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Strategies for multiplying decimals and whole numbers

CCSS.Math:

Video transcript

in this video we're going to further build our intuition for multiplying decimals so let's say that we wanted to figure out what eight times seven tenses pause this video and see if you can figure this out on your own alright now there's several ways that we could approach what eight times seven tenths is we could view this as eight times and we could write seven tenths as a fraction so we can re-express this as seven tenths 7 over 10 is the same thing as zero point seven and we already know how to multiply fractions you could view this as being equal to eight is the same thing as 8 over one or eight holes I guess you could say times 7 over 10 times 7 tenths which is going to be equal to if we multiply our numerator we're going to get 56 and if we multiply our denominators we get tense and that makes sense if I have 8 times 7/10 I end up with 56 tenths now 56 tenths can also be written as this is the same thing as 50 plus 6 over 10 which is the same thing as 50 over 10 plus 6 over 10 and so this is the same thing as this is 5 holes so 5 & 6 tenths 5 & 6 tenths which we can write as 5 & 6 tenths or 5.6 it's always good to do a little bit of a reality check whenever you get an answer when you're multiplying decimals say okay 7 tenths is a little bit less than 1 so we would expect this product if we're multiplying 8 times something a little bit less than 1 we would expect the product to be a little bit less than 8 so 5 point 6 makes sense if for some reason we got the we you computed something and you were to get 60 you'd say wait that doesn't make sense I should get a value less than 8 and similarly if you somehow got a value a product of like 1 you're like well that's a lot less than 8 I should get something that is 7 tenths of 8 now another way that you could approach this is you could view this as the same thing as 8 and once again I'm just gonna write this in a different way eight times seven eight times seven tenths so if you have eight times seven of something what is that going to be equal to well eight times seven that's 56 so you're going to be this is going to be equal to 56 tenths 56 tenths and one way to think about 56 tenths 56 tenths is the same thing as 50 tenths 50 let me color code that differently so this is going to be the same thing as 50 tenths 50 tenths plus 6 tenths plus the six tenths can't write tenths six tenths and 50 tenths is the same thing as five ones so five ones and six tenths which is exactly what we have here five ones and six tenths let's do another example that's a little bit more involved so let's say that we want to figure out we want to figure out what is three times 0.87 pause this video and try to figure that out well once again there's many ways to approach it but we could just start with the way that we just looked at we could say hey this is the same thing as three times and we can re-express this as this is the same thing as 87 hundredths 87 hundredths and so if I have three times 87 of something what am I going to be left with well this is going to be equal to some number of hundredths and to figure out that we do have to figure out what what's three times 87 so 87 times three seven times three is twenty-one we regroup that two it becomes two tens and then 8 times 3 is 24 and that's really 24 tens plus those other two tens so we get 26 tens which is the same thing as two hundred 206 tens there's going to be 261 so the three times 87 of something is going to be 261 of that something and in this case something is hundreds so this is two hundred and sixty-one hundredths so how do we express this as a decimal well there's a couple of ways that you can approach it you can think about it is this is the ones place this is the tenths place this is the hundredths place and so very clearly one hundredths here would be one in the hundreds place if you have 60 hundredths which is what the six represents 60 hundredths is the same thing as six tenths and then last but not least if you have two hundred hundredths that's the same thing as two wholes another way to think about it is you go to the hundredths place and then you start from there but you write out two hundred sixty one one the 60 hundredths and then the two hundred hundreds and you get to point six one now another way that you could have approached this and we saw this in the last example is you could say hey this is going to be the same thing as three times 87 hundredths see these are all equivalent but hopefully one of these are more than one of these register with you of what's really going on well this is going to be the same thing as three wholes times 87 hundredths 87 hundredths and so this is going to be equal to in the numerator we have three times 87 three times 87 and hundredths one times 100 is a hundred three times 87 hundredths well we already know what three times 87 is this is equal to 260 one hundredths and you can see a hundred goes into 261 two times and you're left with 61 hundredths so these are all equivalent representations and just a reminder so it's always good to estimate and so what you have here is you have three times something that's a little bit less than one so you would expect a value a little bit less than three and so 2.61 also meets that sniff test that this seems about right for some reason you got 26 or 261 that would be way off or even if you got 0.26 one that would also feel way off so hopefully this is helpful