I'm going to borrow one from here. So I'm going to have eight zero three. I want to give my answer in standard form. So the first part of my number must be a number between one and 10. So to get a number between one and 10 here, I am going to put my decimal point between my three and my zero. So I'm going to have 3.08 times 10 raised to the power of something. What is that something that when I multiply by 3.08 would give me this number? Well, it's going to be one, two, three, four, five, six. It's going to be times 10 to the power of six. How did you get along with that one? Last one then. What is the total population of Algeria and Sierra lone? Give your answer in standard form. Again, I've written my standard form as an ordinary number for both countries. So I'm just going to add them up together. So again, there'll be zero, zero, zero, zero, zero, six, seven, one. I'm told to give my answer in standard form. So I am going to do that. I need the first part of my standard form to be a number between one and 10. So I'm going to put my decimal point here. So it's going to be a 1.76. And what do I need to raise the power to a number that if when I multiply out, I get back to my original number? So it's going to be one, two, three, four, five, six, seven, times 10 to the power of seven. It is now time for your independent task. If you need to go back through the video and re-watch somethings, to be able to help you answer certain parts of this, of your questions for your independent tasks, then feel free to do so. So pause your screen now and attempt all the questions on your worksheet. And then once you finish your questions, click play, and we would go through the answers together. Okay, let's go over your independent tasks then. The distance between the Earth and Mars is 55 million kilometres. Write the number 55 million in standard form. So for the first one, we're going to ride 5.5 times 10 to the seven. The distance between the Earth and Neptune is 80 times greater than the distance between the Earth and Mars. Calculate the distance between the Earth and Neptune. So if 80 times greater, which is 80 times bigger than the distance between Earth and Mars. So I'm going to be multiplying this by, 80 multiply by 5.5 times 10 to the seven. So 80 times 55, I know is 4,400. So I'm putting my decimal place. I am left with four, four, zero, 440. And then I've got times 10 to the power of seven. So I am going to put my seven, one, two, three, four, five, six, seven. Now I want to write this in standard form. So I am going to get a number between one and 10. I'm just going to write the answer here so that you know the answer is here. So I want a number between one and 10. So I'm going to put my decimal point between four and four, 4.4. And if I raise it to the power of 10, what would be the number I have to raise it to so that when I multiply it out, I would get this ordinary number. And your answers should have been nine. So it'd be one, two, three, four, five, six, seven, eight, nine. So I've find the answer is 4.4 times 10 to the power of nine. Next one then. Work out the value of two times 10 to the negative two divided by eight times 10 to the six. Well I know that when I'm divided, I divide my base and then I subtract my powers. So if I write that again, they'll be two divided by eight. That is a quarter. So that as a decimal, that is 0.25. And then I'm subtracted my power negative two subtract six and that takes me to negative eight. So if I am writing this down, I'm going to write 0.25 times 10 to the negative eight. I'm not quite done though, am I? 'Cause I need to convert, I need to give my answer in standard form and this is not in standard form. So what do I need to do? I need to multiply this by 10, don't I? So that will be 2.5. If I multiply by 10, I need to decrease my value by 10. So that is going to be 10 to the negative nine. Here are three numbers written in standard form, arrange these numbers in order of size. It helps if I can change all the powers to be the same. So I am going to try and change all to be to the power of negative eight. So to get to the power of negative eight, I'm going to divide by 10 and then add 10. So that takes me to there. So this one I am going to multiply by 10 and then I could decrease my power by 10. So I've got 57 times 10 to the negative eight. 0.84 times 10 to the negative eight. And 6.6, four times 10 to the negative eight. So the smallest number there is what? Well, it would be this, this is the smallest number, this is the second smallest and this is the third smallest. So when you have a question like this, try and convert it, try and change it so that all your powers are the same. Work out the value of seven times 10 to the power of seven multiplied by five times 10 to the power of six. Well, I'm multiplying in standard form. What do I have to do? I have to do seven times five, that is 35. And then I'm adding my powers, ain't I? So that'll be times 10 to the 13. And then I am going to put this 35 to be a number between one and 10. So I'm going to divide that by 10. So that would be 3.5. Because I divided by 10, I'm going to increase my power by 10. So that would be 10 to the power of 14. Workout 4.2 times 10 squared, plus 2.4 times 10 to the power four. Well, I am going to write this out as an ordinary number, so that'll be four, two, zero. 'Cause that'll be one, two. And then I am adding two, two, four, zero, zero. So one, two, three and four. And I'm going to be adding these numbers together. I haven't been told to give my answer in standard form. So I am just going to leave my answer like that. Okay. You now have your explore task. Light travels at a speed of three times 10 to the power of eight metres per second. Your job is to convert this to miles per hour. How fast is this in miles per hour? I have put some information on your screen that I think may be very useful to you. So one mile is 1.61 kilometre and one hour is 3,600 seconds. So pause the video now. How would you go about converting this speed in metres per second to miles per hour? And once you finish, press play, and there should be an answer on your screen. So pause the video now and have a go at this task. A very big well done to you for completing, not just this lesson, but the entire unit. So you were stuck right through to the end and started from the beginning straight to the end. So very big well done to you. Do complete your quiz before leaving today, as that shows you what you've learned so far. So again, very big well done and goodbye. Bye.

Mixed problems with standard form

Mixed problems with standard form

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UnitMaths / Indices and standard form

Maths