## Introduction

When dealing with big numbers with many digits, we must remember to round off the numbers before performing any operations. This brings about a great difference when it comes to estimation. When we round off the numbers before a function versus after, we can see that the approximated estimate is very different from the actual sum. But this does not mean your answer is incorrect.

## What is Estimation?

Estimation is defined as a roughly calculated answer that is different from the actual answer but still within the range of doubt. An estimation or an approximate estimation is also called an “educated guess” or an "approximation."

## How To Round Off Numbers

To estimate correctly, one must follow the following rules for rounding off.

Step 1: Identify the place values of the number.

Taking, for example, this number, $190456$

Here, the number 6 is in the units place, 5 is in the tens place, 4 is in the hundreds place, 0 is in the thousands place, 9 is in the ten thousand places, and 1 is in the lakhs place.

Step 2: Circle the number that needs to be rounded off or underline the number that needs to be rounded off.

Continuing the same example,

If you must round off at the tens place

$1904\underline{\underline{5}}6$

If you have to round off at the ten thousand places, we have to circle the bigger number, which is 9

$1\underline{\underline{9}}0456$

Step 3: Check the number on the immediate right of the circled or underlined number.

If that number is $\ge 5$add 1 to the circled number and follow it with zeros till the unit's place.

If that number is 5, do not change it and follow it with zeros until the units are placed.

Continuing the same example,

$1904\underline{\underline{5}}6=190460$ (Rounding up)

$1\underline{\underline{9}}0456=190000$(Rounding down)

## Performing Operations With Estimation

When asked to estimate, one must round off the starting materials to perform any operation, such as addition, subtraction, multiplication, or division.

For example, the question is, divide 234567 by 231 by rounding off the given numbers to their greatest place values.

Here the greatest place value for the number 234567 is $\underline{\underline{2}}34567$

Since$3<5$, the number 2 will remain unchanged, and our estimation will be 200000

The greatest place value for 231 is $\underline{\underline{2}}31$

Since $3<5$, our estimation will be 200

Now, by dividing our estimations to get an approximately estimated answer.

$200\overset{1000}{\overline{\left){200000}\right.}}$

Our approximate estimation is 1000

Whereas, when we perform the actual division, we get the following

$231\overset{1015.44}{\overline{\left){234567}\right.}}$

The actual answer after division is 1015.44

## Solved Problems of Estimation and Rounding Off Numbers

Q1. Divide 164550 by 4500 by rounding off to the nearest thousand.

Solution: Here, the nearest thousands place can differ for both the dividend and divisor. 164550 will be rounded off to the lakhs place, also called the hundred thousand place, whereas 4500 will be rounded off to the thousands place.

Therefore, the two rounded-off values for division are:

$\underline{\underline{1}}64550=200000$($\because 6\ge 5$)

$\underline{\underline{4}}500=5000(\because 5\ge 5)$

Making the estimated division look like the following

$5000\overset{40}{\overline{\left){200000}\right.}}$

Hence the estimated number is 40.

Let’s check the actual product,

$4500\overset{36.56}{\overline{\left){164550}\right.}}$

Actual answer = 36.56, which is rounded off to 40 (nearest 10)

Q2. Add 367840 and 456402 rounded off to the thousands place.

Solution:

Here, the question does not say nearest thousand; therefore, we only need to round off to the thousand places.

$ 36\underline{\underline{7}}840=368000(\because 8\ge 5) $

$ 45\underline{\underline{6}}402=456000(\because 4<5) $

In addition, the estimated sum is

$368000+456000=824000$

Answer= 824000

Q3. Estimate the answer when 34 is multiplied by 67.

Solution:

When we round off $\underline{\underline{3}}4$ and $\underline{\underline{6}}7$ we get 30 and 70

$\therefore 30\times 70=2100$

Answer=2100

Q4. If you have 65489 m of aluminium wire and 45783 m of brass wire, how much do you have when joining the two? Estimate to the nearest thousand.

Solution:

The operation is to be performed here in addition.

On rounding off the two lengths to the nearest thousand, we get 70000 and 50000

$\therefore 70000+50000=12000m$

Answer: You have approximately 12000m of the mixed wire.

## FAQs on Estimation and Rounding Off Numbers

1. Can the estimated answer of an operation be the same if it is rounded off after operating?

Yes, sometimes the answer may be the same when rounded off later. But, it is important to remember that you must round off first to get an accurate answer all the time.

2. Can we estimate in fractions?

To estimate fractions, you must first convert the fraction into decimals or percentages and then round off to the nearest required value or the highest place value. The answer can be changed back into fractions or left in the converted form.

3. Can the estimated answer and the actual answer be the same?

The estimated number and actual answers can be the same on very few rare occasions. Make sure to recheck both calculations to ensure the answers are the same.

For Example: Add 4050 and 450 and round off to the highest place value.

Estimated Answer is $\underline{\underline{4}}050+\underline{\underline{4}}50=4000+500=4500$

The actual answer is $4050+450=4500$