Operations on Whole Numbers — Exercise 2(A)

Fill in the blanks :

(i) 168 + 259 = ............... + 168

(ii) ....... + 317 = 317

(iii) (37 + 68) + ............... = 37 + (............... + 56)

(iv) 8 + 3 x 4 = ...............

(v) 18 x (............... + 23) = (18 x 17) + (18 x ...............)

Answer

(i) According to the Commutative Property of Addition: (a + b = b + a)

168 + 259 = 259 + 168.

(ii) If 0 is added to any number, the number remains unchanged.

0 + 317 = 317.

(iii) According to the Associative Property of Addition: (a + b) + c = a + (b + c).

(37 + 68) + 56 = 37 + (68 + 56).

(iv) According to the DMAS rule, multiplication is performed before addition.

⇒ 8 + (3 x 4)

⇒ 8 + 12

⇒ 20

8 + 3 x 4 = 20.

(v) According to the Distributive property of multiplication over addition:

[a x (b + c) = (a x b) + (a x c)].

18 x (17 + 23) = (18 x 17) + (18 x 23).

Fill in the blanks :

(i) 237 x 1 = ...............

(ii) 56 x ............... = 0

(iii) 0 ÷ 53 = ...............

(iv) 37 x 59 = 59 x ...............

(v) 0 x 138 = ...............

(vi) 73 ÷ 73 = ...............

Answer

(i) According to the Identity Property of Multiplication, any number multiplied by 1 equals itself.

237 x 1 = 237.

(ii) According to the Zero Property of Multiplication, any number multiplied by 0 equals 0.

56 x 0 = 0.

(iii) Zero divided by any non-zero number is 0.

0 ÷ 53 = 0.

(iv) According to the Commutative Property of Multiplication, the order in which you multiply two numbers does not change the result.

37 x 59 = 59 x 37.

(v) According to the Zero Property of Multiplication, any number multiplied by 0 equals 0.

0 x 138 = 0.

(vi) Any non-zero number divided by itself is 1.

73 ÷ 73 = 1.

Divide 3605 by 29 and verify the division algorithm.

Answer

$\begin{array}{l} \phantom{x^2 }{\quad 124} \\ 29\overline{\smash{\big)}3605} \\ \phantom{x}\phantom{}\underline{-29} \\ \phantom{{x^2 }+} 70 \\ \phantom{{x}+}\underline{-58} \\ \phantom{{x^2 } +2} 125 \\ \phantom{{x} +}\underline{-116} \\ \phantom{{x^2 + 3x}} 9 \\ \end{array}$

Dividend = 3605

Divisor = 29

Quotient = 124

Remainder = 9

Verification: Dividend = (Divisor × Quotient) + Remainder

Substituting values we get :

(Divisor × Quotient) + Remainder

= (29 x 124) + 9

= 3596 + 9

= 3605.

Since L.H.S. = R.H.S.

Hence, the result is verified by the division algorithm.

Find the number which when divided by 45 gives 16 as quotient and 9 as remainder.

Answer

Given, Divisor = 45

Quotient = 16

Remainder = 9

Using formula, Dividend = (Divisor × Quotient) + Remainder

= (45 × 16) + 9

= 720 + 9

= 729

Hence, the number = 729.

Find the largest number of 5-digits which is exactly divisible by 57.

Answer

The largest 5-digit number is 99,999.

To find the largest 5-digit number exactly divisible by 57, divide 99,999 by 57 and subtract the remainder from 99,999.

$\begin{array}{l} \phantom{x^2 }{\quad1754} \\ 57\overline{\smash{\big)}99999} \\ \phantom{x)}\phantom{}\underline{-57} \\ \phantom{{x^2 }+} 429 \\ \phantom{{x} )))}\underline{-399} \\ \phantom{{x^2 } + 2x } 309 \\ \phantom{{x} +1)}\underline{-285} \\ \phantom{{x^2 + 3x +)}} 249 \\ \phantom{{x} + 1x +}\underline{-228} \\ \phantom{{x^2 + 3x + 1)}} 21\ \end{array}$

The remainder when 99,999 is divided by 57 is 21.

Therefore, 99,999 − 21 = 99,978.

Hence, the largest 5-digit number which is exactly divisible by 57 = 99,978.

Find the smallest 6-digit number which is exactly divisible by 63.

Answer

The smallest 6-digit number = 1,00,000.

To find the smallest 6-digit number exactly divisible by 63, we divide 1,00,000 by 63 and add the difference between the divisor and remainder to 1,00,000.

$\begin{array}{l} \phantom{x^2 }{\quad 1587} \\ 63\overline{\smash{\big)}100000} \\ \phantom{x}\phantom{)}\underline{-63} \\ \phantom{x^2+} 370 \\ \phantom{{x}+}\underline{-315} \\ \phantom{{x^2 }2x + } 550 \\ \phantom{{x}+ 11}\underline{-504} \\ \phantom{{x^2 + 3x))}} 460 \\ \phantom{{x} + 1x)}\underline{-441} \\ \phantom{{x^2 + 3x)44}} 19\ \end{array}$

Required number to be added = 63 − 19 = 44

Number = 1,00,000 + 44 = 1,00,044.

Hence, the smallest 6-digit number exactly divisible by 63 = 1,00,044.

On dividing 1653 by a certain number, we get 45 as quotient and 33 as remainder. Find the divisor.

Answer

Given:

Dividend = 1653

Quotient = 45

Remainder = 33

Using the formula,

Dividend = (Divisor × Quotient) + Remainder

Substituting the values, we get :

⇒ 1653 = (Divisor × 45) + 33

⇒ Divisor × 45 = 1653 - 33

⇒ Divisor × 45 = 1620

⇒ Divisor = $\dfrac{1620}{45}$

$\begin{array}{l} \phantom{x^2 }{\quad 36} \\ 45\overline{\smash{\big)}1620} \\ \phantom{x}\phantom{)}\underline{-135} \\ \phantom{{x^2 }+} 270 \\ \phantom{{x} +}\underline{-270} \\ \phantom{{x^2 a}+} 0\ \end{array}$

⇒ Divisor = 36

Hence, the divisor = 36.

Use distributive law and evaluate :

(i) 576 x 285 + 576 x 115

(ii) 385 x 178 - 385 x 78

(iii) 365 x 645 + 135 x 645

(iv) 407 x 168 - 307 x 168

Answer

(i) 576 x 285 + 576 x 115

Using distributive law: a x b + a x c = a x (b + c)

⇒ 576 x (285 + 115)

⇒ 576 x 400

⇒ 2,30,400

Hence, 576 x 285 + 576 x 115 = 2,30,400.

(ii) 385 x 178 - 385 x 78

Using distributive law: a x b - a x c = a x (b - c)

⇒ 385 x (178 - 78)

⇒ 385 x 100

⇒ 38,500

Hence, 385 x 178 - 385 x 78 = 38,500.

(iii) 365 x 645 + 135 x 645

⇒ 645 x 365 + 645 x 135

Using distributive law: a x b + a x c = a x (b + c)

⇒ 645 x (365 + 135)

⇒ 645 x 500

⇒ 3,22,500

Hence, 365 x 645 + 135 x 645 = 3,22,500.

(iv) 407 x 168 - 307 x 168

⇒ 168 x 407 - 168 x 307

Using distributive law: a x b - a x c = a x (b - c)

⇒ 168 x (407 - 307)

⇒ 168 x 100

⇒ 16,800

Hence, 407 x 168 - 307 x 168 = 16,800.

Using the most convenient grouping, find each of the following products :

(i) 5 x 648 x 20

(ii) 8 x 329 x 25

(iii) 8 x 12 x 25 x 7

(iv) 125 x 40 x 8 x 25

Answer

(i) 5 x 648 x 20

⇒ 648 x (5 x 20)

⇒ 648 x 100

⇒ 64,800.

Hence, 5 x 648 x 20 = 64,800.

(ii) 8 x 329 x 25

⇒ 329 x (8 x 25)

⇒ 329 x 200

⇒ 65,800.

Hence, 8 x 329 x 25 = 65,800.

(iii) 8 x 12 x 25 x 7

⇒ (8 x 25) x 12 x 7

⇒ 200 x 12 x 7

⇒ 2400 x 7

⇒ 16,800.

Hence, 8 x 12 x 25 x 7 = 16,800.

(iv) 125 x 40 x 8 x 25

⇒ 125 x 40 x (8 x 25)

⇒ 125 x 40 x 200

⇒ 125 x (40 x 200)

⇒ 125 x 8000

⇒ 10,00,000.

Hence, 125 x 40 x 8 x 25 = 10,00,000.

Divide and verify the answer by division algorithm :

(i) 3680 ÷ 87

(ii) 17368 ÷ 327

(iii) 32679 ÷ 265

Answer

(i) 3680 ÷ 87

$\begin{array}{l} \phantom{x^2 }{\quad 42} \\ 87\overline{\smash{\big)}3680} \\ \phantom{x}\phantom{)}\underline{-348} \\ \phantom{{x^2 }+} 200 \\ \phantom{{x} +}\underline{-174} \\ \phantom{{x^2 a}+} 26\ \end{array}$

Dividend = 3680

Divisor = 87

Quotient = 42

Remainder = 26

Verification: Dividend = (Divisor × Quotient) + Remainder

Substituting the values in R.H.S. of the equation :

(Divisor × Quotient) + Remainder

= (87 x 42) + 26

= 3,654 + 26

= 3,680.

Since, L.H.S. = R.H.S. = 3,680

Hence, the result is verified by the division algorithm.

(ii) 17368 ÷ 327

$\begin{array}{l} \phantom{x^2 ++}{\quad 53} \\ 327\overline{\smash{\big)}\quad 17368} \\ \phantom{x^ + )}\phantom{))}\underline{-1635} \\ \phantom{{x^2 } + 2a)} 1018 \\ \phantom{{x} +1a )}\underline{-981} \\ \phantom{{x^2 a} + 2x)} 37\ \end{array}$

Dividend = 17368

Divisor = 327

Quotient = 53

Remainder = 37

Verification: Dividend = (Divisor × Quotient) + Remainder

Substituting the values in R.H.S. of the equation :

(Divisor × Quotient) + Remainder

= (327 x 53) + 37

= 17,331 + 37

= 17,368

Since, L.H.S. = R.H.S. = 17,368

Hence, the result is verified by the division algorithm.

(iii) 32679 ÷ 265

$\begin{array}{l} \phantom{x^2 ))}{\quad 123} \\ 265\overline{\smash{\big)}\quad 32679} \\ \phantom{x^ + )}\phantom{))}\underline{-265} \\ \phantom{{x^2 } + 2a)} 617 \\ \phantom{{x} +1)}\underline{-530} \\ \phantom{{x^2 a} + 2x)} 879 \\ \phantom{{x} +1a ))}\underline{-795} \\ \phantom{{x^2 a} + 2x))} 84\ \end{array}$

Dividend = 32679

Divisor = 265

Quotient = 123

Remainder = 84

Verification: Dividend = (Divisor × Quotient) + Remainder

Substituting the values in R.H.S. of the equation :

(Divisor × Quotient) + Remainder

= (265 x 123) + 84

= 32,595 + 84

= 32,679.

Since, L.H.S. = R.H.S. = 32,679

Hence, the result is verified by the division algorithm.

Verify each of the following :

(i) 2867 + 986 = 986 + 2867

(ii) 368 x 215 = 215 x 368

(iii) (156 + 273) + 74 = 156 + (273 + 74)

(iv) (86 x 55) x 110 = 86 x (55 x 110)

Answer

(i) 2867 + 986 = 986 + 2867

According to the Commutative Property of Addition: a + b = b + a.

Taking L.H.S. = 2867 + 986

= 3,853

Taking R.H.S. = 986 + 2867

= 3,853

Since, L.H.S. = R.H.S.

Hence, proved that 2867 + 986 = 986 + 2867.

(ii) 368 x 215 = 215 x 368

According to the Commutative Property of Multiplication: a x b = b x a.

Taking L.H.S. = 368 x 215

= 79,120

Taking R.H.S. = 215 x 368

= 79,120

Since, L.H.S. = R.H.S.

Hence, proved that 368 x 215 = 215 x 368.

(iii) (156 + 273) + 74 = 156 + (273 + 74)

According to the Associative Property of Addition: (a + b) + c = a + (b + c).

Taking L.H.S. = (156 + 273) + 74

= 429 + 74

= 503

Taking R.H.S. = 156 + (273 + 74)

= 156 + 347

= 503

Since, L.H.S. = R.H.S.

Hence, proved that (156 + 273) + 74 = 156 + (273 + 74).

(iv) (86 x 55) x 110 = 86 x (55 x 110)

According to the Associative Property of Multiplication: (a x b) x c = a x (b x c).

Taking L.H.S. = (86 x 55) x 110

= 4730 x 110

= 5,20,300

Taking R.H.S. = 86 x (55 x 110)

= 86 x 6050

= 5,20,300

Since, L.H.S. = R.H.S.

Hence, proved that (86 x 55) x 110 = 86 x (55 x 110).

Simplify :

(i) 39 - 18 ÷ 3 + 2 x 3

(ii) 8 + 2 x 5

(iii) 5 x 8 - 6 ÷ 2

(iv) 19 - 9 x 2

(v) 15 ÷ 5 x 4 ÷ 2

Answer

(i) 39 - 18 ÷ 3 + 2 x 3

= 39 - 6 + 2 x 3

= 39 - 6 + 6

= 33 + 6 [Addition and subtraction from left to right]

= 39.

Hence, 39 - 18 ÷ 3 + 2 x 3 = 39.

(ii) 8 + 2 x 5

= 8 + 10

= 18.

Hence, 8 + 2 x 5 = 18.

(iii) 5 x 8 - 6 ÷ 2

= 5 x 8 - 3

= 40 - 3

= 37.

Hence, 5 x 8 - 6 ÷ 2 = 37.

(iv) 19 - 9 x 2

= 19 - 18

= 1.

Hence, 19 - 9 x 2 = 1.

(v) 15 ÷ 5 x 4 ÷ 2

= 3 x 4 ÷ 2

= 3 x 2

= 6.

Hence, 15 ÷ 5 x 4 ÷ 2 = 6.