Question 1

Q1: Let us try to find a few more pairs of numbers from their sums and differences: (a) Sum = 27, Difference = 9 (b) Sum = 4, Difference = 12 (c) Sum = 0,...

Accepted Answer

To Find: Pairs of numbers given their sum and difference. Let: The two numbers be $x$ and $y$. Formulas: Given the sum ($S$) and difference ($D$): $$x + y = S$$ $$x - y = D$$ Adding the two equations gives $2x = S + D$, so $x = \frac{S+D}{2}$. Subtracting the second equation from the first gives $2...

Question 2

Q1: Find the values of: (a) $14 \times (-15)$ (b) $-16 \times (-5)$ (c) $36 \div (-18)$ (d) $(-46) \div (-23)$

Accepted Answer

To Find: The values of the given expressions. (a) $14 \times (-15)$ $$14 \times (-15) = -(14 \times 15) = -210$$ Final Answer: $-210$ (b) $-16 \times (-5)$ $$(-16) \times (-5) = +(16 \times 5) = 80$$ Final Answer: $80$ (c) $36 \div (-18)$ $$36 \div (-18) = -(36 \div 18) = -2$$ Final Answer:...

Question 3

Q2: A freezing process requires that the room temperature be lowered from $32^{\circ} \mathrm{C}$ at the rate of $5^{\circ} \mathrm{C}$ every hour. Wh...

Accepted Answer

Given: Initial Temperature = $32^{\circ} \mathrm{C}$ Rate of temperature change = $-5^{\circ} \mathrm{C}$ per hour (since it is lowered) Time = 10 hours To Find: The room temperature after 10 hours. Solution: Total change in temperature = Rate of change $\times$ Time Total change = $(-5^{\circ...

Question 4

Q3: A cement company earns a profit of ₹8 per bag of white cement sold and a loss of ₹5 per bag of grey cement sold. [Represent the profit/ loss as intege...

Accepted Answer

Given: Profit on white cement = +₹8 per bag Loss on grey cement = -₹5 per bag (a) To Find: Total profit or loss Solution: Profit from 3,000 bags of white cement = $3000 \times 8 = \text{₹}24,000$ Loss from 5,000 bags of grey cement = $5000 \times (-5) = -\text{₹}25,000$ Total Profit/Loss = Profi...

Question 5

Q4: Replace the blank with an integer to make a true statement. (a) $(-3) \times \_\_ = 27$ (b) $5 \times \_\_ = (-35)$ (c) $\_\_ \times (-8) = (...

Accepted Answer

To Find: The integer that completes each statement. (a) $(-3) \times \_\_ = 27$ Let the integer be $x$. $(-3) \times x = 27 \implies x = 27 \div (-3) = -9$. Answer: -9 (b) $5 \times \_\_ = (-35)$ Let the integer be $x$. $5 \times x = -35 \implies x = -35 \div 5 = -7$. Answer: -7 (c) $\_...

Question 6

Q1: Find the values of the following expressions: (a) $(-5) \times (18+(-3))$ (b) $(-7) \times 4 \times (-1)$ (c) $(-2) \times (-1) \times (-5) \tim...

Accepted Answer

To Find: The values of the given expressions. (a) $(-5) \times (18+(-3))$ First, solve the expression inside the parentheses: $18 + (-3) = 15$. Then multiply: $(-5) \times 15 = -75$. Final Answer: -75 (b) $(-7) \times 4 \times (-1)$ Multiply from left to right: $(-7) \times 4 = -28$. Then: $(-2...

Question 7

Q2: Find the values of the following expressions: (a) $(-27) \div 9$ (b) $84 \div (-4)$ (c) $(-56) \div (-2)$

Accepted Answer

To Find: The values of the given expressions. (a) $(-27) \div 9$ A negative divided by a positive is a negative. $27 \div 9 = 3$. Final Answer: -3 (b) $84 \div (-4)$ A positive divided by a negative is a negative. $84 \div 4 = 21$. Final Answer: -21 (c) $(-56) \div (-2)$ A negative divided by a...

Question 8

Q1: Using the token interpretation, find the values of: (a) $3 \times (-2)$ (b) $(-5) \times (-2)$ (c) $(-4) \times (-1)$ (d) $(-7) \times 3$

Accepted Answer

To Find: The values of the given multiplications. (a) $3 \times (-2)$ This means placing 2 negative tokens into a bag 3 times. This results in $2+2+2=6$ negative tokens. Final Answer: $3 \times (-2) = -6$ (b) $(-5) \times (-2)$ This means removing 2 negative tokens from a bag 5 times. To do this,...

Question 9

Q2: If $123 \times 456 = 56088$, without calculating, find the value of: (a) $(-123) \times 456$ (b) $(-123) \times (-456)$ (c) $(123) \times (-456)$

Accepted Answer

Given: $123 \times 456 = 56088$ To Find: The values of the given expressions using the rules of integer multiplication. Rules: - The product of a negative and a positive integer is negative. - The product of two negative integers is positive. (a) $(-123) \times 456$ Since one integer is negative a...

Question 10

Q3: Try to frame a simple rule to multiply two integers.

Accepted Answer

Rule for Multiplying Two Integers: 1.  Find the magnitude: Multiply the absolute values of the two integers (ignoring their signs). 2.  Determine the sign:        If the two integers have the same sign (both are positive or both are negative), the product is positive.        If the two integers have...

Question 11

Q1: Find the following products. (a) $4 \times (-3)$ (b) $(-6) \times (-3)$ (c) $(-5) \times (-1)$ (d) $(-8) \times 4$ (e) $(-9) \times 10$ (f) $10 \...

Accepted Answer

To Find: The products of the given integers. (a) $4 \times (-3)$ $4 \times 3 = 12$. A positive times a negative is a negative. Final Answer: $-12$ (b) $(-6) \times (-3)$ $6 \times 3 = 18$. A negative times a negative is a positive. Final Answer: $18$ (c) $(-5) \times (-1)$ $5 \times 1 = 5$. A...

Question 12

Q1: Using tokens, argue out the following statements. (a) $7-18=7+(-18)$ (b) $4-(-12)=4+12$

Accepted Answer

To Argue: The given statements are true based on the concept of additive inverse. (a) $7-18=7+(-18)$ Argument: Subtracting a number is mathematically equivalent to adding its additive inverse. The additive inverse of an integer 'a' is '-a'. In this case, the number being subtracted is 18. Its additi...

Question 13

Q1: If the first movement is -4 and the final position is 5, what is the second movement?

Accepted Answer

Given: First Movement = -4 Final Position = 5 To Find: The second movement. Formula: Final Position = First Movement + Second Movement Solution: Let the second movement be $x$. $$5 = -4 + x$$ $$x = 5 - (-4)$$ $$x = 5 + 4$$ $$x = 9$$ Final Answer: The second movement is 9.

Question 14

Q2: If there are multiple strikes causing movements in the order $1, -2, 3, -4, \ldots, -10$, what is the final position of the coin?

Accepted Answer

Given: A series of movements: $1, -2, 3, -4, \ldots, 9, -10$. To Find: The final position of the coin, which is the sum of all movements. Solution: The total movement is the sum of the series: $S = 1 - 2 + 3 - 4 + 5 - 6 + 7 - 8 + 9 - 10$. We can group the terms in pairs: $$S = (1 - 2) + (3 - 4) + (...

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