The height of a stack of paper is 3.025 centimeters. Given that each sheet of paper is 0.605 millimeters thick, how many sheets of paper are in the stack?
This question is all about trying to work out how many sheets of paper there are in a stack, using only two pieces of information to help us. Firstly, we know the height of the whole stack of paper. And that’s 3.025 centimeters. And we’re also told the thickness of each individual sheet of paper, 0.605 millimeters. To find out the number of sheets of paper that fit into a stack that’s 3.025 centimeters tall, we need to use division. But how would we write the division that we need to work out?
You might think that we need to divide 3.025 by 0.605. In other words, take the two numbers straight out of the problem. But it’s not quite as simple as that. The height of the stack of paper is given to us in centimeters. But the thickness of each sheet of paper is given to us in millimeters. To work out this division correctly, we’re going to have to convert one of the numbers so that they’re both in the same unit of measurement. Let’s convert our first measurement. That’s the height of the stack of paper and let’s change it into millimeters.
We know there are 10 millimeters for every one centimeter. And so to change a measurement in centimeters into millimeters, we need to multiply it by 10. When any numbers are multiplied by 10, the digits move one place to the left. And so watch what happens when we multiply 3.025 by 10. The decimal place will stay where it is. And see what happens to the digits. They’ve shifted one place to the left. Now both measurements are in millimeters. We can think about dividing. We need to find the number of 0.605s that there are in 30.25.
This seems like quite a tricky division to have to work out. Both numbers are decimals. Is there a way that we could change both numbers and turn them into integers or whole numbers? We’ve already said that the digits in a number move one place to the left when they’re multiplied by 10. What if we multiply 0.605 by 10 three times, or in other words, multiply it by 1000? Then, the digits will move three places to the left. One, that’s the same as multiplying by 10. Two, that’s the same as multiplying by 100. Three, that’s the same now as multiplying by 1000. We’ve turned our decimal into a whole number. So we can change our divisor to 605. But we now need to make sure that the answer to our division stays the same.
So because we’ve multiplied the divisor by 1000, we’re going to have to do the same to 30.25. So we start with 30.25. Multiplying by 10 causes the digits to shift one place to the left. Now we’ve multiplied by 100 which gives us a whole number. But we need to carry on because we need to multiply by 1000 to keep the division the same, so one more shift to the left, 30250. By multiplying both numbers by 1000, we’ve managed to turn them both into whole numbers without changing the answer to the division. Now we can get on to find the answer.
We can’t work out how many 605s there are in three or 30 or even 302. So we need to consider the first four digits of our number. How many 605s are there in 3025? Again, this might seem tricky. We don’t know our 605 times table. But as usual, there’s often a method that we can use or a strategy to help. In this case, we can use estimation to help. 3025 is very close to 3000. So let’s round it down to 3000. And 605 is also very close to 600. So we can round both numbers down. Can you see a factor in there that we can use to help us? We know that 30 divided by six equals five. This means 300 divided by 60 is also five. And this means we know that 3000 divided by 600 is five too. So our estimate for 3025 divided by 605 is five.
Let’s multiply 605 by five to see how close we get to 3025. Again, we can make things easier for ourselves here. We know that 605 multiplied by 10 is 6050. So 605 multiplied by five, which is half of 10, will give us an answer that’s half of 6050. Half of 6000 is 3000. Half of 50 is 25, 3025. The number of 605s in 3025 is five exactly. What we do at this point normally with long division is to subtract to show there’s no remainder. 3025 take away 3025 is zero. And we’d also bring down the last digit to divide, but that’s a zero to. So the number of 605s in zero is zero.
This has been quite a tricky question to answer and there are a lot of steps involved. We had to convert our measurements so that they were same units. And finally, we had to use some facts we knew already to help us divide the numbers that we created. Even though a question could look tricky, there’s often a strategy or a fact you know already that you can use to help.
The number of sheets of paper that there are in the whole stack is 50.