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Estimation of Numbers (Rounding Off Method)

Introduction

Using estimation, you can predict or identify a response close to the correct answer. It helps make decisions quickly and generates a range of possible outcomes close enough to be helpful. Estimation is another method for making numbers easier to work with to estimate when we are only required to have a general idea of how many. An estimate is an educated guess based on knowledge or information already known.

This article will define the estimation of numbers and provide examples of how to round off numbers.

What is the Estimation of Numbers?

We can simplify calculations by estimating a number, which is a reasonable guess. It calls for mental math manipulation.

We engage in estimation without even realizing it. For example, a little kid predicts how many candies he can obtain from his parents, estimates a person’s weight, and a stock market analyst forecasts market trend. Estimation creates an approximate judgment or opinion about size, amount, weight, etc. In other words, estimation is to calculate approximately.

We use the estimation of numbers daily; it is a fundamental component of mathematics. Of course, there are situations when an estimate won’t do; we frequently need to know the precise figure.

You can make two different types of estimating errors.

Overestimate. When the estimated number is more than the actual outcome.

Underestimate. When the estimated number is less than the actual outcome.

Why do we perform the estimation of numbers?

To avoid complex calculations, estimating numbers refers to approximating or rounding off the numbers when the value is used for some other purpose. The words accurate and estimation have different meanings.

In mathematics, the terms “exact” and “estimation” relate to “equal” and “approximate,” respectively. Time can be saved by estimating numbers. We can answer the problem without using a calculator once we have estimated the values. But in mathematics, we always need a precise response. Finding a number sufficiently close to the correct answer is the estimation process. But that is not an accurate response.

Estimation of Numbers (Rounding Off Method)

Before even solving the difficulties, we often estimate the solutions in our thoughts regarding mathematics. Although we prefer exact solutions in mathematics, there are occasions when we must approximate the solutions to represent them. In mathematics, one type of estimation that is frequently utilized is rounding off.

The value of a number is not changed when it is rounded off; instead, it is brought closer to the following number while maintaining its value. It is done for whole numbers and decimals at different places of thousands, hundreds, tens, tenths, etc. The significant figures are preserved when numbers are rounded off. Hence, the number of figures known with some certainty constitutes the number of important figures.

The number is rounded off considering the rounding off digit. The rounding off digit retains if the number that follows to its right is less than 5. Add one to the rounded-off digit if the number is 5 or above.

Consider the number 4.2; since 2 is less than 5, it will be rounded off to 4. In contrast, as 6 is greater than 5, the number will be rounded up to 5 if it is 4.6. As a result, we can say: 4.2 ≈ 4 and 4.6 ≈ 5 

The approximation between the two values is represented by the symbol ≈.

On a big scale, we occasionally also approximate the whole numbers while computing or estimating values. Take 578 as an example; it would become 580, whereas 431 would become 430. Once more, you can see that the approximation is based on the last digit of the whole integer.

What are Rounding Off Numbers?

Rounding off numbers is a method of simplifying numbers to make them easier to understand or work with. When an exact answer isn’t required, and an approximation will do, rounding can be used.

Rounding Off Whole Numbers

The following are the basic steps in rounding off whole numbers.

Step 1: Determine the round off digit.

Step 2: Look at the digit that follows the rounded off digit to the right. Do not change the round-off digit if the number is less than 5. Add one to the rounded off digit if the number is 5 or above.

Step 3: Replace all the digits with zeros to the right of the round-off digit.

Rounding Down
Do not change the rounding off digit if the number immediately to the right of it is less than 5. Then, replace all the digits with zeros to the right of the rounding off digit.

Rounding Up
Add one to the rounding off digit if the number immediately to the right of it is greater or equal to 5. Then, replace all the digits with zeros to the right of the rounding off digit.

For example, let us say that we must round off 4,378 to the nearest hundreds. Following the steps, we have, 

Step 1: Determine the round off digit.

The rounding off digit is the hundreds place. In 4,378, the number 3 is in the hundreds place.

Step 2: Look at the digit that follows the rounded off digit to the right. Do not change the round-off digit if the number is less than 5. Add one to the rounded off digit if the number is 5 or above.

In 4,378, the number to the right of 3 is 7. Since 7 is greater than 5, we add 1 to 3. Thus, the digit in the hundreds place becomes 4.

Step 3: Replace all the digits with zeros to the right of the round-off digit.

The digits 7 and 8 will be replaced with zeros.

Thus, 4378 is 4400 when rounded off to the nearest hundreds. 

Also, on the number line, 4378 is more than halfway from 4300 to 4400.   

Rounding Off to Nearest Tens

Step 1: Identify the digit in the tens place.

Step 2: Look at the digit in the ones place. If the number is less than five, do not change the digit in the tens place. Add one to the tens place if the number is 5 or above.

Step 3: Replace all the digits with zeros to the right of the tens place.

Examples

Use the method of rounding off numbers to solve the following:

( a ) Round off 567 to the nearest tens

( b ) Round off 46, 983 to the nearest tens

( c ) Round off 126, 879 to the nearest tens

Solution

( a ) Round off 567 to the nearest tens

Step 1: Identify the digit in the tens place. 

In 567, The digit in the tens place is 6.

Step 2: Look at the digit in the ones place. If the number is less than five, do not change the digit in the tens place. Add one to the tens place if the number is 5 or above.

The digit to the right of 6 is 7, greater than 5. We must add 1 to the digit in the tens place. Hence, 6 + 1 = 7.

Step 3: Replace all the digits with zeros to the right of the tens place.

Only the number 7 will be replaced with zero. 

Thus, 567 will be 570 when rounded off to the nearest tens.  

( b ) Round off 46, 983 to the nearest tens

Step 1: Identify the digit in the tens place. 

In 46, 983, the digit in the tens place is 8.

Step 2: Look at the digit in the ones place. If the number is less than five, do not change the digit in the tens place. Add one to the tens place if the number is 5 or above.

The tens digit retains since the digit to the right of the tens place is 3, which is less than 5.

Step 3: Replace all the digits with zeros to the right of the tens place.

Only the number 3 will be replaced with zero. 

Thus, 46, 983 will be 46, 980  when rounded off to the nearest tens.  

( c ) Round off 126, 879 to the nearest tens

Step 1: Identify the digit in the tens place. 

In 126, 879, the digit in the tens place is 7.

Step 2: Look at the digit in the ones place. If the number is less than five, do not change the digit in the tens place. Add one to the tens place if the number is 5 or above.

We must add 1 to the digit in the tens place since 9 is greater than 5. Hence, 7 + 1 = 8.

Step 3: Replace all the digits with zeros to the right of the tens place.

Only the number 9 will be replaced with zero. 

Thus, 126, 879 will be 126, 880 when rounded off to the nearest tens.  

Rounding Off to Nearest Hundreds

The following are the basic steps in rounding off to nearest hundreds.

Step 1: Identify the digit in the hundreds place.

Step 2: Look at the digit in the tens place. Do not change the digit in the hundreds place if the number is less than 5. Add one to the hundreds place if the number is 5 or above.

Step 3: Replace all the digits with zeros to the right of the hundreds place.

Examples

Use the method of rounding off numbers to solve the following:

( a ) Round off 125 to the nearest hundreds

( b ) Round off 15,275 to the nearest hundreds 

( c ) Round off 574, 869 to the nearest hundreds

Solution

( a ) Round off 125 to the nearest hundreds

Step 1: Identify the digit in the hundreds place.

In 125, the digit in the hundreds place is 1.

Step 2: Look at the digit in the tens place. Do not change the digit in the hundreds place if the number is less than 5. Add one to the hundreds place if the number is 5 or above.

Since the digit to the right of 1 is 2, which is less than 5, the digit in the hundreds place retains.

Step 3: Replace all the digits with zeros to the right of the hundreds place.

The numbers 2 and 5 will be replaced with zero. 

Thus, 125 will be 100 when rounded off to the nearest hundreds.  

( b ) Round off 15,275 to the nearest hundreds 

Step 1: Identify the digit in the hundreds place.

In 15,275, the digit in the hundreds place is 2.

Step 2: Look at the digit in the tens place. Do not change the digit in the hundreds place if the number is less than 5. Add one to the hundreds place if the number is 5 or above.

We must add 1 to the digit in the hundreds place since the digit to the right of the hundreds place is 7, which is greater than 5; hence, 2 + 1 = 3.

Step 3: Replace all the digits with zeros to the right of the hundreds place.

The numbers 7 and 5 in the tens and ones place will be replaced with zero. 

Thus, 15,275 will be 15,300 when rounded off to the nearest hundreds.  

( c ) Round off 574, 869 to the nearest hundreds

Step 1: Identify the digit in the hundreds place.

In 574, 869, the digit in the hundreds place is 8.

Step 2: Look at the digit in the tens place. Do not change the digit in the hundreds place if the number is less than 5. Add one to the hundreds place if the number is 5 or above.

We must add 1 to the digit in the hundreds place since the digit to the right of the hundreds place is 6, which is greater than 5; hence, 8 + 1 = 9.

Step 3: Replace all the digits with zeros to the right of the hundreds place.

The numbers 6 and 9 in the tens and ones places will be replaced with zero. 

Thus, 574 869 will be 574 900  when rounded off to the nearest hundreds.  

Rounding Off to Nearest Thousands

The following are the basic steps in rounding off to nearest thousands.

Step 1: Identify the digit in the thousands place.

Step 2: Look at the digit in the hundreds place. If the number is less than 5, do not change the digit in the thousands place. Add one to the thousands place if the number is 5 or above.

Step 3: Replace all the digits with zeros to the right of the thousands place.

Examples

Use the method of rounding off numbers to solve the following:

( a ) Round off 2,367 to the nearest thousands

( b ) Round off 45,872 to the nearest thousands

( c ) Round off 768, 578 to the nearest thousands

Solution

( a ) Round off 2,367 to the nearest thousands

Step 1: Identify the digit in the thousands place.

In 2,367, the digit in the thousands place is 2.

Step 2: Look at the digit in the hundreds place. If the number is less than 5 do not change the digit in the thousands place. Add one to the thousands place if the number is 5 or above.

The digit in the thousands place retains since the digit to the right of 2 is 3, which is less than 5.

Step 3: Replace all the digits with zeros to the right of the thousands place.

The numbers 3, 6, and 7 will be replaced with zero. 

Thus, 2 367 will be 2 000 when rounded off to the nearest thousands.  

( b ) Round off 45,872 to the nearest thousands

Step 1: Identify the digit in the thousands place.

In 45,872, the digit in the thousands place is 5.

Step 2: Look at the digit in the hundreds place. If the number is less than 5, do not change the digit in the thousands place. Add one to the thousands place if the number is 5 or above.

We must add 1 to the digit in the thousands place Since the digit to the right of the thousands place is 8, which is greater than 5; hence, 5 + 1 = 6.

Step 3: Replace all the digits with zeros to the right of the thousands place.

The numbers 8, 7, and 2 will be replaced with zero. 

Thus, 45,872 will be 46,000 when rounded off to the nearest thousand.  

( c ) Round off 768, 578 to the nearest thousands

Step 1: Identify the digit in the thousands place.

In 768, 578, the digit in the thousands place is 8.

Step 2: Look at the digit in the hundreds place. If the number is less than 5, Do not change the digit in the thousands place. Add one to the thousands place if the number is 5 or above.

We must add 1 to the digit in the thousands place since the digit to the right of the thousands place is 5. Hence, 8 + 1 = 9.

Step 3: Replace all the digits with zeros to the right of the thousands place.

The numbers 5, 7, and 8 will be replaced with zero. 

Thus, 768,578 will be 769,000 when rounded off to the nearest thousand.  

Rounding Off Decimal Numbers

The following are the basic steps in rounding off decimal numbers.

Step 1: Find the place value of the number you are rounding to (round off digit).

Step 2: Look at the digit that follows the rounded off digit to the right. Do not change the round-off digit if the number is less than 5. Add one to the rounded off digit if the number is 5 or above.

Step 3: Replace all the digits with zeros to the right of the round-off digit. 

Examples

Use the method of rounding off numbers to solve the following:

( a ) Round off 3425.567 to the nearest tenths

( b ) Round off 93.0635 to the nearest hundredths

( c ) Round off 3,787. 89586 to the nearest thousandths

Solution

( a ) Round off 342.567 to the nearest tenths

Step 1: Identify the digit in the tenths place.

In 342.567, the digit in the tenths place is 5.

Step 2: Look at the digit in the hundredths place. Do not change the digit in the tenths place if the number is less than 5. Add one to the tenths place if the number is 5 or above.

We must add 1 to the digit in the tenths place since the digit to the right of the tenths place is 6, which is greater than 5; hence, 5 + 1 = 6.

Step 3: Replace all the digits with zeros to the right of the thousands place.

The numbers 6 and 7 that are in the hundredths and thousandths place will be replaced with zero. 

Thus, 342.567 will be 342.600 or 342.6 when rounded off to the nearest tenths.  

( b ) Round off 93.0635 to the nearest hundredths

Step 1: Identify the digit in the hundredths place.

In 93.0635, the digit in the hundredths place is 6.

Step 2: Look at the digit in the thousandths place. Do not change the digit in the hundredths place if the number is less than 5. Add one to the hundredths place if the number is 5 or above.

The digit in the hundredths place retains since the digit to the right of the 6 is less than 5. 

Step 3: Replace all the digits with zeros to the right of the hundredths place.

The numbers 3 and 5 in the thousandths and ten thousandths place will be replaced with zero. 

Thus, 93.0635 will be 93.0600 or 93.06 when rounded off to the nearest hundredths.  

( c ) Round off 3,787. 89486 to the nearest thousandths

Step 1: Identify the digit in the thousandths place.

In 3,787. 89486, the digit in the thousandths place is 4.

Step 2: Look at the digit in the ten thousandths place. Do not change the digit in the thousandths place if the number is less than 5. Add one to the thousandths place if the number is 5 or above.

8 is the digit in the ten thousandths place and is greater than 5. Hence, 4 + 1 = 5.

Step 3: Replace all the digits with zeros to the right of the thousandths place.

The numbers 8 and 6 in the ten thousandths and hundred thousandths place will be replaced with zero. 

Thus, 3,787. 89486 will be 3,787. 89500 or 3,787. 895 when rounded off to the nearest thousandths.

Summary

Estimation of Numbers

Estimation creates an approximate judgment or opinion about size, amount, weight, etc. In other words, estimation is to calculate approximately.

You can make two different types of estimating errors.

Overestimate. When the estimated number is more than the actual outcome.

Underestimate. When the estimated number is less than the actual outcome.

Rounding Off Numbers

Rounding off numbers is a method of simplifying numbers to make them easier to understand or work with. When an exact answer isn’t required, and an approximation will do, rounding can be used.

The following are the basic steps in rounding off numbers.

Step 1: Determine the round off digit.

Step 2: Look at the digit that follows the rounded off digit to the right. Do not change the round-off digit if the number is less than 5. Add one to the rounded off digit if the number is 5 or above.

Rounding Down

Do not change the rounding off digit if the number immediately to the right of it is less than 5. Then, replace all the digits with zeros to the right of the rounding off digit.

Rounding Up

Add one to the rounding off digit if the number immediately to the right of it is greater or equal to 5. Then, replace all the digits with zeros to the right of the rounding off digit. Step 3: Replace all the digits with zeros to the right of the round-off digit.

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