SOLVED EXAMPLES

 SOLVED EXAMPLES

 Ex. 1. 9587? = 7429 – 4358.

Sol let 9587 – X = 7429 – 4358. Then,

9587 – X = 3071 → X = 95 87 – 3071 = 6516

Ex 2. 5793405 x 9999

Sol. 5793405 x 9999= 5793405 x (10000 – 1)

              =57934050000 – 5793405

              = 57928256595

Ex. 3. 839478 x 625 = 839478 x 5²

      Sol        = 839478 x 625 = 839478 x 5⁴

                  = 839478 x (10/2)⁴ = 839478x 10⁴/2⁵

                  =8394780000/16 = 524673750

Ex.4. 976x 236 +976 x 763=?

Sol. Using distributive law, we get:

976 x 237 + 976+ 763 = 976 x (237 +763)

=976 x 1000 = 976000.

Ex 5. 986 x 307 – 986 x 207 =?

Sol. By distributive law, we get.

986 x 307 -986x 207 =986 x (307 – 207)

= 986 x 100 = 98600

2560000                 +  49              +22400

1582449

Ex 6. 1607 x 1607 =?

Sol. 1607 x 1607 = (1607)²

  = (1600 + 7)²= (1600)² + 7² + 2 x 1600 x 7

  = 2560000 + 49 + 22400 = 2582449.

  

1960016       -11200

1948816

 Ex. 7. 1396 x 1396 =?

Sol. 1396 x 1396 = (1396)²

      = (1400 – 4)² = (1400)² + 4² - 2 x 14000x4

     = 1960000 + 49- 11200 = 1948816

Ex. 8. (475 x 475 + 125 x 125) = ?

Sol. We have (a² + b²) = ½ [(a + b)² + (a – b )²]

      (475)²+ (125)² = 1/2 [(475 + 125)²(475 – 125)²]

                                =1/2 x 482500 =241250.

[(a² -b²) = (a + b) (a – B)]

Ex. 9. (9 x 96 – 204 x 204) = ?

Sol. 796 + 9 – 204 x 204 = (796)² - (204)²

  = (796 + 204) (796 – 204)

= (1000 x 593) = 592000.

Ex 10. (387x 387 + 113 x113 + 2 x 387 x 113)=?

Sol. Given exp. = (387)² + (113)² +2x 387 + 113

(a² +b²+ 2ab), where a = 387 and b = 113

= (a +b)² = (387+ 113)²= (500)² = 250000

Ex. 11. (87x 87 + 61- 2 x 87 x 61)= ?

TESTS OF DIVISIBLITY

TESTS OF DIVISIBLITY

I.                    Divisibility by 2 :

 A number is divisible by 2 if its unit digit is any of 0, 2, 4, 6, 8.

Ex. 58694 is divisible by 3.

II.                  Divisibility by 9:

 A number is divisible by 3 only when the sum of its digits is divisible by 3.

 In the numbers 695421, the sum of digits =27, which is divisisible by3.

 695421 are divisible by 3.

(III)             In the number 948653, the sum of digits = 35, which is not divisible by 3.

           948653 is not divisible by 3.

Iv. Divisibility by 4:

A number is divisible by 4 is the sum of its last two digits is divisible by 4.

Ex. (I) 6879376 is divisible by 4, since 76 is divisible by 4.

(II) 496138 is not divisible by 4, since 38 is not divisible by 4.

               V.            Divisibility by 8:

 A number is divisible by 8 if the number formed by hundred’s ten’s and unit’s digit of the given number is divisible by 8.

Ex. (I) in the number 16789552, the number formed by last 3 digits namely 352 is divisible by 8.

16789652 is divisible by 8.

(II) in the number 576484, the number formed by last 3 digits, namely 484 is not divisible by 8.

576484 is not divisible by 8.

             VI.            Divisibility by 10:

 A number is divisible by 10 only when its unit digit is 0.

OPERATIONS ON NUMBERS

SECTION-1: 

1.       NUMBERS:  in Hindu- Arabic system, we have ten digits, namely 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, called zero, one two, three, four, five, six, seven, eight and nine respectively.

 A number is denoted by a group of digits, called numeral.

For denoting a numeral, we use the place-value chart, given below,

EX. 1.  Write each of the following numberless in words.

SOL.  The given numerals in words are.

                     I.            Six lac thirty-eight thousand five dared fort-nine.

                   II.            Twenty-three lac eighty thousand nine hundred seventeen.

                 III.            Eight crore fifty-four lac sixteen thousand eight.

                 IV.            Fifty-six crore thirteen lac seven thousand ninety.

    EX.2. write each of the following numbers in figures.

                 I.            Nine crore four lac six thousand two

               II.            Twelve crore seven lac nine thousand two hundred seven.

             III.            Four lac four thousand forty.

             IV.            Twenty-one crore sixty lac five thousand fourteen.

Sol. using the place value chart, we may write.

2.        2. Face  value and place value (or local value) of a digit in a numeral

(I)                 The face value of a digit in a numeral is its own value, at whatever place it may be 

 EX. In the numeral   6872, the face value of 2 is 2, the face value of 7, the face value of 8 is 8 and the face value of 6 is 6.

(II)               In a given numeral:

Place value of unit digit=(unit digit) x 1,

Place value of tens digit = (tens digit) x 10,

Place value of hundred’s digit = (hundred’s digit) x 100 and so on.

Ex. In the numeral 70984, we have

Place value of 4= (4x1) =4

Place value of 8 = (8 x 10) =80.

Place value of  9 = (9 x 100 )= 900,

Place value of 7 = (7 x 10000) =70000.