Question 11 Mark
In a test, $+3$ marks are given for every correct answer and $-1$ mark are given for every incorrect answer. Sona attempted all the questions and scored $+20$ marks though she got $10$ correct answers. How many questions were given in the test$?$
AnswerLet $x$ be the correct answers and $y$ be the incorrect answers, given by Sona. It is given that, if she gives $10$ correct answers and her score is $20.$ Since, for every correct answer, $+3$ is given and for every incorrect answer, $-1.$ is given. Hence, Total number od question $=$ Correct anwser + Incorrect answer $= x + y = 10 + 10 = 20$
View full question & answer→Question 21 Mark
$(-157) \times (-19) + 157 =$ _____.
Answer$(-157) \times (-19) + 157 = (-1) \times (157) \times (-19) + 157.$
Solution:
$(-157) \times (-19) + 157 = (-1) \times (157) \times (-19) + 157 = 157\{-(-19) + 1\}$
$ [$taking $157$ as common$] = 157\{19 + 1\} = 157 \times 20 = 3140$
View full question & answer→Question 31 Mark
$a ÷ (-b) = - (a ÷ b).$
AnswerDivision of a negative integer and a positive integer is always a negative integer $\frac{\text{a}}{\text{-b}}=\frac{\text{-b}}{\text{a}}=-\big(\frac{\text{a}}{\text{b}}\big)$ i.e where, $a$ and $b$ are integers.
View full question & answer→Question 41 Mark
What’s the Error? Ramu evaluated the expression $-7 - (-3)$ and came up with the answer $-10.$ What did Ramu do wrong?
AnswerRamu went wrong in solving $- (-3)$ and took it as $-3$ only. Correct answer $= -7 - (-3) = -7 + 3 = -4$
View full question & answer→Question 51 Mark
If $x, y$ and $z$ are integers then $(x+\_\_\_\_\_ ) + z = \_\_\_\_\_ + (y + \_\_\_\_\_).$
AnswerIf $x, y$ and $z$ are integers then $(x + y) + z = x + (y + z).$
Solution:
Addition is associative for integers, $i.e. (a + b) + c = a + (b + c).$
View full question & answer→Question 61 Mark
Product of two negative integers is a negative integer.
AnswerProduct of two negative integers is a positive integer, i.e. $(-a) × (-b) = ab$ where, $a$ and $b$ are two positive integers.
View full question & answer→Question 71 Mark
$a ÷ b = b ÷ a.$
AnswerDivision is not commutative for integers, i.e. $\text{a}\div\text{b}\neq\text{b}\div\text{a}$ where, $a$ and $b$ are integers.
View full question & answer→Question 81 Mark
$-3 × 3 = -12 - (-3)$
Answer$\because LHS = (-3) \times 3 = (-9) $
$RHS = (-12) - (-3) = (-12) + 3 = (-9)$
Hence, $LHS = RHS.$
View full question & answer→Question 91 Mark
When $-16$ is divided by _____ the quotient is $4$
AnswerWhen $-16$ is divided by negative integer the quotient is $4$
Solution:
When $(-16)$ is divided by negative integer, i.e. $-4$ the quotient is $4$ as both signs are cancelled out.
View full question & answer→Question 101 Mark
$(-9) \times 20 =$ _____.
Answer$(-9) × 20 = -180.$
Solution:
$(-9) × 20 = -180 [\because$ in multiplication of integers, if both the numbers have different signs, then the result is a negative number$].$
View full question & answer→Question 111 Mark
$[(-32) ÷ 8 ] ÷ 2 = -32 ÷ [ 8 ÷ 2]$
Answer$\because\text{LHS}=[(-32)\div8]\div2$
$= \bigg[\frac{-32}{8}\bigg]+2=-4+2=-2$
$\text{and}\ \text{RHS}=(-32)\div[8\div2]$
$\bigg[\frac{-32}{8}\bigg]+2=-4+2=-2$
$\text{and}\ \text{RHS}=(-32)\div[8\div2]$
$=(-32)\div\bigg[\frac{8}{2}\bigg]$
$=(-32)\div4=\frac{(-32)}{4}=-8$
Hence, $LHS \neq RHS.$
View full question & answer→Question 121 Mark
$(-20) × (5 - 3) = (-20) × (-2)$
Answer$\because LHS = (-20) \times (5 - 3) = (-20) \times 2 = (-40)$
$RHS = (-20) \times (-2) = 40$
Hence, $LHS = RHS.$
View full question & answer→Question 131 Mark
$(-69) ÷ (69) =$ _____.
Answer$(-69) ÷ (69) = (-1)$
View full question & answer→Question 141 Mark
$65 ÷ (-13) =$ _____.
Answer$65\div(-13)\ 65\times\frac{1}{(-13)}[\because$ division is inverse of multiplication$] = -5$
View full question & answer→Question 151 Mark
When we divide a negative integer by a positive integer, we divide them as whole numbers and put _____ a sign before quotient.
AnswerWhen we divide a negative integer by a positive integer, we divide them as whole numbers and put negative a sign before quotient.Solution:
When we divide a negative integer by a positive integer or a positive integer by a negative integer, we divide them as whole numbers and put a negative sign before quotient.
View full question & answer→Question 161 Mark
_____ $\times (-23) = -920$
Answer$(40) \times (-23) = -920$
$\bigg[\because\bigg(\frac{-920}{-23}\bigg)=40\bigg]$
View full question & answer→Question 171 Mark
_____ $\times (-93) = 93$
Answer$(-1) \times (-93) = 93$
$\bigg[\because\frac{93}{-93}=(-1)\bigg]$
View full question & answer→Question 181 Mark
Integers are closed under division.
AnswerFalse. Solution: Because, when we divide two integers, we may or may not get an integer. e.g. $\frac{2}{1}=2(\text{integer})$ and $\frac{2}{3}(\text{not an integer}).$
View full question & answer→Question 191 Mark
$88 \times $_____ $= -88$
Answer$88 \times (-1) = -88$
$\bigg[\because\frac{-88}{88}=(-1)\bigg]$
View full question & answer→Question 201 Mark
$(-225) ÷ 5 =$ _____.
Answer$(-225) ÷ 5$
$=-225\times\frac{1}{5}[\because$ division is inverse of multiplication$] = -45$
View full question & answer→Question 211 Mark
If we multiply six negative integers and six positive integers, then the resulting integer is _____.
Answer If we multiply six negative integers and six positive integers, then the resulting integer is positive. Solution:
If we multiply six negative integers and six positive integers, then the resulting integer is positive, because even numbers of negative integers, in multiplication becomes positive. View full question & answer→Question 221 Mark
$(a + b) \times c = a \times c + a \times b.$
AnswerIntegers show distributive property of multiplication over addition, i.e.
$a \times (b + c)$
$= a \times b + a \times c$
where a, b and c are integers.
View full question & answer→Question 231 Mark
_____ is the multiplicative identity for integers.
Answer$1$ is the multiplicative identity for integers,
i.e. $a \times 1 = 1 \times a = a$ for any integer a.
View full question & answer→Question 241 Mark
The successor of $0 \times (-25)$ is $1 \times (-25)$
AnswerWe know that, successor means adding $1$ to the given number.
Here, given number is $0 × (-25) = 0 [$on multiplying by $0$ to any number the result is zero$]$
Hence, the successor of $0 =0+1$ $=1\ \text{but}\ 1$ $\neq\ 1\times(-25)$
View full question & answer→Question 251 Mark
$4 \times (-5) = (-10) \times (-2)$
Answer$\because LHS = 4 \times (-5) = -20$
$RHS = (-10) \times (-2) = 20$
Hence, $LHS = RHS.$
View full question & answer→Question 261 Mark
When we multiply two integers their product is always greater than both the integers
Answere.g. Let two integers are $(-5)$ and $2$
So, $(-5) \times 2 = -10 \Rightarrow (-10) < (-5)$ and $(-10) < 2$
View full question & answer→Question 271 Mark
$(-23) + 47$ is same as $47 + (-23)$
AnswerBecause, addition is commutative,
i.e. $a + b = b + a$
$\Rightarrow (-23) + 47$
$= 47 + (-23)$
View full question & answer→Question 281 Mark
Integers are closed under multiplication.
AnswerTrue. Solution: If we multiply two integers, we get an integer.
View full question & answer→Question 291 Mark
Difference of two negative integers cannot be a positive integer.
Answere.g. Taking two negative integers, i.e. $-4$ and -$5 - 4 -(-5) = -4 + 5 = 1$ [positive integer]
View full question & answer→Question 301 Mark
$[(-8) + $______ $] + $________$ = $________$ + [(-3) + $________ $] = -3$
Answer$[(-8) + (-3)] + 8 = (-8) + [(-3) + 8]$
$[\because$ addition is associative, i.e. $a + (to + c) = (a + b) + c] = -8 + 5 = -3$
View full question & answer→Question 311 Mark
_____$ \times (-1) = -35$
Answer$-35 \times (-1) = -35$
$\bigg[\because\bigg(\frac{-35}{-1}\bigg)=35\bigg]$
View full question & answer→Question 321 Mark
When we change the order of integers, their sum remains the same.
AnswerBecause, sum of two integers is commutative, i.e. $a + b = b + a$ for two integers $a$ and $b.$
View full question & answer→Question 331 Mark
$113 ÷$_____$ = -1$
Answer$113 ÷ (-113) = -1$
$\big[\because\frac{113}{-1}=(-113)$
View full question & answer→Question 341 Mark
$(-5) \times (-6) \times (-7) =$ _____.
AnswerOdd negative integers make the resultant integer, negative.
$(-5) \times (-6) \times (-7) = 30 \times (-7) = -210.$
View full question & answer→Question 351 Mark
Multiplication is not commutative for integers.
AnswerMultiplication is commutative for integers, i.e. $a × b = b × a$ for any two integers $a$ and $b.$
View full question & answer→Question 361 Mark
$a - b = b - a.$
AnswerSubtraction is not commutative for integers, i.e. $\text{a}-\text{b}\neq\text{b}-\text{a}$ where, $a$ and $b$ are integers.
View full question & answer→Question 371 Mark
We can write a pair of integers whose sum is not an integer.
AnswerFalse. Solution: Because, sum of two integers, is always be an integer.
View full question & answer→Question 381 Mark
We get additive inverse of an integer a when we multiply it by _____.
AnswerAdditive inverse of an integer is the same integer value, with opposite sign. So, we get additive inverse of integer a, when we multiply it by $(-1).$
View full question & answer→Question 391 Mark
_____ $\div (-1 ) = -83$
Answer$83 ÷ (-1 ) = -83$
$\big[\because\frac{-83}{-1}=83\big]$
View full question & answer→Question 401 Mark
$(-95) ÷ $_____$ = 95$
Answer$(-95) ÷ (-1) = 95$
$\big[\because\frac{-95}{95}=(-1)\big]$
View full question & answer→Question 411 Mark
_____ $\times (-1) = 47$
Answer$(-47) \times (-1) = 47$
$\bigg[\because\frac{47}{-1}=(-47)\bigg]$
View full question & answer→Question 421 Mark
$(-23) \times (42) = (-42) \times $_____.
Answer$(-23) \times (42) = (-1) \times (23) \times (42) = (-1) \times (42) \times (23)$
$[\because$ multiplication is commutative, i.e. $a \times b = b \times a] = (-42) \times (23)$
View full question & answer→Question 431 Mark
On the following number line, $(-4) \times 3$ is represented by the point _____.
Answer$(-4) \times 3 = (-12)$ On the number line, each division has epual spacing pf $2$ units.
So, $A = -20 + 2 = -18 $
$B = -18 + 2 = -16 $
$C = -16 + 2 = -14 $
$D = -14 + 2 = -12$
Hence, $(-4) \times 3$ is represented by the point $D.$
View full question & answer→Question 441 Mark
$11 × (-5) = -( \_\_\_\_\_ × \_\_\_\_\_ ) = \_\_\_\_\_.$
AnswerWe can write the equation as, $11 × (-5) = -(11 × 5) = -55$
View full question & answer→Question 451 Mark
$99 \times 101$ can be written as $(100 - 1) \times (100 + 1).$
Answer$\because 99 \times 101 = 9999$ and $(100 - 1) \times (100 + 1)$
$= 100 \times (100 + 1) -1 \times (100 + 1)$
$= 100 \times 100 + 1 \times 100 - 1 \times 100 -1 \times 1 [$using distributive property$]$
$= 10000 + 100 - 100 -1 = 9999$
View full question & answer→Question 461 Mark
$3 × (-1 ) × (-15) = $_____.
AnswerTwo negative integers and one positive integer make the resultant integer, positive.
$3 \times (-1) \times (-15) = (-3) \times (-15)= 45c.$
View full question & answer→Question 471 Mark
If $a, b, c$ are integers and $b \neq 0$ then, $a \times (b - c) = a \times b - a \times c$
AnswerMultiplication can be distributive over subtraction, i.e. $a \times (b - c) = a \times b - a \times c.$
View full question & answer→Question 481 Mark
$(-28) ÷ (-28) =$ _____.
Answer$(-28) ÷ (-28) = 1$
View full question & answer→Question 491 Mark
$51 ÷$ _____$ = -51$
Answer$51 ÷ (-1) = -51$
$\big[\because\frac{51}{-51}=(-1)\big]$
View full question & answer→Question 501 Mark
$(-a) + b = b\ +$ Additive inverse of _____.
AnswerAdditive inverse is the negation of a number. As we know, addition is commutative for integers, i.e. $-a + b = b + (-a)$
Now $‘- a’$ is the additive inverse of $a.$
So, a will be the answer.
View full question & answer→Question 511 Mark
While multiplying a positive integer and a negative integer, we multiply them as _____ numbers and put a _____ sign before the product.
Answer While multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a negative sign before the product. Solution:
When multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a negative sign before the product. View full question & answer→Question 521 Mark
$(-8) + (-8) + (-8) = \_\_\_\_\_ \times (-8)$
AnswerLet $x$ be the missing number.
Then, $-8 - 8 - 8 = x \times (-8)$
$\Rightarrow-24=\text{x}\times(-8)$
$\Rightarrow\frac{-24}{-8}=\text{x}$
$\Rightarrow\text{x}=3$
Hence, $(-8) + (-8) + (-8) = 3 \times (-8)$
View full question & answer→Question 531 Mark
$[(-8) + \_\_\_\_\_\_ ] + \_\_\_\_\_\_\_\_ = \_\_\_\_\_\_\_\_ + [(-3) + \_\_\_\_\_\_\_\_ ] = -3$
Answer$[(-8) + (-3)] + 8 = (-8) + [(-3) + 8]$
$[\because$ addition is associative, i.e. $a + (to + c) = (a + b) + c] = -8 + 5 = -3$
View full question & answer→Question 541 Mark
$a \times b = b \times a.$
AnswerMultiplication is commutative for integers, i.e. $a \times b = b \times a$ where, $a$ and $b$ are integers.
View full question & answer→Question 551 Mark
$(-9) + (-11)$ is greater than $(-9) - (-11).$
Answer$(-9) + (-11) = -9 - 11 = -20$ and $(-9) - (-11) = -9 + 11 = 2$ So, $(-9) - (-11)$ is greater than $(-9) + (-11).$
View full question & answer→Question 561 Mark
$[12 \times (-7)] \times 5 = $_____$ \times [(-7) \times $_____$ ]$
AnswerMultiplication is associative for integers, i.e. $(a \times b) \times c = a \times (b \times c)$
So, $[12 \times (-7)] \times 5 = 12 \times [(-7) \times 5]$
View full question & answer→Question 571 Mark
Integers are closed under subtraction.
AnswerTrue. Solution: Because, if we subtract two integers we get another integer.
View full question & answer→Question 581 Mark
$(-100) ÷ (-10) = $______.
Answer$(-100)\div(-10)$ $=(-100)\times\frac{1}{(-10)}[\because$ division is inverse of multiplication$] = (-10)$
View full question & answer→Question 591 Mark
$5 - (-8)$ is same as $5 + 8$
View full question & answer→Question 601 Mark
The sum of an integer and its additive inverse is zero $(0).$
AnswerAdditive inverse is the number, that when added to a given number yields zero.
View full question & answer→Question 611 Mark
$23 \times (-99) =$_____$\times (-100 +$_____$ ) = 23 \times $_____$ + 23 \times $_____.
AnswerWe can write the equation as, $23 \times (-99) = 23 \times (-100 + 1) = 23 \times (-100) + 23 \times 1$
$ [\because$ integers show distributive property of multiplication over addition,
i.e. $a \times (b + c) = a \times b + a \times c]$
View full question & answer→Question 621 Mark
_____$ ÷ (-1) = 75$
Answer$-75 ÷ (-1) = 75$
$\big[\because\frac{75}{-1}=(-75)$
View full question & answer→Question 631 Mark
What’s the Error? Reeta evaluated $-4\ + d$ for $d = -6$ and gave an answer of $2.$ What might Reeta have done wrong$?$
AnswerReeta went wrong in solving $+\ (-6)$ and took it as $+ 6.$ Correct answer $= -4 + d = -4 + (-6) = -4 - 6 = -10$
View full question & answer→Question 641 Mark
$(-237) \times 0$ is same as $0 \times (-39)$
AnswerWhen we multiply a number with $0,$
we always get $0$
$\Rightarrow (-237) \times 0 = 0$
View full question & answer→Question 651 Mark
$(-5) \times (33) = 5 \times (-33)$
Answer$\therefore LHS = (-5) \times 33 = (-165)$ and $RHS = 5 \times (-33) = (-165)$ Hence, $LHS = RHS$.
View full question & answer→Question 661 Mark
Product of a negative integer and a positive integer is a positive integer.
AnswerProduct of a negative integer and a positive integer is a negative integer, i.e.
$a \times (-b)$
$= -ab$
where, $a$ and $b$ are two positive integers.
View full question & answer→Question 671 Mark
When we change the order of integers their difference remains the same.
AnswerSubtraction of two integers is not commutative,
i.e. $\text{a}-\text{b}\neq\text{b}-\text{a}$ for two integers $a$ and $b.$
View full question & answer→Question 681 Mark
Division is the inverse operation of _____.
AnswerDivision is the inverse operation of multiplication.Solution:
Division is the inverse operation of multiplication.
View full question & answer→Question 691 Mark
$(-25) \times (-2) = $_____.
AnswerTwo negative integers make the resultant integer,
positive. $(-25) \times (-2) = 50$
View full question & answer→Question 701 Mark
$(-43) +$_____$ = -43$
AnswerZero $(0)$ is an additive identity for integers,
i.e. $a + 0 = 0 + a = a$ for any integer $a.$
View full question & answer→Question 711 Mark
Social Studies Application: Remembering that $1AD$ came immediately after $1BC$, while solving these problems take $1BC$ as $-1$ and $1AD$ as $+1$ Greek mathematician Archimedes lived between $287BC$ and $212BC$ and Aristotle lived between $380BC$ and $322BC.$ Who lived during an earlier period$?$
AnswerAristotle lived in an earlier period, as $380BC$ and $322BC$ is earlier than $287BC$ and $212BC.$
View full question & answer→Question 721 Mark
If we multiply five positive integers and one negative integer, then the resulting integer is _____.
AnswerIf we multiply $5$ positive integers and one negative integer, then the resulting integer is negative.
View full question & answer→Question 731 Mark
$a ÷ (-1) = -a.$
Answer$a + (-1) = \frac{\text{a}}{(-1)} = -a$
$[$as division of a negative and positive integer is always negative$]$
View full question & answer→Question 741 Mark
Going $500\ m$ towards east first and then $200\ m$ back is same as going $200\ m$ towards west first and then going $500\ m$ back.
AnswerCase I Going $500\ m$ towards East first, i.e. point $A$ to $B$ and then $200\ m$ back, i.e. $B$ to $C.$
As per the above figure shown, final position is $C,$ i.e. $300\ m$ in East. View full question & answer→Question 751 Mark
Product of three negative integers is a negative integer.
AnswerProduct of three negative integers is a negative integer, i.e. $(-a) × (-0) × (-c) = (-abc)$ where, $a,b$ and $c$ are three positive integers.
View full question & answer→Question 761 Mark
________ $÷ (-10) = 0$
AnswerDivision of $0$ by any number, results as zero. So, the answer is $0$.
View full question & answer→Question 771 Mark
$(-1) \times (-2) \times (-3) = 1 \times 2 \times 3$
Answer$\because LHS = (-1) \times (-2) \times (-3) = (-6)$
$RHS = 1 \times 2 \times 3 = 6$
Hence, $LHS = RHS.$
View full question & answer→Question 781 Mark
Multiplication fact $(-8) \times (-10) = 80$ is same as division fact $80 ÷ (-8) = (-10)$
Answer
| Multiplication fact |
Division fact |
| $(-8) × (-10) = 80$ |
$80 + (-8) = (-10)$ |
| $LHS = (-1) × 8 × (-1) × 10 = (-1) (-1) × 8 × 10$ |
$LHS = 80 + (-8) = \frac{80}{-8}$ |
| $= 1 × 80 = 20 = RHS.$ |
$= (-10) = RHS.$ |
View full question & answer→Question 791 Mark
$(-19) \times (-11) = 19 \times 11$
AnswerProduct of two negative integers is a positive integer,
i.e. $(-a) \times (-b) = a \times b$ where, $a$ and $b$ are positive integers.
$\Rightarrow LHS = (-19) \times (-11) = 209 $
$RHS = 19 \times 11 = 209$
Hence, $LHS = RHS.$
View full question & answer→Question 801 Mark
If we multiply _____ number of negative integers, then the resulting integer is positive.
AnswerIf we multiply even number of negative integers, then the resulting integer is positive.Solution:
If we multiply even numbers of negative integers, then the resulting integer is positive.
View full question & answer→Question 811 Mark
$(-40) \times $_____$ = 80$
Answer$(-40) \times (-2) = 80$
$\bigg[\because\frac{93}{-40}=(-2)\bigg]$
View full question & answer→Question 821 Mark
Sum of two negative integers always gives a number smaller than both the integers
Answere.g. Taking two negative integers, i.e. $(-5)$ and $(-3) + (-5) + (-3) = -5 - 3 = -8 = -8 < -5$ and $-8 < -3$
View full question & answer→