**Introduction**

Numbers are very commonly used in our day to day life. In general, we need not state the exact number as an answer to the question which requires us to tell about how many. In other words, in order to ease our calculations, we often approximate the numbers to some extent. It helps us give approximate answers where exact values are not required. For instance, we may say that there are around 600 students in a school. That does not necessarily mean that there are exactly 600 students in a school. The exact number of students in the school may be somewhere around 600 which, for convenience, we have rounded off to 600. Rounding off is also termed as a shortcut method. Another example can be if you plan an event at home you can just estimate the number of guests that are supposed to attend the event. It can be a rough estimate of the number of people say 200. This does not mean that exactly 200 people will attend the event. This means we can round off numbers to the nearest place value, say 10, 100, 1000 etc.

**What does rounding off mean?**

Rounding off is a type of estimation where a number is made simpler by keeping its value intact but closer to the next number. It can be done for whole numbers as well as for decimals. There are two types of rounding off –

**Rounding Up**– Rounding up means the round off the given number to the next whole number.**Rounding Down**– Rounding down means the round off the given number to the previous whole number.

**How do we round off a number to the nearest 10?**

Consider two numbers 22 and 29. If we plot these numbers on a number line we find that both 22 and 29 lie between 20 and 30. Also, 22 is nearer to 20 while 29 is nearer to 30. So, we round off 22 to 20. Correct to the nearest tens. If we consider 29, we round off it to the nearest 10 which is 30. Hence, we round off 29 to the nearest 30.

**So, we can say that rounding off to the nearest 10 means that we wish to round off the last two digits of a number, i.e. up to the tens place of a number.**

From the above discussion, we can see that there are two ways of rounding off a number to the nearest 10. The two methods are –

- Rounding Up
- Rounding Down

Let us understand both methods one by one.

**Rounding Up**

**If the number at the tens place is followed by 5, 6, 7, 8, or 9, the number is rounded up. **

**Example – We want to round off 5489 to the nearest 10.**

- Look for the digit at the ten’s place. It is 8.

- Now look for the digit to the right of the digit at the ten’s place. The number is 9

3. Since this number is greater than 5, we round up the number, i.e. we round it off to the nearest 10 that comes next to this number. So, 9 becomes 10 which by adding to 8 at the ten’s place, becomes 90.

Hence, the number 5489 after being rounded off to the nearest 10 is **5490**

**Note: Here we moved up the number to the nearest ten, which is why it is called rounding up.**

Now that we have understood what we mean by rounding up, let us understand what is meant by rounding down.

**Rounding Down**

**If the number at the hundred’s place is followed by 0, 1, 2, 3 or 4, the number is rounded up. **

**Example – We want to round off 2461 to the nearest 10.**

- Look for the digit at the ten’s place. It is
**6**.

2. Now look for the digit to the right of the digit at the ten’s place. The number is **1.**

3. Since this number is less than 5, we round down the number, i.e. we round it off to the nearest 10 that comes before this number. So, 1 becomes 0 which by adding to 6 at the ten’s place, becomes 60.

Hence, the number 2461 after being rounded off to the nearest 10 is **2460.**

**Note: Here we moved down the number to the nearest ten, which is why it is called rounding down.**

**How do we round off a number to the nearest 100?**

We shall again use the two methods – Rounding up and rounding down to understand how to round off a number to the nearest 100.

**Rounding Up**

**If the number at the hundred’s place is followed by 5, 6, 7, 8, or 9, the number is rounded up. **

**Example – We want to round off 54897 to the nearest 100.**

- Look for the digit at the hundred’s place. It is 8.

- Now look for the digit to the right of the digit at the hundred’s place. The number is 9

3. Since this number is greater than 5, we round up the number, i.e. we round it off to the nearest 100 that comes next to this number. So, 97 becomes 100 which on adding to 8 at the hundred’s places, becomes 900.

Hence, the number 54897 after being rounded off to the nearest 100 is **54900**

**Note: Here we moved up the number to the nearest hundred, which is why it is called rounding up.**

**Rounding Down**

**If the number at the hundred’s place is followed by 0, 1, 2, 3 or 4, the number is rounded up. **

**Example – We want to round off 24615 to the nearest 100.**

- Look for the digit at the hundred’s place. It is
**6**.

2. Now look for the digit to the right of the digit at the hundred’s place. The number is **1.**

3. Since this number is less than 5, we round down the number, i.e. we round it off to the nearest 100 that comes next to this number. So, 15 becomes 00 which on adding to 6 at the hundred’s places, becomes 600.

Hence, the number 24615 after being rounded off to the nearest 100 is **24600.**

**Note : Here we moved down the number to the nearest hundred, which is why it is called rounding down.**

**How do we round off a number to the nearest 1000?**

We shall again use the two methods – Rounding up and rounding down to understand how to round off a number to the nearest 1000.

**Rounding Up**

**If the number at the thousand’s place is followed by 5, 6, 7, 8, or 9, the number is rounded up. **

**Example – We want to round off 613629 to the nearest 1000.**

- Look for the digit at the thousand’s place. It is
**3**.

- Now look for the digit to the right of the digit at the thousand’s place i.e. the hundred’s place of the number. The number is
**6**

3. Since this number is greater than 5, we round up the number, i.e. we round it off to the nearest 1000 that comes next to this number. So, 629 becomes 1000 which on adding to 3 at the thousand’s place, becomes 4000.

Hence, the number 613629 after being rounded off to the nearest 100 is **614000**

**Note : Here we moved up the number to the nearest thousand, which is why it is called rounding up.**

**Rounding Down**

**If the number at the hundred’s place is followed by 0, 1, 2, 3 or 4, the number is rounded up. **

**Example – We want to round off 104328 to the nearest 1000.**

- Look for the digit at the thousand’s place. It is
**4**.

2. Now look for the digit to the right of the digit at the thousand’s place. The number is **3.**

3. Since this number is less than 5, we round down the number, i.e. we round it off to the nearest 1000 that comes next to this number. So, 328 becomes 000 which on adding to 4 at the thousand’s place, becomes 4000.

Hence, the number 24615 after being rounded off to the nearest 100 is **104000.**

**Note : Here we moved down the number to the nearest hundred, which is why it is called rounding down.**

**Rounding off Decimal Numbers**

Let us recall what we mean by a decimal. A decimal is another way of expressing a fraction. For example the fraction 110 in decimal form is written as 0.1. This dot between the two digits is called the decimal point. So, the decimal number 1.8 will be read as one point eight. Now, just like the place values of whole numbers such as ones, tens and hundreds, the decimal numbers also have their own place value system. This is extremely important to understand as unless we are not aware of the place of a digit in a decimal, we cannot round it off.

**Round Off to the Nearest Tenth**

Rounding off to the nearest tenth is also known as rounding off to one decimal place. So, how do we round off a number to the nearest tenth? Let us understand this by an example.

Suppose a tree is 6.84 m tall. Let us round off its height to the nearest tenth. To round off this number to the nearest tenth, let us draw a number line and show the decimals as below

In the above number line, we can see that 6.84 is closer to 6.80 than to 6.90. Hence, 6.84 will be rounded off the nearest decimal place 6.8.

**Round Off to the Nearest Hundredth**

Rounding off to the nearest hundredth is also known as rounding off to two decimal places. This is the most common form of rounding off that we use in solving everyday problems, especially when related to money. So, how do we round off a number to the nearest hundredth?

The following are the rules that must be followed for rounding off a number to the nearest hundredth –

- Identify the place you want to the round to. ( In case we have the hundredths place )
- Examine the digit to the right of the rounding place digit.
- If that digit is equal to 5 or greater than 5. Add 1 to the rounding place digit. If the digit is less than 5, let the rounding place digit remain as such.
- In both the cases, drop all the digits to the right of the rounding place digit.

**Round Off to the Nearest Thousandth **

Rounding off to the nearest thousandth is also known as rounding off to three decimal places. So, how do we round off a number to the nearest thousandth? Let us understand this by an example.

The following are the rules that must be followed for rounding off a number to the nearest thousandth –

- Identify the place you want to the round to. ( In case we have the thousandth place )
- Examine the digit to the right of the rounding place digit.
- If that digit is equal to 5 or greater than 5. Add 1 to the rounding place digit. If the digit is less than 5, let the rounding place digit remain as such.
- In both the cases, drop all the digits to the right of the rounding place digit.

**Graphical representation of Rounding Off Numbers**

Let us consider some examples to understand the graphical representation of rounding off numbers.

**Example – Rounding off a number to the nearest 100**

Let us take a number 74

This number lies between 70 and 80. To round off this number, either we take it back to 70 or move it forward to 80. **How do we decide where to go?**

**Look at the number line below – **

The numbers that lie in the **red areas** will be rounded down to 70. Similarly, the numbers that lie in the blue areas will be rounded up to 80. This is because the numbers that lie in red areas, i.e. the numbers between 71 and 74 are closer to 70 as compared to 80. Hence they will be rounded down.

However, the numbers that lie between 75 and 79 are closer to 80 due to which they would be rounded up to 80.

Let us take another example.

**Example – Rounding off a number to the nearest 100**

Let us take a number 748

This number lies between 700 and 800. To round off this number, either we take it back to 700 or move it forward to 800. **How do we decide where to go?**

**Look at the number line below – **

The numbers that lie in the **red areas** will be rounded down to 700. Similarly, the numbers that lie in the blue areas will be rounded up to 800. This is because the numbers that lie in red areas, i.e. the numbers between 701 and 749 are closer to 700 as compared to 800. Hence they will be rounded down.

However, the numbers that lie between 750 and 799 are closer to 800 due to which they would be rounded up to 800.

Let us take another example

**Example – Rounding off a number to the nearest 1000**

Let us take a number 7348

This number lies between 7000 and 8000. To round off this number, either we take it back to 700 or move it forward to 800. **How do we decide where to go?**

**Look at the number line below – **

The numbers that lie in the **red areas** will be rounded down to 7000. Similarly, the numbers that lie in the blue areas will be rounded up to 8000. This is because the numbers that lie in red areas, i.e. the numbers between 7001 and 7499 are closer to 7000 as compared to 8000. Hence they will be rounded down.

However, the numbers that lie between 7500 and 7999 are closer to 8000 due to which they would be rounded up to 8000.

**Solved Examples**

**Example 1** Round off the number 6,852,718 to nearest 10, 100, 1000 and million

**Solution** 6,852,783 rounded to the nearest ten is 6,852,780

6,852,783. rounded to the nearest hundred is 6,852,800

6,852,783 rounded to the nearest thousand is 6,853,000

6,852,783 rounded to the nearest million is 7,000,000

**Example 2** Round off the whole number 23 to the nearest 10

**Solution** We have been given the whole number 23 and we need to round it off to the nearest 10. Let us see how to do it based on what we have learnt above.

Examine the number 23. Look for the digit at the ten’s place. The number is 2. Now look for the digit to the right of the digit at the ten’s place. The number is **3.**

Since this number is less than 5, we round down the number, i.e. we round it off to the nearest 10 that comes before this number. So, 3 becomes 0 which by adding to 2 at the ten’s place, becomes 20.

Hence, the number 23 after being rounded off to the nearest 10 is **20.**

**Key Facts and Summary**

- Rounding off is a type of estimation where a number is made simpler by keeping its value intact but closer to the next number.
- Rounding off to the nearest tenth is also known as rounding off to one decimal place.
- Rounding off to the nearest hundredth is also known as rounding off to two decimal places.
- Rounding off to the nearest thousandth is also known as rounding off to three decimal places.
- A decimal is another way of expressing a fraction.
- Rounding off to the nearest tenth is also known as rounding off to one decimal place.
- If the number at the tenth place is followed by 5, 6, 7, 8, or 9, the number is rounded up.
- If the number at the hundredth’s place is followed by 0, 1, 2, 3 or 4, the number is rounded up.
- Rounding off to the nearest tenth is also known as rounding off to two decimal places.
- Rounding off to the nearest thousandth is also known as rounding off to three decimal places.

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