Chapter 2 Operations with Integers (Class 7 - Latest Maths NCERT (Ganita Prakash II) Solutions)

Question. 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 $= 0$, Difference $= 10$

(d) Sum $= 0$, Difference $= -10$

(e) Sum $= -7$, Difference $= -1$

(f) Sum $= -7$, Difference $= -13$

Question 1. Using the token interpretation, find the values of:

(a) $3 \times (-2)$

(b) $(-5) \times (-2)$

(c) $(-4) \times (-1)$

(d) $(-7) \times 3$

Question 2. If $123 \times 456 = 56088$, without calculating, find the value of:

(a) $(-123) \times 456$

(b) $(-123) \times (-456)$

(c) $123 \times (-456)$

Question 3. Try to frame a simple rule to multiply two integers.

Consider the numbers represented by the following tokens:

(a)

(b)

(c)

Question. 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) $10 \times (-17)$

Question 1. Find the values of:

(a) $14 \times (-15)$

(b) $-16 \times (-5)$

(c) $36 \div (-18)$

(d) $(-46) \div (-23)$

Question 2. A freezing process requires that the room temperature be lowered from $32^\circ\text{C}$ at the rate of $5^\circ\text{C}$ every hour. What will be the room temperature $10$ hours after the process begins?

Question 3. A cement company earns a profit of $\textsf{₹} 8$ per bag of white cement sold and a loss of $\textsf{₹} 5$ per bag of grey cement sold. [Represent the profit/ loss as integers.]

(a) The company sells $3,000$ bags of white cement and $5,000$ bags of grey cement in a month. What is its profit or loss?

(b) If the number of bags of grey cement sold is $6,400$ bags, what is the number of bags of white cement the company must sell to have neither profit nor loss.

Question 4. Replace the blank with an integer to make a true statement.

(a) $(-3) \times \text{_____} = 27$

(b) $5 \times \text{_____} = (-35)$

(c) $\text{_____} \times (-8) = (-56)$

(d) $\text{_____} \times (-12) = 132$

(e) $\text{_____} \div (-8) = 7$

(f) $\text{_____} \div 12 = -11$

Question 1. Find the values of the following expressions:

(a) $(-5) \times (18 + (-3))$

(b) $(-7) \times 4 \times (-1)$

(c) $(-2) \times (-1) \times (-5) \times (-3)$

Question 2. Find the values of the following expressions:

(a) $(-27) \div 9$

(b) $84 \div (-4)$

(c) $(-56) \div (-2)$

Question 3. Find the integer whose product with $(-1)$ is:

(a) $27$

(b) $-31$

(c) $-1$

(d) $1$

(e) $0$

Question 4. If $47 - 56 + 14 - 8 + 2 - 8 + 5 = -4$, then find the value of $-47 + 56 - 14 + 8 - 2 + 8 - 5$ without calculating the full expression.

Question 5. Do you remember the Collatz Conjecture from last year? Try a modified version with integers. The rule is — start with any number; if the number is even, take half of it; if the number is odd, multiply it by $-3$ and add $1$; repeat. An example sequence is shown below.

Try this with different starting numbers: $(-21)$, $(-6)$, and so on. Describe the patterns you observe.

Question 6. In a test, $(+4)$ marks are given for every correct answer and $(-2)$ marks are given for every incorrect answer.

(a) Anita answered all the questions in the test. She scored $40$ marks even though $15$ of her answers were correct. How many of her answers were incorrect? How many questions are in the test?

(b) Anil scored $(-10)$ marks even though he had $5$ correct answers. How many of his answers were incorrect? Did he leave any questions unanswered?

Question 7. Pick the pattern — find the operations done by the machine shown below.

Question 8. Imagine you’re in a place where the temperature drops by $5^\circ\text{C}$ each hour. If the temperature is currently at $8^\circ\text{C}$, write an expression which denotes the temperature after $4$ hours.

Question 9. Find $3$ consecutive numbers with a product of (a) $-6$, (b) $120$.

Question 10. An alien society uses a peculiar currency called ‘pibs’ with just two denominations of coins — a $+13$ pibs coin and a $-9$ pibs coin. You have several of these coins. Is it possible to purchase an item that costs $+85$ pibs?

Yes, we can use $10$ coins of $+13$ pibs and $5$ coins of $-9$ pibs to make a total of $+85$. Using the two denominations, try to get the following totals:

(a) $+20$

(b) $+40$

(c) $-50$

(d) $+8$

(e) $+10$

(f) $-2$

(g) $+1$

[Hint: Writing down a few multiples of $13$ and $9$ can help.]

(h) Is it possible to purchase an item that costs $1568$ pibs?

Question 11. Find the values of:

(a) $(32 \times (-18)) \div ((-36))$

(b) $(32) \div ((-36) \times (-18))$

(c) $(25 \times (-12)) \div ((45) \times (-27))$

(d) $(280 \times (-7)) \div ((-8) \times (-35))$

Question 12. Arrange the expressions given below in increasing order.

(a) $(-348) + (-1064)$

(b) $(-348) - (-1064)$

(c) $348 - (-1064)$

(d) $(-348) \times (-1064)$

(e) $348 \times (-1064)$

(f) $348 \times 964$

Question 13. Given that $(-548) \times 972 = -532656$, write the values of:

(a) $(-547) \times 972$

(b) $(-548) \times 971$

(c) $(-547) \times 971$

Question 14. Given that $207 \times (-33 + 7) = -5382$, write the value of $-207 \times (33 - 7) = \text{_________}$.

Question 15. Use the numbers $3, -2, 5, -6$ exactly once and the operations ‘$+$’, ‘$-$’, and ‘$\times$’ exactly once and brackets as necessary to write an expression such that —

(a) the result is the maximum possible

(b) the result is the minimum possible

Question 16. Fill in the blanks in at least $5$ different ways with integers:

(a) $\text{_____} + \text{_____} \times \text{_____} = -36$

(b) $(\text{_____} - \text{_____}) \times \text{_____} = 12$

(c) $(\text{_____} - (\text{_____} - \text{_____})) = -1$