Forming Numbers, Rounding off Numbers

# Forming Numbers, Rounding off Numbers

## Forming Numbers

To form a 2-digit number, we use 2 digits; to form a 3-digit number, we use 3 digits and to form a 4-digit number, we use 4 digits. For example, 17, 276 and 6825, etc.
The smallest 4-digit number is formed by adding 1 to the largest 3-digit number.
999 + 1 = 1000
The largest 4-digit number is 9999.
The smallest 5-digit number is formed by adding 1 to the largest 4-digit number.
9999 + 1 = 10000

## Forming Largest Number

Let us form the largest 5-digit number using 5, 4, 2, 9 and 7.
To form the largest 5-digit number, we put the largest digit in the ten thousands column, then the next largest digit in thousands column and so on. We put the digits in descending order to get 97542 which is the largest 5-digit number with the given digits.

TTh  Th    H    T   O
9      7      5     4    2

## Forming Smallest Number

Let us form the smallest 5-digit number using 5, 4, 2, 9 and 7.
To form the smallest number, we put the digits in the ascending order to get the number 24579. This is the smallest 5-digit number with the given digits.

TTh  Th    H    T    O
2        4      5     7     9

If the digits have zero, then to form the smallest number, we first put the smallest digit except 0 and then we put zero on the second position and all other digits in ascending order. For example, if the digits are 3, 0, 6, 4, 1, then the smallest number is 10346
Similarly, 6-digit, 7-digit, etc. numbers can be formed.

Example 1: Form the largest and smallest 6-digit numbers with the digits 7, 0, 6, 4, 8 and 9.

Solution: The largest number is 987640 (descending order of digits)
The smallest number is 406789 (ascending order of digits except 0)

Example 2: Form the largest and smallest 7-digit numbers using the digits 8, 3, 5, 0, 1, 0 and 9.

Solution: The largest 7-digit number will be 9853100.
The smallest 7-digit number will be 1003589.

## Expanded Form

In expanded form, a number is written as the sum of the place values of each digit.

Example 3: Write the following numbers in expanded form.
a.      5343279                       b. 832945431
Solution:
a.    5343278 = 5000000 + 300000 + 40000 + 3000 + 200 + 70 + 8
b.      832945430 = 800000000 + 30000000 + 2000000 + 900000 + 40000 + 5000 + 400 + 30

## Standard Form

The standard form of a number is just opposite to writing a number in expanded form.
To write the expanded form 70000000 + 30000 + 5000 + 600 + 2 in standard form, arrange the numbers in appropriate columns and then add.
The standard form of the number is 70035602.

## Rounding off Numbers or Estimation of Numbers

Sometimes it is necessary to give an approximate value to make comparison. For example, a teacher may say that there were 5,000 people at the school concert. This does not mean that exactly 5000 people were present at the school concert. It is merely the approximate number of people who were there.
Similarly, the number of tickets sold for an ODI cricket match may have been 19975. The news channels may say that 20000 tickets were sold.
This process of expressing a number to the nearest convenient figure to make it easier to understand is called ‘Rounding off a Number’ or ‘Estimation of Numbers’.

#### We follow these rules to round off numbers:

1.      Find the rounding digit. If we have to round off a number to the nearest 10, then the rounding digit is the digit at the tens place of the number. Similarly, if we have to round off a number to the nearest 100, then the rounding digit is the digit at the hundreds place, and so on.

2.      If the digit just to the right of the rounding digit is 0, 1, 2, 3 or 4, keep the rounding digit as it is and change all the digits to the right of the rounding digit to zero. For example, 234 rounded off to the nearest ten is 230 and 5127 rounded off to the nearest hundred is 5100.

3.      If the digit just to the right of the rounding digit is 5, 6, 7, 8 or 9, then add 1 to the rounding digit and change all the digits to the right of the rounding digit to zero. For example, 5768 rounded off to the nearest hundred is 5800 and 403681 rounded off to the nearest thousand is 404000.

Example 4: Round off the following numbers to the nearest 10.

a.      63       b.  78          c.  361        d.   125

Solution:
a.      63 rounded off to the nearest 10 is 60.
b.      78 rounded off to the nearest 10 is 80.
c.       361 rounded off to the nearest 10 is 360
d.      125 rounded off to the nearest 10 is 130.

Example 5: Round off the following numbers to the nearest 100.

a.       642       b.  784          c.  568        d.   351
Solution:
a.      642 rounded off to the nearest 100 is 600.
b.      784 rounded off to the nearest 100 is 800.
c.       568 rounded off to the nearest 100 is 600
d.      351 rounded off to the nearest 100 is 400.

Example 6: Round off the following numbers to the nearest 1000.

a.        26573       b.  17487          c.  53619        d.   71250
Solution:
a.      26573 rounded off to the nearest 1000 is 27000.
b.      17487 rounded off to the nearest 1000 is 17000.
c.       53619 rounded off to the nearest 1000 is 54000
d.      71250 rounded off to the nearest 1000 is 71000.

Example 7: Round off the following numbers to the nearest 10000.

a.      46526       b.  36486          c.  296135        d.   381281
Solution:
a.      46526 rounded off to the nearest 10000 is 50000.
b.      36486 rounded off to the nearest 10000 is 40000.
c.       296135 rounded off to the nearest 10000 is 300000
d.      381281 rounded off to the nearest 10000 is 380000.

## Estimation in Operations on Numbers

In our day-to-day life, we need not use the exact numbers, but estimate the numbers to get an approximate value.
Mr. Shweta spent Rs 3976 on groceries. In discussing with others, she said the expenditure as approximately Rs 4000 on the groceries. What did she do? She rounded off the number to the nearest thousand.

### Estimating Sum

Example 8: A book shopkeeper sold 8786 books in the first six months of the year 2019 and 7638 books in the last six months of the year. How many books were sold by the shopkeeper in the year 2019?
Solution: Let us find the actual and estimated number of books sold by the shopkeeper.

Actual sum               Estimated sum
8786                              9000          (Rounding off to the nearest thousand)
+ 7238                           + 7000          (Rounding off to the nearest thousand)
--------                          ---------
16024                            16000
---------                         ----------

### Estimating Difference

Example 9: A factory produced 42548 toys, out of which 28946 were sent to wholesalers. How many toys are still left in the factory?
Solution: Let us find the actual and estimated difference between the number of toys produced and the number of toys sent to wholesalers.

Actual                    Estimated
42548                        43000          (Rounding off to the nearest thousand)
- 28946                      - 29000          (Rounding off to the nearest thousand)
-----------                     -----------
13602                         14000
------------                    ------------

Estimating Products

Example 10: A jar can hold 14860 beads. If 8 such jars are to be filled, what is the actual and estimated number of beads required?
Solution: Let us find the product of number of beads in a jar and number of jars.
Actual            Estimated
14860               17000                (Rounding off to the nearest thousand)
× 8                   × 10                (Rounding off to the nearest ten)
----------          -----------
118880             170000
-----------          -----------

### Estimating Quotient

Example 11: 59668 tiles are to be sent to Bihar. There are only 7 loading vehicles. How many actual and estimated tiles should be loaded on each vehicle?

Solution:

Actual Quotient

Estimated Quotient

Total number of tiles = 60000             (Rounding off to the nearest thousand)
Number of vehicles = 10                     (Rounding off to the nearest ten)
Tiles on each vehicle = 60000 ÷ 10 = 6000

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