Estimation is a way of rounding numbers off to some place value for some reason. For example, when a reporter wants to report the number of spectators in a stadium, he does not have to count every single person all the way up to the last person and then say for example, “There are 53, 476 people in the stadium right now.” Nobody would required that number with that much accuracy. The reporter can simply round the number of spectators to the nearest 10,000’s and simply say that there are about 50,000 people in the stadium today. Estimation is used in those kinds of situations.
There are some rules for rounding numbers but it’s useful to know why those rules make sense first. For example, when you want to round 13 off to the nearest tens, you’d simply round it off to 10. On the other hand, when you want to round 17 off to the nearest 10’s, you’d simply round it off to 20. The rule that we use here is that since you’re rounding off to the nearest 10’s, you have to take a look at the place value to the right of the 10’s place value, 1’s, since that it that place value is 10 times smaller than the 10’s place value. If the digit in the 1’s place value is greater than 5, then you round the digit in the 10’s place value up to 2 and use a 0 for the 1’s place value. On the other hand, if the digit in the 1’s place value is less than 5, then you don’t have to make any changes to the 10’s place. The only thing that you’d have to do is use a 0 for the 1’s place value. So in short, you either round up to 20 or round down to 10. There are other rules if the 1’s place value’s digit is exactly 5 but we’ll talk about that shortly.
Now, these rules would make sense to you if you looked at the number on a number line. I cannot show a number line here but imagine a number line going in both directions but on the right you have the number 20 and on the left hand side of 20, you have the number 10. The distance between 10 and 20 has been divided into 10 equal distances and so you have 9 numbers between them, 11 through 19. The number that comes exactly between 10 and 20 is 15. 13 is much closer to 10 than to 20, so when you’re rounding off to the nearest 10’s, you round down to 10. 17, on the other hand, is much closer to 20, so when you’re rounding off to the nearest 10’s, you round up to 20. Please draw a number line and see this for yourself and then the rules would not be just rules. They’ll make logical sense to you.
Now, let’s say that you have the number 15 and want to round to the nearest 10’s. The place value before or smaller than 10’s in 15 is the 1’s place value and the digit there is 5. It’s common practice to round up to 20. On a number line, 15 would be equidistant from 10 and 20. It’s neither closer to 10 nor to 20. So that way you cannot decide. Mathematicians have used statistics and have found that, in such cases, you should round up.
Rounding numbers has many applications. Even without knowing it, you’re already doing it in you day-to-day life.
Applications of Rounding Numbers
Rounding Numbers Off to Estimate Addition or Subtraction
One application of rounding numbers is in estimating the outcome of adding and subtracting numbers. For example, let’s say, you were to have to two parties simultaneously at two different locations and you were to take care of food and drinks for the two parties. For that, you need to have an estimation of the number of people attending both the parties together. In other words, you would have to add the number the attendees. But since there’s usually no way to know who shows up and who not, you always have to go with the odds. So let’s say that you have told 38 people about party A. About party B, you have talked to 126 people. To add the number together, you could do two things. You could round both the number to the nearest 10’s and add, in which case, you would get 130 + 40 = 170. 170 is an overestimate of the actual number of attendees. You’d gotten 164 if you had added the number together without rounding them off. You might still have to return some food or drinks though.
Another way of adding the two numbers is by rounding the two number to the nearest 100’s and then add them together. 126 rounded to the nearest 100’s would be 100 and 38 rounded to the nearest 100’s would be 0. And 100 + 0 = 100. This clearly would not be a good way of rounding the numbers since you really don’t want to lose that much accuracy. Many people would have no food or drinks if you did so. You could still round to the nearest 100’s in a different situation where you’d invited say exactly 865 people to a wedding party. In this situation, rounding to the nearest 100’s would be fine.
So, whether you round your numbers to the 10’s or 100’s or whatever place value depends on the nature of your number, where it’s come from (how you have gotten the number) and how much accuracy you need.
Rounding Numbers Off to Estimate Multiplication
Rounding numbers off is also very helpful in estimating the result of multiplication. For example let’s say that you were multiplying 23786 by 5487. Once you have done the calculation, assuming that you’re doing that by hand, you should have idea what the result should be close to. If you do this calculation, you’d get exactly 130,513,782. So this number, rounded off to the nearest millions would be 131,000,000. Rounded off to the nearest thousands 130,514,000 and rounded off to the nearest 100’s, it’d be 130,500,000. So if you somehow could know that you should get some number close to 130,500,000, then you could check the result of your calculation. Meaning that if you got some number like 250,000,000, you definitely know that there was something wrong with your calculation. So how could you check your calculation quickly?
The numbers that you’re multiplying are, 23,786 and 5,487. There are some ways to do the same thing:
- If I round both of these numbers to the nearest 1000’s, I’d get, 24,000 and 5000 and 24,000 * 5,000 = 120,000,000 which is a good estimation compared to the actual value which was 130,500,000.
- To save a little time, I could also, round 23,786 to the nearest 10,000’s where I’d get 20,000. Then I could round 5,487 to the nearest 1000’s where I’d get 5000. Now I could multiply 20,000 * 5,000 = 100,000,000. Multiplying these two numbers takes less time than doing the multiplication in the previous step but the estimate is not as close to actual value compared to the previous case.
What this means is that as you’re rounding your number off, the more you move towards the largest place values in your numbers, the more information and accuracy you’re losing in your number and when you use those rounded numbers in calculation, the result will be farther away from the actual result. Meaning that the more you move towards the larger place values in your number while rounding, your estimate is going to be farther away from the actual value of multiplication. To understand this point compare the following examples. For the sake of simplicity, I’ll multiply two numbers together but I’ll take one of them already a simple number so that I’ll round only one number in each step.
Let’s say that you want to estimate 5,000 * 23,469. I’m not going to round 5,000. In each step, I’ll round 23,469 to different place values:
- 5,000 * 23,469 = 117,345,000 (actual value of the multiplication)
- 5,000 * 23,470 = 117,350,000 (second number rounded to the nearest 10’s)
- 5,000 * 23,500 = 117,500,000 (second number rounded to the nearest 100’s)
- 5,000 * 23,000 = 115,000,000 (second number rounded to the nearest 1000’s)
- 5,000 * 20,000 = 100,000,000 (second number rounded to the nearest 10,000’s)
So you see that the more I move towards the greater place values in the second number while rounding off, the more information I’m losing related to the number and the result of the calculation is getting farther and farther away from the actual value of the multiplication.
So you want to find a good balance between saving time and result accuracy. Based on the accuracy that you need in each case, you can round your number off accordingly but the important thing to know is that the less your number is rounded off (rounded off to smaller place values), a better estimate you’re going to obtain.
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