Video Transcripts: Decimals and Fractions

Rewriting decimals as fractions 

Write 2.75 as a simplified fraction. So once you get some practice here. You're going to find it  pretty straightforward to do. But we're really going to think through it and get the intuition for  why this makes sense. So if we were to write this down, the 2, that literally just represents two  1's, I'll just write it down like that. Then we have the 7 in another color. We have a 7 one place to  the right of the decimal. It's in the tenths place, with a T-H-S at the end. So it literally  represents 7 over 10. And then finally, we have the 5 in the hundredths place, so it represents 5  over 100. Now, if I want to write this as a simplified fraction, or really as a mixed number, I have  to merge these fraction parts right here. And to add two fractions, you have to have a common  denominator. And to figure out the common denominator, you just have to think about the  least common multiple of 10 and 100. And that's 100. 100 is divisible by both 100 and 10. So  let's get this 10 to be 100. So we can do that by multiplying it by 10. So when you multiply  something by 10, you add a zero at the end of it. But you can't just do that to the denominator.  We also have to do that to the numerator. So we multiplied the denominator by 10. Let's also  multiply the numerator by 10. 7 times 10 is 70, or 70 over 100. It's the exact same thing as 7/10.  Now we can add these two. What is 70 plus 5? 70 plus 5 is 75. And our denominator is 100, so  this can be rewritten as 2 and 75/100. And we saw that in the last video, you would read this as  two and seventy-five hundredths. Now we aren't in a completely simplified fraction yet  because 75 and 100 have common factors. And the largest number that goes into both, if you're familiar with quarters, is 25. Three quarters is $0.75, four quarters is 100 cents, or four quarters  is $1.00. So you divide both of them by 25. So 75 divided by 25 is 3, and 100 divided by 25 is 4. So as a simplified mixed number, this becomes 2 and 3/4. And after you do a lot of practice here,  and you just see a lot of numbers like this, it will be almost second nature for you to say, oh,  2.75 is the same thing as 2 and 75/100, is the same thing as 2 and 3/4. 

Worked example: Converting a fraction (7/8) to a decimal 

Write 7/8 as a decimal. And so the main realization here is that 7/8 is the same thing as 7 divided by 8, which is the same thing as 7 divided by 8. These are all different ways of writing the same  thing. So let's actually divide 8 into 7. And I'll do it down here just so I have some more real  estate to work with. I'm going to divide 8 into 7. And I'm going to add a decimal point here, just  because we know that this value is going to be less than 1. 7/8 is less than 1. We're going to have some digits to the right of the decimal point. And let me put the decimal point right up here,  right above the decimal point in 7. And then we start dividing. And now this really turns into a  long division problem. And we just have to make sure we keep track of the decimal sign. So 8  goes into-- it doesn't go into 7 at all, but it does go into 70. So 8 goes into 70 eight times. So it  goes into 70 eight times. 8 times 8 is 64. And then you subtract. 70 minus 64 is 6. And then  bring down another 0 because we still have a remainder. We want to get to the point that we  have no remainders. Assuming that this thing doesn't repeat forever. And there's other ways  we can deal with that. 8 goes into 60? Well, let's see. It doesn't go into it eight times because  that's 64. 8 goes into 60 seven times. 7 times 8 is 56. And then we subtract again. 60 minus 56 is 4. And now, we can bring down another 0 right over here. And 8 goes into 40? Well, it goes into  40 exactly five times. 5 times 8 is 40. And we have nothing. We have nothing left over. And so  we're done. 7 divided by 8 or 7/8 is equal to 7 divided by 8, which is equal to 0.875. But I'll put a 

leading 0 here just so it makes it clear that this is where the decimal is. 0.875. And we are done. 

Fraction to decimal: 11/25 

Let's see if we can write the fraction 11 over 25, or we could call 11/25, to see if we can write that as a decimal. And we're going to round it to the nearest thousandths place. And so another way of viewing this is 11 over 25, this is the same thing as 11 divided by 25. So we can literally divide  

25 into 11, and whatever we get, that is going to be the decimal representation of 11/25. And  since we're going to go into the places less than the ones place, we're going to go into the  tenths place, the hundredths place, and the thousandths place, let's add some zeroes to this 11  right over here after the decimal, and now let's start to divide. 25 doesn't go into 1. 25 doesn't  go into 11. 25 does go into 110. So 25 goes into 110 four times. 4 times 25 is 100. So it goes into  it four times. Let's keep the decimal up here. So we'll write 0.4. 4 times 25 is 100. And now we  can subtract 110 minus 100 is 10. And now we can bring down another 0. 25 goes into 100  exactly four times. 4 times 25 is 100, and then you subtract, and you get 0. So we actually didn't even have to round this one. This fraction is exactly 0.44. 

Fraction to decimal with rounding 

Let's see if we can express 16/21 as a decimal. Or we could call this 16 twenty-firsts. This is also  16 divided by 21. So we can literally just divide 21 into 16. And because 21 is larger than 16,  we're going to get something less than 1. So let's just literally divide 21 into 16. And we're going to have something less than 1. So let's add some decimal places here. We're going to round to  the nearest thousandths in case our digits keep going on, and on, and on. And let's start  dividing. 21 goes into 1 zero times. 21 goes into 16 zero times. 21 goes into 160-- well, 20 would  go into 160 eight times. So let's try 7. Let's see if 7 is the right thing. So 7 times 1 is 7. 7 times 2 is 14. And then when we subtract it, we should get a remainder less than 21. If we pick the largest  number here where, if I multiply it by 21, I get close to 160 without going over. And so if we  subtract, we do get 13. So that worked. 13 is less than 21. And you could just subtract it. I did it  in my head right there. But you could regroup. You could say this is a 10. And then this would be a 5. 10 minus 7 is 3. 5 minus 4 is 1. 1 minus 1 is 0. Now let's bring down a 0. 21 goes into 130. So  let's see. Would 6 work? It looks like 6 would work. 6 times 21 is 126. So that looks like it works.  So let's put a 6 there. 6 times 1 is 6. 6 times 2 is 120. There's a little bit of an art to this. All right,  now let's subtract. And once again, we can regroup. This would be a 10. We've taken 10 from  essentially this 30. So now this becomes a 2. 10 minus 6 is 4. 2 minus 2 is 0. 1 minus 1 is 0. Now  let's bring down another 0. 21 goes into 40, well, almost two times, but not quite, so only one  time. 1 times 21 is 21. And now let's subtract. This is a 10. This becomes a 3. 10 minus 1 is 9. 3  minus 2 is 1. And we're going have to get this digit. Because we want to round to the nearest  thousandth. So if this is 5 or over, we're going round up. If this is less than 5, we're going to  round down. So let's bring another 0 down here. And 21 goes into 190. Let's see, I think 9 will  work. Let's try 9. 9 times 1 is 9. 9 times 2 is 18. When you subtract, 190 minus 189 is 1. And we  could keep going on, and on, and on. But we already have enough digits to round to the nearest thousandth. This digit right over here is greater than or equal to 5. So we will round up in the  thousandths place. So if we round to the nearest thousandths, we can say that this is 0.76. And  then this is where we're going around up-- 762.