Video Transcript
A roll of cloth, with a length of two hundred and eighty point five four meters, is divided into pieces of equal length. Given that each piece is one point six nine meters long, how many pieces was the roll divided into?
So this question is really asking how many times does one point six nine go into two hundred and eighty point five four. In other words, what is two hundred and eighty point five four divided by one point six nine.
Now I can express two hundred and eighty point five four divided by one point six nine, actually, as a fraction. Two hundred and eighty point five four over one point six nine. But we don’t normally have decimals in fractions, so I’m gonna multiply the numerator and the denominator by a hundred in order to get rid of those decimals. Now because I multiply the numerator and denominator by the same number, I have an entirely equivalent fraction.
Now we’re gonna look at two different ways of approaching this question and see which you like the best. So first of all, we’re just gonna do some long division. We’re gonna do two eight zero five four divided by one six nine. And in order to make life easier, I’m just gonna write up my hundred and sixty-nine times table first. One times a hundred and sixty-nine is a hundred and sixty-nine. And then if I add a hundred and sixty-nine to itself, I get three hundred and thirty-eight. Then if I add a hundred and sixty nine again, I’m gonna get five hundred and seven. Then adding a hundred and sixty-nine again, gives me six hundred and seventy-six. Adding a hundred and sixty-nine again, gives me eight hundred and forty-five. Adding a hundred and sixty-nine again, gives me one thousand and fourteen. Adding it again, gives me one thousand one hundred and eighty-three. And again, gives one thousand three hundred and fifty-two. And again, gives me one thousand five hundred and twenty-one.
Now the next one will be an interesting check. If I add a hundred and sixty-nine to one thousand five hundred and twenty-one, I should get the same as if I just multiply a hundred and sixty-nine by ten, so get one six nine zero. And luckily, I do! So this is a useful check. Now I’ve got my hundred and sixty-nine times table. It’s gonna make the long division a little bit more easy.
So first up, how many times does a hundred and sixty-nine go into two hundred and eighty? Well looking along at my times table, two times a hundred and sixty-nine will be three hundred and thirty-eight. That’ll be too big. So it’s gonna be one. And one times a hundred and sixty-nine is a hundred and sixty-nine. Then if we find the difference between those two, well zero take away nine I can’t do. So I’m gonna need to borrow one from the next column, so seven and ten. So ten take away nine is one, seven take away six is one, and two take away one is one. Now we can bring down the next digit, and ask how many times does one hundred and sixty nine go into one thousand one hundred and fifteen. Well seven times a hundred and sixty-nine will be one thousand one hundred and eighty-three, which should be too big. So it’s gonna be six. And six times a hundred and sixty-nine is one thousand and fourteen. So now I need to find the difference between one thousand one hundred and fifteen and one thousand and fourteen. Well five take away four is one, one take away one is zero, one take away zero is one, and one take away one is nothing. So I wro- won’t write anything there. Now I can bring down my next digit, and I need to ask how many times does a hundred and sixty-nine go into one thousand and fourteen. I know the answer to this, it’s six. I’ve just worked that out. And six times a hundred and sixty-nine is one thousand and fourteen. So the difference between those is nothing. So we’ve got an exact answer, a hundred and sixty-six. So the answer is exactly a hundred and sixty-six pieces.
Well let’s try and think of another way of simplifying and evaluating this expression here, two eight zero five four divided by one six nine. Well if I can find the prime factor decomposition of each of those numbers, I’m gonna see if anything will cancel down. So first of all, I’m gonna decompose twenty-eight thousand and fifty-four. Well it’s an even number, so I know it will divide by two. And it’s a relatively easy number to divide by two. So instead of twenty-eight thousand, it’s gonna be fourteen thousand. And instead of fifty-four, it’s gonna be twenty-seven. And two is a prime number. So now fourteen thousand and twenty-seven, what’s that divisible by? Well it’s certainly not divisible by two. And is not divisible by three, all the digits don’t add up to three. So at this point, I’m thinking this might be a bit of a tricky one to answer. So let’s look at a hundred and sixty-nine and try to do the decomposition on a hundred and sixty-nine.
Well I happen to know that a hundred and sixty-nine is a square number. It’s thirteen squared, so thirteen times thirteen. And thirteen is a prime number. Now if this is gonna cancel down, thirteen would need to be a factor of fourteen thousand and twenty-seven. So let’s just try that. Thirteens into fourteen go once, and one times thirteen is thirteen. The difference between fourteen and thirteen is one. Let’s bring down the next digit. Thirteens into ten don’t go. So let’s bring down the next digit. So how many times does thirteen go into a hundred and two? Well I know that ten times thirteen would be a hundred and thirty, so that’s too big. Five times thirteen would be sixty-five and that’s too small. So let’s try seven. Seven times thirteen is ninety-one. Well that’s getting closer. And eight times thirteen is just a bit too big, that’s a hundred and four. So we’re gonna have to go with seven. Seven times thirteen, as we said, is ninety-one. The difference between two and one is one. And then the difference between zero and nine, we can’t do. So we’ll have to borrow one. Ten an- ten take away nine is one. And then let’s bring down the next digit. So the next question is, how many times does thirteen go into a hundred and seventeen. Well eight times thirteen was a hundred and four, so nine times thirteen is exactly equal to one one seven. So the answer is nine. So fourteen thousand and twenty-seven divided by thirteen, which is a prime number, is equal to one thousand and seventy-nine.
Well we’ve still got a pretty big number to find factors of. But look, a hundred and sixty nine is thirteen times thirteen. So let’s just, on the off chance, try to see if thirteen is a factor of one thousand and seventy-nine. And then if it is, that means we’re gonna be able to cancel both of those out and get a nice number. So let’s just try it. Well thirteens don’t go into one or ten, so how many thirteens go into a hundred and seven. But we just said that eight times thirteen is a hundred and four, so anymore thirteens and that’ll be too big. So eight thirteens are a hundred and four. A hundred and seven take away a hundred and four is three. Let’s bring down the next digit, the nine. And thirteens go into thirty-nine exactly three times. So one thousand and seventy-nine is thirteen times eighty-three. And thirteen is a prime number. And thinking about eighty-three, that’s a prime number as well. So I can rewrite twenty-eight thousand and fifty-four as two times thirteen times thirteen times eighty-three. And now my fraction will cancel. Thirteen is going to thirteen once, thirteen is going to thirteen once, and I’m left with two times eighty-three on the top and just one on the bottom. And two times eighty-three is one hundred and sixty-six.
So two different ways of answering the same question, both pretty horrible cause they’re not very nice numbers. But sometimes, prime factor decomposition makes your life a lot easier. It’s sad that it didn’t in this particular case.
