The Use of letters in place of Numbers

The Use of letters in place of Numbers

        We use symbols in mathematics. Using symbols, we write ‘the sum of five and nine’ in short as ‘5 + 9’.

         

Using symbols makes our writing short and clear. Use of letters, too makes mathematical writing easier.

         

Letters are used in place of numbers in two ways.

1.    The use of a letter for an unknown number.

‘ what is the number that is bigger than 8 by 4 ?” To find the answer to this question, let us add ‘8 + 4’. The sum 12 is the answer to the question.

      

 Using a sign, we write the information number that is bigger than four by eight’ as ‘8 + 4’.

      

 Now, let us see how we could write the information ‘a number that is bigger than another number by 4’ in signs.

      

We do not know exactly which number we are speaking of still, we can write this information using signs. Let us show the unknown number with the letter ‘a’. using signs, we can write ‘a’ number bigger than another number by 4’ as ‘a

+ 4’.

        

Similarly, ‘a number less than another number by 7’ can be written as ‘b – 7’. Here, b is written in place of the unknown number.

        

You will notice that when we add, subtract, multiply or divide two or more numbers, we get only one number in the end. But, the expression we get by using the sign for an operation between a letter and a number cannot be simplified any further.

     

   ‘The product  of a number and 2’ can be written as ‘m × 2’ or          2 × m by writing ‘m’ in place of ‘a number’. However, the product of a number and a letter is conventionally written with the number first and without the multiplication sign. Thus, the product m × 2 or 2 × m is written as 2m. similarly, 10 × k or k × 10 is written as 10k.

         

Some more examples of the use of a letter in place of an unknown number are given below. Study them. Note that we can choose to write any of the letters a,b,c,…,z in place of numbers.

   Information                                                      Expressed using letters

(1)  Sum of 10 and another number                              10 + p

(2)  Product of 23 and another number                         23 × d or 23d

(3) The quotient of 18 divided by a number              18 ÷ y or 18/y

(4)  The quotient of a number divided by 18           x/18 or x ÷ 18

(5)  The number obtained by subtracting                  a - 15           

15 from a number

(6)  The number obtained by subtracting                      15 – b          

a number from 15

(7)  A number less than another by 2                             (d – 2)

(8) The mangoes left after eating 6 from a                    (m – 6)               

Certain number of mangoes       

 

(9)  The number of guavas in each share if                           

3 equal shares are made of a certain                        p/3             

Number of guavas.

Order of Operations and the Use of Brackets

Order of Operations and

 the Use of Brackets

You have learned to carry out the operations of addition, subtraction, multiplication, and division. Let us study these operations further. Consider the following examples :

Ex (1) Simplify : 75 + 25 x 10

Addition and multiplication are the two operations in this example.

If we carry out the addition 75 + 25 first, the value of this expression will be 100 x 10 = 1000

If we carry out the multiplication 25 x 10 first, then the value of the expression w

ill be 75 + 250 = 325.

We shall have to say that the same expression has two different values. It is obvious that this will cause confusion.

Use of Brackets

It is necessary to know which operation should be carried out first in order that the expression yields a single value. Brackets are used for this purpose. The operation to be carried out first is written inside brackets.

             

For example, if the addition is to be carried out first in the expression 75 + 25 x 10, then the expression is written as                   (75 + 25) x 10.

 If the expression is 75 +(25x10), multiplication will be carried out first.

The value of the expression (75 + 25 ) x 10 is 1000.

And the value of 75 + (25 x 10 ) is 325.

Similarly, ( 48 ÷ 8 ) ÷ 2 = 3 and 48 ÷ ( 8 ÷ 2 ) = 12.

The following rules are generally

 followed to minimize the use of 

brackets in an expression.

(1)  Operations in brackets should be carried out first.

(2)  After that, multiplication and division must be carried out in the order in which they occur from left to right.

(3)  Then, addition and subtraction should be carried out in the order in which they occur from left to right.

(4) If there are more than one operations inside a bracket, they are carried out first, following rules 2 and 3.

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Operations on Numbers : Multiplication and division Multiplication

Operations on Numbers : Multiplication and division

Multiplication

To multiply numbers having upto five digits by a three-digit number.

          

Study the following multiplication.

Ex. 1. 879 × 248

In the multiplication 879 × 248 = 217992; 879 is the multiplicand, 248 is the multiplier and 217992 is the product.

1.    A pump draws 1240 litres of water per minute. How much water will be drawn in 3 hours?

place your answers by comments:

Operations on Numbers : Addition and Subtraction

READ THE NUMBERS

FIRST WATCH THE READ  NUMBERS VIDEOS AND THAN FOLLOW THE OTHER LESSONS.

SUBTRACT THE NUMBERS

Operations on Numbers : Addition and Subtraction

Addition

          

We have learnt how to add numbers having up to five digits. We can add numbers having more than five digits using the same method. Study the following examples.

Ex. 1. Add                                                 

658003 + 804017                                    

	TL	L	TTH	TH	H	T	U
+		6 8	5 0	8 4	0 0	0 1	3 7
	1		1			1
	1	4	6	2	0	2	0

Ex. 2. Add

890067 + 9989744

	1	1	1			1	1
	C	TL	L	TH	TH	H	T	U
			8	9	0	0	6	7
+		9	9	8	9	7	4	4
	1	0	8	7	9	8	1	1

Make a habit of keeping the carried over number in your mind rather than writing it down.

Numbers

1.    Numbers

The numbers we use for counting objects are called counting numbers.These numbers have names like one,two,three,and so       on. In English,they are written using numberls such as 1,2,3, -----These numberls are known as International numberls.The numberls in the Devanagari script are 1,2,3,and so on.

Numbers greater then ten lakh : reading and writing

•           In the previous standard, we learnt how to write numbers
• upto the lakhs place. 
• 
  
• The smallest six-digit number is 1,00,000 and 9,99,999 (nine lakh ninety-nine thousand nine hundred and ninety-nine) is the largest. 
• 
  
• Whenwe add I to this number,we get the next number, which is 10,00,000 or ten lakh. Ten lakh is also called a million.
•          
•  Ten lakh is the smallest seven-digit number and 99,99,999 (ninety-nine lakh ninety-nine thousand nine hundred and ninety-nine) is the largest seven-digit number. 
• 
  
• When we add I to it, we get the eight-digit number 1,00,00,000.

          C       TL      L        TTH   TH     H       T        U

          1       0       0       0       0       0       0       0

In this number, the place to the left of ten lakhs is called ‘crores’.One crore is the smallest eight-digit number.

          

The place to the left of the crores place is that of ten crores. Thus, one ten-crores and 5 crore, that is, 15 crore is written as 15,00,00,000.

          

Now, let us take the example of 42,35,78,959. The group of TC (ten crores) and C is together read as forty-two crore;

the group of TL and L is read as thirty-five lakh; the group of TTH and TH is read as seventy-eight thousand and the group of H, T and U is read as nine hundred and fifty-nine. 

Hence, the number is read and written in words as ‘ forty-two crore,thirty-five lakh seventy-eight thousand nine hundred and fifty-nine’.

         

If we add I to 99,99,99,999; that is ,to the largest nine-digit number, we get the next number 1000000000  which is the smallest ten-digit number. It is read as one hundred crore. It is also called a billion.