Integers

Evaluate :

(i) 427 × 8 + 2 × 427

(ii) 394 × 12 + 394 × (-2)

(iii) 558 × 27 + 3 × 558

Answer

Using the distributive property a × b + a × c = a × (b + c) :

(i) Solving,

427 × 8 + 2 × 427

= 427 × (8 + 2)

= 427 × 10

= 4270.

∴ 427 × 8 + 2 × 427 = 4270.

(ii) Solving,

394 × 12 + 394 × (-2)

= 394 × {12 + (-2)}

= 394 × 10

= 3940.

∴ 394 × 12 + 394 × (-2) = 3940.

(iii) Solving,

558 × 27 + 3 × 558

= 558 × (27 + 3)

= 558 × 30

= 16740.

∴ 558 × 27 + 3 × 558 = 16740.

Evaluate :

(i) 673 × 9 + 673

(ii) 1925 × 101 - 1925

Answer

(i) Solving,

673 × 9 + 673

= 673 × (9 + 1)

= 673 × 10

= 6730.

∴ 673 × 9 + 673 = 6730.

(ii) Solving,

1925 × 101 - 1925

= 1925 × (101 - 1)

= 1925 × 100

= 192500.

∴ 1925 × 101 - 1925 = 192500.

Verify :

(i) 37 × {8 + (-3)} = 37 × 8 + 37 × (-3)

(ii) (-82) × {(-4) + 19} = (-82) × (-4) + (-82) × 19

(iii) {7 - (-7)} × 7 = 7 × 7 - (-7) × 7

(iv) {(-15) - 8} × -6 = (-15) × (-6) - 8 × (-6)

Answer

(i) 37 × {8 + (-3)} = 37 × 8 + 37 × (-3)

L.H.S. = 37 × {8 + (-3)} = 37 × 5 = 185

R.H.S. = 37 × 8 + 37 × (-3) = 296 + (-111) = 296 - 111 = 185

∴ L.H.S. = R.H.S.

Hence, verified.

(ii) (-82) × {(-4) + 19} = (-82) × (-4) + (-82) × 19

L.H.S. = (-82) × {(-4) + 19} = (-82) × 15 = -1230

R.H.S. = (-82) × (-4) + (-82) × 19 = 328 + (-1558) = 328 - 1558 = -1230

∴ L.H.S. = R.H.S.

Hence, verified.

(iii) {7 - (-7)} × 7 = 7 × 7 - (-7) × 7

L.H.S. = {7 - (-7)} × 7 = (7 + 7) × 7 = 14 × 7 = 98

R.H.S. = 7 × 7 - (-7) × 7 = 49 - (-49) = 49 + 49 = 98

∴ L.H.S. = R.H.S.

Hence, verified.

(iv) {(-15) - 8} × -6 = (-15) × (-6) - 8 × (-6)

L.H.S. = {(-15) - 8} × (-6) = (-23) × (-6) = 138

R.H.S. = (-15) × (-6) - 8 × (-6) = 90 - (-48) = 90 + 48 = 138

∴ L.H.S. = R.H.S.

Hence, verified.

Evaluate :

(i) 15 × 8

(ii) 15 × (-8)

(iii) (-15) × 8

(iv) (-15) × -8

Answer

(i) 15 × 8 = 120.

(ii) 15 × (-8) = -120.

(iii) (-15) × 8 = -120.

(iv) (-15) × (-8) = 120.

Evaluate :

(i) 4 × 6 × 8

(ii) 4 × 6 × (-8)

(iii) 4 × (-6) × 8

(iv) (-4) × 6 × 8

(v) 4 × (-6) × (-8)

(vi) (-4) × (-6) × 8

(vii) (-4) × 6 × (-8)

(viii) (-4) × (-6) × (-8)

Answer

(i) 4 × 6 × 8 = 24 × 8 = 192.

(ii) 4 × 6 × (-8) = 24 × (-8) = -192.

(iii) 4 × (-6) × 8 = (-24) × 8 = -192.

(iv) (-4) × 6 × 8 = (-24) × 8 = -192.

(v) 4 × (-6) × (-8) = (-24) × (-8) = 192.

(vi) (-4) × (-6) × 8 = 24 × 8 = 192.

(vii) (-4) × 6 × (-8) = (-24) × (-8) = 192.

(viii) (-4) × (-6) × (-8) = 24 × (-8) = -192.

Evaluate :

(i) 2 × 4 × 6 × 8

(ii) 2 × (-4) × 6 × 8

(iii) (-2) × 4 × (-6) × 8

(iv) (-2) × (-4) × 6 × (-8)

(v) (-2) × (-4) × (-6) × (-8)

Answer

(i) 2 × 4 × 6 × 8 = 8 × 48 = 384.

(ii) 2 × (-4) × 6 × 8 = (-8) × 48 = -384.

(iii) (-2) × 4 × (-6) × 8 = (-8) × (-48) = 384.

(iv) (-2) × (-4) × 6 × (-8) = 8 × (-48) = -384.

(v) (-2) × (-4) × (-6) × (-8) = 8 × 48 = 384.

Determine the integer whose product with '-1' is :

(i) -47

(ii) 63

(iii) -1

(iv) 0

Answer

We will use the property:

(-1) x a = -a

Since multiplying by -1 changes only the sign and not the value of an integer, the required integer has the same value as the given product but with the opposite sign .

(i) Product = -47

(-1) x 47 = -47

Hence, required integer = 47.

(ii) Product = 63

(-1) x -63 = 63

Hence, required integer = -63.

(iii) Product = -1

(-1) x 1 = -1

Hence, required integer = 1.

(iv) Product = 0

(-1) x 0 = 0

Hence, required integer = 0.

Eighteen integers are multiplied together. What will be the sign of their product, if :

(i) 15 of them are negative and 3 are positive ?

(ii) 12 of them are negative and 6 are positive ?

(iii) 9 of them are positive and the remaining are negative ?

(iv) all are negative ?

Answer

The sign of the product depends on the number of negative integers. If the number of negative integers is even, the product is positive; if it is odd, the product is negative.

(i) Number of negative integers = 15, which is odd.

∴ The product will be negative.

(ii) Number of negative integers = 12, which is even.

∴ The product will be positive.

(iii) Number of negative integers = 18 - 9 = 9, which is odd.

∴ The product will be negative.

(iv) Number of negative integers = 18, which is even.

∴ The product will be positive.

Find which is greater ?

(i) (8 + 10) × 15 or 8 + 10 × 15

(ii) 12 × (6 - 8) or 12 × 6 - 8

(iii) {(-3) - 4} × (-5) or (-3) - 4 × (-5)

Answer

(i) (8 + 10) × 15 = 18 × 15 = 270

8 + 10 × 15 = 8 + 150 = 158

Since 270 > 158,

∴ (8 + 10) × 15 is greater.

(ii) 12 × (6 - 8) = 12 × (-2) = -24

12 × 6 - 8 = 72 - 8 = 64

Since 64 > -24,

∴ 12 × 6 - 8 is greater.

(iii) {(-3) - 4} × (-5) = (-7) × (-5) = 35

(-3) - 4 × (-5) = -3 - (-20) = -3 + 20 = 17

Since 35 > 17,

∴ {(-3) - 4} × (-5) is greater.

State true or false :

(i) product of two different integers can be zero.

(ii) product of 120 negative integers and 121 positive integers is negative.

(iii) a × (b + c) = a × b + c

(iv) (b - c) × a = b - c × a.

Answer

(i) True. If one of the two different integers is 0, their product is 0. For example, 0 × 5 = 0.

(ii) False. The number of negative integers is 120, which is even, so the product is positive (positive integers do not change the sign).

(iii) False. By the distributive property, a × (b + c) = a × b + a × c, not a × b + c.

(iv) False. By the distributive property, (b - c) × a = b × a - c × a, not b - c × a.

Divide :

(i) 117 by 9

(ii) (-117) by 9

(iii) 117 by (-9)

(iv) (-117) by (-9)

(v) 225 by (-15)

(vi) (-552) ÷ 24

(vii) (-798) by (-21)

(viii) (-910) ÷ 26

Answer

(i) $117 \div 9 = \dfrac{117}{9} = \dfrac{13 \times 9}{9}$ = 13.

(ii) $(-117) \div 9 = \dfrac{-117}{9} = \dfrac{-13 \times 9}{9}$ = -13.

(iii) $117 \div (-9) = \dfrac{117}{-9} = \dfrac{-117}{9} = \dfrac{-13 \times 9}{9}$ = -13.

(iv) $(-117) \div (-9) = \dfrac{-117}{-9} = \dfrac{117}{9} = \dfrac{13 \times 9}{9}$ = 13.

(v) $225 \div (-15) = \dfrac{225}{-15} = \dfrac{-225}{15} = \dfrac{-15 \times 15}{15}$ = -15.

(vi) $(-552) \div 24 = \dfrac{-552}{24} = \dfrac{-23 \times 24}{24}$ = -23.

(vii) $(-798) \div (-21) = \dfrac{-798}{-21} = \dfrac{798}{21} = \dfrac{38 \times 21}{21}$ = 38.

(viii) $(-910) \div 26 = \dfrac{-910}{26} = \dfrac{-35 \times 26}{26}$ = -35.

Evaluate :

(-234) ÷ 13

Answer

$(-234) \div 13 \\[1em] = \dfrac{-234}{13} \\[1em] = \dfrac{-18 \times 13}{13} \\[1em] = \mathbf{-18}$

Evaluate :

234 ÷ (-13)

Answer

$234 \div (-13) \\[1em] = \dfrac{234}{-13} \\[1em] = \dfrac{-234}{13} \\[1em] = \dfrac{-18 \times 13}{13} \\[1em] = \mathbf{-18}$

Evaluate :

(-234) ÷ (-13)

Answer

$(-234) \div (-13) \\[1em] = \dfrac{-234}{-13} \\[1em] = \dfrac{234}{13} \\[1em] = \dfrac{18 \times 13}{13} \\[1em] = \mathbf{18}$

Evaluate :

374 ÷ (-17)

Answer

$374 \div (-17) \\[1em] = \dfrac{374}{-17} \\[1em] = \dfrac{-374}{17} \\[1em] = \dfrac{-22 \times 17}{17} \\[1em] = \mathbf{-22}$

Evaluate :

(-374) ÷ 17

Answer

$(-374) \div 17 \\[1em] = \dfrac{-374}{17} \\[1em] = \dfrac{-22 \times 17}{17} \\[1em] = \mathbf{-22}$

Evaluate :

(-374) ÷ (-17)

Answer

$(-374) \div (-17) \\[1em] = \dfrac{-374}{-17} \\[1em] = \dfrac{374}{17} \\[1em] = \dfrac{22 \times 17}{17} \\[1em] = \mathbf{22}$

Evaluate :

(-728) ÷ 14

Answer

$(-728) \div 14 \\[1em] = \dfrac{-728}{14} \\[1em] = \dfrac{-52 \times 14}{14} \\[1em] = \mathbf{-52}$

Evaluate :

272 ÷ (-17)

Answer

$272 \div (-17) \\[1em] = \dfrac{272}{-17} \\[1em] = \dfrac{-272}{17} \\[1em] = \dfrac{-16 \times 17}{17} \\[1em] = \mathbf{-16}$

Find the quotient in each of the following divisions :

(i) 299 ÷ 23

(ii) 299 ÷ (-23)

(iii) (-384) ÷ 16

(iv) (-572) ÷ (-22)

(v) 408 ÷ (-17)

Answer

(i) Required quotient = $\dfrac{299}{23} = \dfrac{13 \times 23}{23}$ = 13.

(ii) Required quotient = $\dfrac{299}{-23} = \dfrac{-299}{23} = \dfrac{-13 \times 23}{23}$ = -13.

(iii) Required quotient = $\dfrac{-384}{16} = \dfrac{-24 \times 16}{16}$ = -24.

(iv) Required quotient = $\dfrac{-572}{-22} = \dfrac{572}{22} = \dfrac{26 \times 22}{22}$ = 26.

(v) Required quotient = $\dfrac{408}{-17} = \dfrac{-408}{17} = \dfrac{-24 \times 17}{17}$ = -24.

Divide :

(i) 204 by 17

(ii) 152 by -19

(iii) 0 by 35

(iv) 0 by (-82)

(v) 5490 by 10

(vi) 762800 by 100

Answer

(i) $204 \div 17 = \dfrac{204}{17} = \dfrac{12 \times 17}{17}$ = 12.

(ii) $152 \div (-19) = \dfrac{152}{-19} = \dfrac{-152}{19} = \dfrac{-8 \times 19}{19}$ = -8.

(iii) $0 \div 35 = \dfrac{0}{35}$ = 0.

(iv) $0 \div (-82) = \dfrac{0}{-82}$ = 0.

(v) $5490 \div 10 = \dfrac{5490}{10} = \dfrac{549 \times 10}{10}$ = 549.

(vi) $762800 \div 100 = \dfrac{762800}{100} = \dfrac{7628 \times 100}{100}$ = 7628.

State, true or false :

(i) 0 ÷ 32 = 0

(ii) 0 ÷ (-9) = 0

(iii) (-37) ÷ 0 = 0

(iv) 0 ÷ 0 = 0

Answer

(i) True. Zero divided by any non-zero integer is 0.

(ii) True. Zero divided by any non-zero integer is 0.

(iii) False. Division of any integer by zero is not defined.

(iv) False. Division of zero by zero is not defined.

Evaluate :

(i) 42 ÷ 7 + 4

(ii) 12 + 18 ÷ 3

(iii) 19 - 20 ÷ 4

(iv) 16 - 5 × 3 + 4

(v) 6 - 8 - (-6) ÷ 2

(vi) 13 - 12 ÷ 4 × 2

(vii) 16 + 8 ÷ 4 - 2 × 3

(viii) 16 ÷ 8 + 4 - 2 × 3

(ix) 16 - 8 + 4 ÷ 2 × 3

(x) (-4) + (-12) ÷ (-6)

(xi) (-18) + 6 ÷ 3 + 5

(xii) (-20) × (-1) + 14 ÷ 7

Answer

Solving using the BODMAS rule (Division, Multiplication, Addition, Subtraction) :

(i) Solving,

42 ÷ 7 + 4

= 6 + 4

= 10

∴ 42 ÷ 7 + 4 = 10

(ii) Solving,

12 + 18 ÷ 3

= 12 + 6

= 18

∴ 12 + 18 ÷ 3 = 18

(iii) Solving,

19 - 20 ÷ 4

= 19 - 5

= 14

∴ 19 - 20 ÷ 4 = 14

(iv) Solving,

16 - 5 × 3 + 4

= 16 - 15 + 4

= 20 - 15

= 5

∴ 16 - 5 × 3 + 4 = 5

(v) Solving,

6 - 8 - (-6) ÷ 2

= 6 - 8 - (-3)

= 6 - 8 + 3

= 9 - 8

= 1

∴ 6 - 8 - (-6) ÷ 2 = 1

(vi) Solving,

13 - 12 ÷ 4 × 2

= 13 - 3 × 2

= 13 - 6

= 7

∴ 13 - 12 ÷ 4 × 2 = 7

(vii) Solving,

16 + 8 ÷ 4 - 2 × 3

= 16 + 2 - 6

= 18 - 6

= 12

∴ 16 + 8 ÷ 4 - 2 × 3 = 12

(viii) Solving,

16 ÷ 8 + 4 - 2 × 3

= 2 + 4 - 6

= 6 - 6

= 0

∴ 16 ÷ 8 + 4 - 2 × 3 = 0

(ix) Solving,

16 - 8 + 4 ÷ 2 × 3

= 16 - 8 + 2 × 3

= 16 - 8 + 6

= 22 - 8

= 14

∴ 16 - 8 + 4 ÷ 2 × 3 = 14

(x) Solving,

(-4) + (-12) ÷ (-6)

= (-4) + 2

= -2

∴ (-4) + (-12) ÷ (-6) = -2

(xi) Solving,

(-18) + 6 ÷ 3 + 5

= (-18) + 2 + 5

= -18 + 7

= -11

∴ (-18) + 6 ÷ 3 + 5 = -11

(xii) Solving,

(-20) × (-1) + 14 ÷ 7

= 20 + 2

= 22

∴ (-20) × (-1) + 14 ÷ 7 = 22

The product of two integers is -90. If one of them is 6, find the other integer.

Answer

Let the other integer be x.

6 × x = -90

x = $\dfrac{-90}{6} = \dfrac{-15 \times 6}{6}$

x = -15.

∴ The other integer is -15.

The product of two integers is 210. If one of them is -15, find the other.

Answer

Let the other integer be x.

(-15) × x = 210

x = $\dfrac{210}{-15} = \dfrac{-14 \times 15}{15}$

x = -14

∴ The other integer is -14.

Evaluate :

18 - (20 - 15 ÷ 3).

Answer

Solving,

18 - (20 - 15 ÷ 3)

= 18 - (20 - 5)

= 18 - 15

= 3

∴ 18 - (20 - 15 ÷ 3) = 3

-15 + 24 ÷ (15 - 13).

Answer

Solving,

-15 + 24 ÷ (15 - 13)

= -15 + 24 ÷ 2

= -15 + 12

= -3

∴ -15 + 24 ÷ (15 - 13) = -3

35 - {15 + 14 - (13 + $\overline{2 - 1 + 3}$)}

Answer

Solving,

35 - {15 + 14 - (13 + $\overline{2 - 1 + 3}$)}

= 35 - {15 + 14 - (13 + 4)}

= 35 - {15 + 14 - 17}

= 35 - {29 - 17}

= 35 - 12

= 23

∴ 35 - {15 + 14 - (13 + $\overline{2 - 1 + 3}$)} = 23

27 - {13 + 4 - (8 + 4 - $\overline{1 + 3}$)}

Answer

Solving,

27 - {13 + 4 - (8 + 4 - $\overline{1 + 3}$)}

= 27 - {13 + 4 - (8 + 4 - 4)}

= 27 - {13 + 4 - 8}

= 27 - {17 - 8}

= 27 - 9

= 18

∴ 27 - {13 + 4 - (8 + 4 - $\overline{1 + 3}$)} = 18

32 - [43 - {51 - (20 - $\overline{18 - 7}$)}]

Answer

Solving,

32 - [43 - {51 - (20 - $\overline{18 - 7}$)}]

= 32 - [43 - {51 - (20 - 11)}]

= 32 - [43 - {51 - 9}]

= 32 - [43 - 42]

= 32 - 1

= 31

∴ 32 - [43 - {51 - (20 - $\overline{18 - 7}$)}] = 31

46 - [26 - {14 - (15 - 4 ÷ 2 × 2)}]

Answer

Solving,

46 - [26 - {14 - (15 - 4 ÷ 2 × 2)}]

= 46 - [26 - {14 - (15 - 2 × 2)}]

= 46 - [26 - {14 - (15 - 4)}]

= 46 - [26 - {14 - 11}]

= 46 - [26 - 3]

= 46 - 23

= 23

∴ 46 - [26 - {14 - (15 - 4 ÷ 2 × 2)}] = 23

45 - [38 - {60 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}]

Answer

Solving,

45 - [38 - {60 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}]

= 45 - [38 - {60 ÷ 3 - (6 - 3) ÷ 3}]

= 45 - [38 - {60 ÷ 3 - 3 ÷ 3}]

= 45 - [38 - {20 - 1}]

= 45 - [38 - 19]

= 45 - 19

= 26

∴ 45 - [38 - {60 ÷ 3 - (6 - 9 ÷ 3) ÷ 3}] = 26

17 - [17 - {17 - (17 - $\overline{17 - 17}$)}]

Answer

Solving,

17 - [17 - {17 - (17 - $\overline{17 - 17}$)}]

= 17 - [17 - {17 - (17 - 0)}]

= 17 - [17 - {17 - 17}]

= 17 - [17 - 0]

= 17 - 17

= 0

∴ 17 - [17 - {17 - (17 - $\overline{17 - 17}$)}] = 0

2550 - [510 - {270 - (90 - $\overline{80 + 7}$)}]

Answer

Solving,

2550 - [510 - {270 - (90 - $\overline{80 + 7}$)}]

= 2550 - [510 - {270 - (90 - 87)}]

= 2550 - [510 - {270 - 3}]

= 2550 - [510 - 267]

= 2550 - 243

= 2307

∴ 2550 - [510 - {270 - (90 - $\overline{80 + 7}$)}] = 2307

30 + [{-2 × (25 - $\overline{13 - 3}$)}]

Answer

Solving,

30 + [{-2 × (25 - $\overline{13 - 3}$)}]

= 30 + [{-2 × (25 - 10)}]

= 30 + [{-2 × 15}]

= 30 + [-30]

= 0

∴ 30 + [{-2 × (25 - $\overline{13 - 3}$)}] = 0

88 - {5 - (-48) ÷ (-16)}

Answer

Solving,

88 - {5 - (-48) ÷ (-16)}

= 88 - {5 - 3}

= 88 - 2

= 86

∴ 88 - {5 - (-48) ÷ (-16)} = 86

9 × (8 - $\overline{3 + 2}$) - 2(2 + $\overline{3 + 3}$)

Answer

Solving,

9 × (8 - $\overline{3 + 2}$) - 2(2 + $\overline{3 + 3}$)

= 9 × (8 - 5) - 2(2 + 6)

= 9 × 3 - 2 × 8

= 27 - 16

= 11

∴ 9 × (8 - $\overline{3 + 2}$) - 2(2 + $\overline{3 + 3}$) = 11

2 - [3 - {6 - (5 - $\overline{4 - 3}$)}]

Answer

Solving,

2 - [3 - {6 - (5 - $\overline{4 - 3}$)}]

= 2 - [3 - {6 - (5 - 1)}]

= 2 - [3 - {6 - 4}]

= 2 - [3 - 2]

= 2 - 1

= 1

∴ 2 - [3 - {6 - (5 - $\overline{4 - 3}$)}] = 1

The sum of two integers is -15. If one of them is 9, find the other.

Answer

Let the other integer be x.

9 + x = -15

x = -15 - 9

x = -24

∴ The other integer is -24.

The difference between integers x and -6 is -5. Find the values of x.

x - (-6) = -5 or -6 - x = -5

Answer

The difference between x and -6 is -5.

x - (-6) = -5 or -6 - x = -5

x + 6 = -5 or -x = -5 + 6

x = -5 - 6 or -x = 1

x = -11 or x = -1

∴ x = -11 or -1

The sum of two integers is 28. If one integer is -45, find the other.

Answer

Let the other integer be x.

(-45) + x = 28

x = 28 - (-45)

x = 28 + 45

x = 73

∴ The other integer is 73.

The sum of two integers is -56. If one integer is -42, find the other.

Answer

Let the other integer be x.

(-42) + x = -56

x = -56 - (-42)

x = -56 + 42

x = -14

∴ The other integer is -14.

The difference between an integer x and (-9) is 6. Find all possible values of x.

Answer

The difference between x and -9 is 6.

x - (-9) = 6 or -9 - x = 6

x + 9 = 6 or -x = 6 + 9

x = 6 - 9 or -x = 15

x = -3 or x = -15

∴ x = -3 or -15

Write all the integers between -15 and 15, which are divisible by 2 and 3.

Answer

An integer divisible by both 2 and 3 is divisible by 6.

The multiples of 6 lying between -15 and 15 are :

-12, -6, 0, 6, 12

Write all the integers between -5 and 5, which are divisible by 2 or 3.

Answer

The integers between -5 and 5 are -4, -3, -2, -1, 0, 1, 2, 3, 4.

Integers divisible by 2 : -4, -2, 0, 2, 4

Integers divisible by 3 : -3, 0, 3

Combining (taking each integer once), the integers divisible by 2 or 3 are :

-4, -3, -2, 0, 2, 3, 4

Find the result of subtracting the sum of all integers between 20 and 30 from the sum of all integers from 20 to 30.

Answer

Sum of all integers from 20 to 30 (both included) :

= 20 + 21 + 22 + ... + 30

= 275

Sum of all integers between 20 and 30 (both excluded) :

= 21 + 22 + ... + 29

= 225

Required result = 275 - 225 = 50

Hence, the required result is 50.

Add the product of (-13) and (-17) to the quotient of (-187) and 11.

Answer

Product of (-13) and (-17) = (-13) × (-17) = 221

Quotient of (-187) and 11 = $(-187) \div 11 = \dfrac{-187}{11} = \dfrac{-17 \times 11}{11} = -17$

Required sum = 221 + (-17) = 221 - 17 = 204

Hence, the required result is 204.

The product of two integers is -180. If one of them is 12, find the other.

Answer

Let the other integer be x.

12 × x = -180

x = $\dfrac{-180}{12}$

x = $\dfrac{-15 \times 12}{12}$

x = -15

∴ The other integer is -15.

(i) A number changes from -20 to 30. What is the increase or decrease in the number ?

(ii) A number changes from 40 to -30. What is the increase or decrease in the number ?

Answer

(i) Change = 30 - (-20)

= 30 + 20

= 50

Since the change is positive,

∴ There is an increase of 50.

(ii) Change = -30 - 40

= -70

Since the change is negative,

∴ There is a decrease of 70.

If a = -12 and b = -10, verify that : a + b = b + a.

Answer

Given a = -12 and b = -10.

L.H.S. = a + b = (-12) + (-10) = -22

R.H.S. = b + a = (-10) + (-12) = -22

∴ L.H.S. = R.H.S.

Hence, verified.

If a = -7, b = -5 and c = 8, verify that : a + (b + c) = (a + b) + c.

Answer

Given a = -7, b = -5 and c = 8.

L.H.S. = a + (b + c) = -7 + {(-5) + 8} = -7 + 3 = -4

R.H.S. = (a + b) + c = {(-7) + (-5)} + 8 = -12 + 8 = -4

∴ L.H.S. = R.H.S.

Hence, verified.

If a = 7, b = 5 and c = -8, verify that : a - (b - c) ≠ (a - b) - c.

Answer

Given a = 7, b = 5 and c = -8.

L.H.S. = a - (b - c) = 7 - {5 - (-8)} = 7 - (5 + 8) = 7 - 13 = -6

R.H.S. = (a - b) - c = (7 - 5) - (-8) = 2 + 8 = 10

Since -6 ≠ 10,

∴ L.H.S. ≠ R.H.S.

Hence, verified.

The difference between two integers x and -12 is -15. Find the value(s) of x.

Answer

The difference between x and -12 is -15.

x - (-12) = -15 or -12 - x = -15

x + 12 = -15 or -x = -15 + 12

x = -15 - 12 or -x = -3

x = -27 or x = 3

∴ x = -27 or 3.

An aeroplane is 800 m vertically above the head of a boy. After sometime, it is 1125 m vertically above the head of the same boy. What is the change in the height of the aeroplane ?

Answer

Initial height = 800 m

Final height = 1125 m

Change in height = 1125 - 800 = 325 m

Since the change is positive, the height has increased.

∴ The change in the height of the aeroplane is an increase of 325 m.

Write a pair of integers whose : (i) sum = -7 (ii) difference = -7

Answer

(i) A pair of integers whose sum is -7 :

(-3) + (-4) = -7

∴ The required pair is (-3, -4).

(ii) A pair of integers whose difference is -7 :

2 - 9 = -7

∴ The required pair is (2, 9).

Write two integers each of which is smaller than -3 and their difference is greater than -3.

Answer

Let the two integers be -4 and -5. Both are smaller than -3.

Their difference = (-4) - (-5) = -4 + 5 = 1, which is greater than -3.

∴ The required integers are -4 and -5.

Evaluate :

(i) (-1) × (-1) × (-1) × .......... 60 times.

(ii) (-1) × (-1) × (-1) × (-1) × .......... 75 times.

Answer

(i) Here, (-1) is multiplied 60 times. Since 60 is even, the product is positive.

(-1) × (-1) × (-1) × ... 60 times = 1

(ii) Here, (-1) is multiplied 75 times. Since 75 is odd, the product is negative.

(-1) × (-1) × (-1) × ... 75 times = -1

Evaluate :

(i) (-2) × (-3) × (-4) × (-5) × (-6)

(ii) (-3) × (-6) × (-9) × (-12)

(iii) (-11) × (-15) + (-11) × (-25)

(iv) 10 × (-12) + 5 × (-12)

Answer

(i) There are 5 negative integers (odd), so the product is negative.

(-2) × (-3) × (-4) × (-5) × (-6)

= -(2 × 3 × 4 × 5 × 6)

= -720

∴ (-2) × (-3) × (-4) × (-5) × (-6) = -720

(ii) There are 4 negative integers (even), so the product is positive.

(-3) × (-6) × (-9) × (-12)

= 3 × 6 × 9 × 12

= 1944

∴ (-3) × (-6) × (-9) × (-12) = 1944

(iii) Using the distributive property :

(-11) × (-15) + (-11) × (-25)

= (-11) × {(-15) + (-25)}

= (-11) × (-40)

= 440

∴ (-11) × (-15) + (-11) × (-25) = 440

(iv) Using the distributive property :

10 × (-12) + 5 × (-12)

= (-12) × (10 + 5)

= (-12) × 15

= -180

∴ 10 × (-12) + 5 × (-12) = -180

(i) If X × (-1) = -36, is X positive or negative ?

(ii) If X × (-1) = 36, is X positive or negative ?

Answer

(i) X × (-1) = -36

⇒ X = (-36) ÷ (-1) = 36

∴ X = 36, which is positive.

(ii) X × (-1) = 36

⇒ X = 36 ÷ (-1) = -36

∴ X = -36, which is negative.

Evaluate :

(-20) + (-8) ÷ (-2) × 3

Answer

Solving,

(-20) + (-8) ÷ (-2) × 3

= (-20) + 4 × 3

= (-20) + 12

= -8

∴ (-20) + (-8) ÷ (-2) × 3 = -8

Evaluate :

(-5) - (-48) ÷ (-16) + (-2) × 6

Answer

Solving,

(-5) - (-48) ÷ (-16) + (-2) × 6

= (-5) - 3 + (-12)

= -5 - 3 - 12

= -20

∴ (-5) - (-48) ÷ (-16) + (-2) × 6 = -20

Evaluate :

16 + 8 ÷ 4 - 2 × 3

Answer

Solving,

16 + 8 ÷ 4 - 2 × 3

= 16 + 2 - 6

= 18 - 6

= 12

∴ 16 + 8 ÷ 4 - 2 × 3 = 12

Evaluate :

16 ÷ 8 × 4 - 2 × 3

Answer

Solving,

16 ÷ 8 × 4 - 2 × 3

= 2 × 4 - 6

= 8 - 6

= 2

∴ 16 ÷ 8 × 4 - 2 × 3 = 2

Evaluate :

27 - [5 + {28 - (29 - 7)}]

Answer

Solving,

27 - [5 + {28 - (29 - 7)}]

= 27 - [5 + {28 - 22}]

= 27 - [5 + 6]

= 27 - 11

= 16

∴ 27 - [5 + {28 - (29 - 7)}] = 16

Evaluate :

48 - [18 - {16 - (5 - $\overline{4 + 1}$)}]

Answer

Solving,

48 - [18 - {16 - (5 - $\overline{4 + 1}$)}]

= 48 - [18 - {16 - (5 - 5)}]

= 48 - [18 - {16 - 0}]

= 48 - [18 - 16]

= 48 - 2

= 46

∴ 48 - [18 - {16 - (5 - $\overline{4 + 1}$)}] = 46

Evaluate :

-8 - {-6 (9 - 11) + 18 ÷ -3}

Answer

Solving,

-8 - {-6 (9 - 11) + 18 ÷ -3}

= -8 - {-6 × (-2) + (-6)}

= -8 - {12 - 6}

= -8 - 6

= -14

∴ -8 - {-6 (9 - 11) + 18 ÷ -3} = -14

Evaluate :

(24 ÷ $\overline{12 - 9}$ - 12) - (3 × 8 ÷ 4 + 1)

Answer

Solving,

(24 ÷ $\overline{12 - 9}$ - 12) - (3 × 8 ÷ 4 + 1)

= (24 ÷ 3 - 12) - (24 ÷ 4 + 1)

= (8 - 12) - (6 + 1)

= (-4) - 7

= -11

∴ (24 ÷ $\overline{12 - 9}$ - 12) - (3 × 8 ÷ 4 + 1) = -11

Find the difference between 8 and -8.

Answer

The difference between two integers 8 and -8 :

Difference = 8 - (-8) = 8 + 8 = 16

or

Difference = -8 - 8 = -16

∴ The difference between 8 and -8 is 16 or -16.

Subtract the sum of -107 and 72 from the sum of 55 and -32.

Answer

Sum of 55 and -32 = 55 + (-32) = 23

Sum of -107 and 72 = (-107) + 72 = -35

Required result = 23 - (-35) = 23 + 35 = 58

∴ The required result is 58.

Write three consecutive integers

(i) succeeding -14

(ii) preceding -22

(iii) which are even and succeed 24

(iv) which are odd and precede 8

Answer

(i) Three consecutive integers succeeding -14 are :

-13, -12, -11

(ii) Three consecutive integers preceding -22 are :

-23, -24, -25

(iii) Three consecutive even integers succeeding 24 are :

26, 28, 30

(iv) Three consecutive odd integers preceding 8 are :

7, 5, 3

Points A, B, C and D are marked on a number line as shown below :

Find :

(i) A - B
(ii) D + C
(iii) C + A
(iv) B - C

Answer

From the number line, A = 3, B = 8, C = -9 and D = -3.

(i) A - B = 3 - 8 = -5

(ii) D + C = (-3) + (-9) = -12

(iii) C + A = (-9) + 3 = -6

(iv) B - C = 8 - (-9) = 8 + 9 = 17

At a place, the temperature on Monday was 30°C. It rose by 3°C on Tuesday and then dropped by 8°C on Wednesday. Find the temperature of this place on

(i) Tuesday (ii) Wednesday

Answer

(i) Temperature on Tuesday = 30°C + 3°C = 33°C

∴ The temperature on Tuesday was 33°C.

(ii) Temperature on Wednesday = 33°C - 8°C = 25°C

∴ The temperature on Wednesday was 25°C.

How much does 35 exceed (-35) ?

Answer

Required = 35 - (-35) = 35 + 35 = 70

∴ 35 exceeds (-35) by 70.

How much is -12 less than 3 ?

Answer

Required = 3 - (-12) = 3 + 12 = 15

∴ -12 is less than 3 by 15.

Subtract (-34) from the sum of 57 and (-51).

Answer

Sum of 57 and (-51) = 57 + (-51) = 6

Required result = 6 - (-34) = 6 + 34 = 40

∴ The required result is 40.

Write a pair of negative integers whose difference is 8.

Answer

Consider the negative integers -2 and -10.

Difference = (-2) - (-10) = -2 + 10 = 8

∴ The required pair is (-2, -10).

Write a positive integer and a negative integer whose sum is -15.

Answer

Consider the positive integer 5 and the negative integer -20.

Sum = 5 + (-20) = -15

∴ The required pair is (5, -20).

Evaluate :

(i) (-2) × (-2) × (-5) × 7

(ii) (-1) × (-5) × (-4) × (-6)

Answer

(i) There are 3 negative integers (odd), so the product is negative.

(-2) × (-2) × (-5) × 7

= -(2 × 2 × 5 × 7)

= -140

∴ (-2) × (-2) × (-5) × 7 = -140

(ii) There are 4 negative integers (even), so the product is positive.

(-1) × (-5) × (-4) × (-6)

= 1 × 5 × 4 × 6

= 120

∴ (-1) × (-5) × (-4) × (-6) = 120

In a class test, containing 20 questions, 5 marks are given for every correct answer and -3 marks are given for each incorrect answer. A student of this class attempted all the questions out of which only 12 were correct. Find the score of this student.

Answer

Number of correct answers = 12

Number of incorrect answers = 20 - 12 = 8

Marks for correct answers = 12 × 5 = 60

Marks for incorrect answers = 8 × (-3) = -24

Total score = 60 + (-24) = 60 - 24 = 36

∴ The score of this student is 36.

Smitha starts moving from a point A and takes 20 steps towards North, each step being 40 cm in length. Then she moves by taking 30 steps towards South, each step being 28 cm long. If Smitha finally reaches at point B, find the distance between A and B.

Answer

Taking the direction towards North as positive and towards South as negative.

Let Smitha start from point A towards North and reach point C.

Distance moved towards North = 20 × 40 = 800 cm

Smitha now starts from point C towards South and reach point B.

Distance moved towards South = 30 × 28 = 840 cm

Net displacement from A = 800 + (-840) = -40 cm

The negative sign shows that B is 40 cm to the South of A.

∴ The distance between A and B is 40 cm.

The addition of -30, -15, -10, 20 and 15 is :

1. -20

2. 25

3. -15

4. 15

Answer

Solving,

(-30) + (-15) + (-10) + 20 + 15

= -55 + 35

= -20

Hence, option 1 is the correct option.

Two integers a and b are such that a ÷ b = -2; then (a, b) is :

1. (-8, -4)

2. (-2, 4)

3. (6, -3)

4. (10, 5)

Answer

Checking each option for a ÷ b = -2 :

For (6, -3) : 6 ÷ (-3) = -2

Hence, option 3 is the correct option.

15 ÷ (-5) - (2 × -3) is equal to :

1. 9

2. -9

3. 3

4. -3

Answer

Solving,

15 ÷ (-5) - (2 × -3)

= -3 - (-6)

= -3 + 6

= 3

Hence, option 3 is the correct option.

7 - 8 ÷ (-2) + 3 × (-4) is equal to :

1. 1

2. -13

3. 13

4. -1

Answer

Solving,

7 - 8 ÷ (-2) + 3 × (-4)

= 7 - (-4) + (-12)

= 7 + 4 - 12

= -1

Hence, option 4 is the correct option.

-[8 - {11 + 30 ÷ (4 - $\overline{7 - 5}$)}] is equal to :

1. -18

2. 18

3. 16

4. -14

Answer

Solving,

-[8 - {11 + 30 ÷ (4 - $\overline{7 - 5})$}]

= -[8 - {11 + 30 ÷ (4 - 2)}]

= -[8 - {11 + 30 ÷ 2}]

= -[8 - {11 + 15}]

= -[8 - 26]

= -[-18]

= 18.

Hence, option 2 is the correct option.

The number that must be subtracted from -2 to get -15 is :

1. -17

2. 17

3. -13

4. 13

Answer

Let the number to be subtracted be x.

-2 - x = -15

-x = -15 + 2

-x = -13

x = 13

Hence, option 4 is the correct option.

$\Bigg\lbrace -2 \div \left(-\dfrac{1}{2}\right)\Bigg\rbrace \Bigg\lbrace -3 \div \left(-\dfrac{1}{3}\right)\Bigg\rbrace\Bigg\lbrace -4 \div \left(-\dfrac{1}{4}\right)\Bigg\rbrace$ is equal to :

1. (-4) × (-9) × (-16)

2. 4 × 9 × 16

3. $\left(-\dfrac{1}{4}\right)\left(-\dfrac{1}{9}\right)\left(-\dfrac{1}{16}\right)$

4. $\dfrac{1}{4\times9\times16}$

Answer

$-2 \div \left(-\dfrac{1}{2}\right) = -2 × (-2) = 4 \\[1em] -3 \div \left(-\dfrac{1}{3}\right) = -3 × (-3) = 9 \\[1em] -4 \div \left(-\dfrac{1}{4}\right) = -4 × (-4) = 16$

So, the expression = 4 × 9 × 16

Hence, option 2 is the correct option.

(-45) × 0 + (-45) ÷ 0 is equal to :

1. 1

2. 0

3. -90

4. none of these

Answer

The term (-45) ÷ 0 involves division by zero, which is not defined. Hence, the value of the expression is not defined.

Hence, option 4 is the correct option.

5 ÷ [5 + {5 - (5 - $\overline{5 - 5}$)}] is :

1. 0

2. 1

3. -1

4. 5

Answer

5 ÷ [5 + {5 - (5 - $\overline{5 - 5})$}]

= 5 ÷ [5 + {5 - (5 - 0)}]

= 5 ÷ [5 + {5 - 5}]

= 5 ÷ [5 + 0]

= 5 ÷ 5

= 1.

Hence, option 2 is the correct option.

$12 \div \dfrac{2}{9} \times 2\dfrac{1}{4}$ is :

1. $\dfrac{32}{9}$

2. $\dfrac{243}{2}$

3. 8

4. none of these

Answer

$12 \div \dfrac{2}{9} \times 2\dfrac{1}{4} \\[1em] = 12 \times \dfrac{9}{2} \times \dfrac{9}{4} \\[1em] = \dfrac{12 \times 9 \times 9}{2 \times 4} \\[1em] = \dfrac{972}{8} \\[1em] = \dfrac{243}{2}$

Hence, option 2 is the correct option.

(-3) × [5 + (-9)] is equal to :

1. -12

2. 12

3. 6

4. -6

Answer

Solving,

(-3) × [5 + (-9)]

= (-3) × (-4)

= 12

Hence, option 2 is the correct option.

Liquid at 15°C is cooled to temperature -5°C, the fall in temperature is :

1. 10°C

2. -10°C

3. 20°C

4. -20°C

Answer

Fall in temperature = 15°C - (-5°C) = 15°C + 5°C = 20°C

Hence, option 3 is the correct option.

The number 25 is decreased by 6 in every second. The number by which 25 will decrease in 6 seconds :

1. 5

2. -5

3. 30

4. 36

Answer

Decrease in 6 seconds = 6 × 6 = 36

Hence, option 4 is the correct option.

A = -9, B = -12 and C = 10; then A + B - C is equal to :

1. 31

2. -31

3. -11

4. 11

Answer

Solving,

A + B - C

= (-9) + (-12) - 10

= -21 - 10

= -31

Hence, option 2 is the correct option.

Statement 1 : The additive inverse of (-a) is a.

Statement 2 : Multiplicative inverse of an integer a is $\dfrac{1}{a}$.

Which of the following options is correct ?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

Statement 1 : For any integer, the sum of the integer and its additive inverse is 0. Since (-a) + a = 0, the additive inverse of (-a) is a. Hence, Statement 1 is true.

Statement 2 : The multiplicative inverse exists only for non-zero integers, not for any integer (it does not exist for a = 0). Hence, Statement 2 is false.

Hence, option 3 is the correct option.

Statement 1 : 16 ÷ (8 ÷ 4) = (16 ÷ 8) ÷ 4

Statement 2 : Subtraction of integers is not associative.

Which of the following options is correct ?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

Statement 1 : L.H.S. = 16 ÷ (8 ÷ 4) = 16 ÷ 2 = 8 and R.H.S. = (16 ÷ 8) ÷ 4 = 2 ÷ 4 = $\dfrac{1}{2}$.

Since 8 ≠ $\dfrac{1}{2}$, Statement 1 is false.

Statement 2 : Subtraction of integers is not associative, i.e. a - (b - c) ≠ (a - b) - c in general. Hence, Statement 2 is true.

Hence, option 4 is the correct option.

Assertion (A) : 155 × 277 + 155 × (-45) = 155 × [277 + (-45)]

Reason (R) : Distributive property holds for any three integers. i.e. a × (b + c) = a × b + a × c, for all integers a, b, c.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Assertion (A) is the distributive property applied to 155, 277 and (-45), so it is true.

Reason (R) correctly states the distributive property, which holds for all integers, so it is true. Also, R is the correct explanation of A.

So, both A and R are true.

Hence, option 3 is the correct option.

Assertion (A) : (-59) × (-47) ≠ (47) × (59)

Reason (R) : For any integers a and b, a × b = b × a

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Assertion (A) : (-59) × (-47) = 59 × 47 = 2773 and (47) × (59) = 2773. Since both are equal, the statement (-59) × (-47) ≠ (47) × (59) is false. So A is false.

Reason (R) : Multiplication of integers is commutative, i.e. a × b = b × a. So R is true.

So, A is false and R is true.

Hence, option 2 is the correct option.

Assertion (A) : The product of two integers is -775. If one of them is 25, the other integer is 31.

Reason (R) : If a and b are two integers, then a ÷ b or b ÷ a may or may not be an integer.

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true.

4. Both A and R are false.

Answer

Assertion (A) : The other integer = (-775) ÷ 25 = -31, not 31. So A is false.

Reason (R) : For two integers a and b, the quotient a ÷ b or b ÷ a may or may not be an integer. So R is true.

So, A is false and R is true.

Hence, option 2 is the correct option.