### Estimation in Addition and Subtraction

Numbers to be added or subtracted are rounded to specified or appropriate place values, and the operation is performed; your result is an estimate. It may be greater or smaller than the actual result, or it may be the same. The purpose of rounding in operations is to ignore the small parts, obtain simpler numbers that are easier to work with, and find the result more quickly. The goal of estimating by rounding is not to deal with small parts but to find a quick general result.

Prerequisite: Rounding in Natural Numbers

## 85+18=

Let's estimate the operation by rounding to the nearest tens.

We can round the number 85 to 90, and 18 to 20. In this case, our number will result in a value close to 110.

Example:

# 381 - 126

Let's perform the operation by rounding to the nearest tens.

## 380 - 130 = 250

The actual result should be a number close to 250.

What is meant by 'close'?

Close to 250' doesn't necessarily mean that the result must be in the 250s, such as 251, 252, 253, etc. It could be in the 240s, 250s, 260s, etc.

Example:

# 381+1256

381 is a number close to 400, so we can consider it as 400. 1256 is a number close to 1300, so in that case;

400+1300=1700

Our result will be a number close to 1700.

Whether to round to the nearest ten, hundred, or thousand depends on the situation of your numbers. Rounding a number in the millions to the nearest million, a number in the thousands to the nearest thousand or hundred, or a number in the hundreds to the nearest hundred or ten is more purposeful.

For example, rounding a number in the millions or hundreds of thousands to the nearest ten when adding is absurd. A ten is a very small number next to a million... We can even ignore the hundreds and thousands, let alone the tens. There is no point in rounding a number in the millions to the nearest ten for addition.

Our goal is to perform quick, simple addition or subtraction, but we must do so without straying too far from the actual result.

Example;

We want to find the sum of 13,176,348, '13 million 476 thousand 348,' and 25,820,749, '25 million 820 thousand 749,' approximately in our heads. Since our goal is to perform this addition quickly and possibly mentally, it makes much more sense to round to the millions.

The number 13,176,348 is much closer to 13 million, so let's consider it as 13 million.

The number 25,820,749 is very close to 26 million, so let's consider it as 26 million.

13 million + 26 million = 39 million... our result should be a number close to 39 million.

## Example;

Is it more logical to perform the operation 951 - 336 by rounding to the nearest ten or hundred? Comment on this from all aspects.

Is it more logical to perform the operation 2023 + 1453 by rounding to the nearest ten, hundred, or thousand? Comment on this from all aspects.

# Limits in Estimated Operations

When performing estimated operations, knowing the range within which the actual result should fall provides us clues in interpreting the real result.

## 385 + 866

Let's round 385 to 400,
And let's round 866 to 900,

400 + 900 = 1300

I performed the operation by enlarging both numbers, so our actual numbers are actually smaller, therefore the real result cannot be greater than 1300.

Let's round 385 to 300,
And let's round 866 to 800,
300 + 800 = 1100

I performed the operation by reducing both numbers, so our actual numbers are larger, therefore the real result must be greater than 1100.

The real result should be something between 1100 and 1300.
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