200.9 divided by 97 and a half is approximately equal to what to the nearest thousandth.
In this problem, we’re being asked to divide a decimal 200.9 by a mixed number 97 and a half. The first thing that we can do to make these numbers easier to work with is to look at the value 97 and a half. How could we write this as a decimal? Well, when a number is worth something and a half, we write the half as 0.5. So 97 and a half can be written as 97 and five tenths or 0.5. And so we can write our division as 200.9 divided by 97.5. Here we’re dividing two decimals. Wouldn’t it be much easier if we were dividing whole numbers? We’d much rather that. Well, we can turn these decimals into whole numbers. Remember that when we multiply a number by 10, the digits shift one place to the left. And so 200.9 becomes 2009 and 97.5 becomes 975.
Now, because we’ve multiplied both numbers by the same amount, the answer to the division will stay the same. So we don’t need to adjust our answer at the end. We’re still going to get the same answer. Now, all we need to do is to divide 2009 by 975. And we can use long division to do this. Before we start, look carefully at the number sentence in the question. There’s what looks like an equal sign, but it’s made up of wobbly lines. This symbol means “is approximately equal to.” And the reason why we use this number in our question and not an equal sign is that we’re being asked to find the answer to the nearest thousandth.
In other words, we’re not being asked to find the exact answer, but we need to round it to the nearest thousandth. And remember, when we’re rounding a number up or down, we need to see the digit in the place that we’re rounding to, in this case, the thousandth digit. But we also need to see the digit in the place to the right because this will tell us whether we need to round up or down. To the right of the thousandth place is the ten thousandth place. But we can see that our number that we’re dividing 2009 doesn’t have any decimal places. So it might make sense before we start to put them in. They’re going to be zero cause it’s a whole number, so 2009 and zero tenths, zero hundredths, zero thousandths, and zero ten thousandths.
We’ll see how these decimal places become useful later on. Now, the first thing we can say about the number 975 is that it’s very close to 1000. So we don’t really need to think too much about it to be able to say that there are two lots of 975 in 2009. But what are two lots of 975 worth? Five twos are 10. Seven twos are 14 — we’ve got one we’ve exchanged that makes 15. Nine twos are 18, and the one that we’ve exchanged takes us to 19. So as we estimated, there are two lots of 975 in 2009. Let’s write two at the top and we can see that there’s going to be a remainder. So we’ll start by writing down the value of two lots of 975 which we’ve said was 1950 and we’ll subtract this from 2009. Nine take away zero equals nine.
Now, we can’t take five away from zero. So we’re going to need to exchange. We can’t exchange from the hundreds place. So we’re gonna have to go all the way to the thousands. We’ve now got one thousand exchange that for ten hundreds. We can now take away one of those hundreds and exchange it for 10 tens. 10 take away five equals five. And if we look carefully at the other two columns they’re both going to equal zero, nine take away nine and one take away one. So we have a remainder of 59. Now, there aren’t any 975s in 59. So we need to bring down another digit to help us. How many 975s are there in 590? Well, the answer is zero. And we need to recall this at the top. But remember that our answer too so far is in the ones place. So we need to put a decimal place in and put our zero in the tenth place.
Now, we need to continue to try to divide. We’re gonna have to bring down another digit to make our number larger. You can see now why we included all those zeros at the start. Now, we can ask ourselves, how many 975s are there in 5900? Well, we’ve said already that 975 is around about 1000. Because we’re dividing 5900. It looks like we might have five lots of 975, possibly even six. Let’s work out what five lots of 975 is to begin with. Five multiplied by five equals 25. If five fives are 25, then we know seven fives are another two lots of five or 10 more than 25. The answer is 35. We’ve exchanged two. So the answer is now 37.
What are nine fives? We know 10 fives are 50, so nine fives are five less than 50, which is 45. We’ve exchanged three, so the answer is 48. We can see that there are definitely five lots of 975 in 5900 But are there six lots of 975? Let’s add one more lot of 975 to see how close we can get. Five plus five equals 10. Seven plus seven equals 14 plus the one we’ve exchanged equals 15. Eight plus nine equals 17 plus the one we exchanged equals 18. And four plus the one we’ve exchanged equals five. Six lots of 975 are 5850, and they fit into 5900. So let’s write six in the correct column at the top. This is now the hundredths column. And we can see once again there’s going to be a remainder. So we’ll write down the value of six lots of 975 which we said was 5850. And we’ll subtract this from 5900.
Now, we don’t really need to use column subtraction to do this. We can just count up from 5850 and we can see that the difference between the two numbers is 50. We still can’t round to our nearest thousandth yet, so we need to keep dividing. We can’t divide 50 by 975, so we need to bring down another digit. Remember, these are all zeros we’re bringing down because it’s a whole number. Now, there are zero lots of 975 in 500. So we’ll write zero at the top. This is our thousandth digit. But we can’t end the calculation there because we need to know whether to keep the digit as zero or whether we round it up to the next thousandth. That’s why we need to bring down one more zero and divide one more time.
How many lots of 975 are there in 5000? Well, we know the answer to this already. If we look across at our calculations, we can see that we worked out that five lots of 975 was 4875 which is very close to 5000. So there are five lots of 975 in 5000. And we don’t need to do anything else with our division. We just need to look at our answer. At the moment, it’s 2.0605. But remember, we’re supposed to round our number to the nearest thousandth. So it’s only supposed to have three decimal places.
At the moment, we have a zero in the thousandth place. But the digit to the right of this is a five. And remember, when a digit is five, six, seven, eight, or nine, we always round up. And so instead of zero thousandths, we can round up to one thousandth. Although we converted the calculation to dealing with whole numbers, the answer to the division is exactly the same as in the question. 200.9 divided by 97 and a half is approximately equal to 2.061 to the nearest thousandth.