Operations with Integers

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 = −13

If the first movement is −4 and the final position is 5, what is the second movement?

If there are multiple strikes causing movements in the order 1, −2, 3, −4, ..., −10, what is the final position of the coin?

Using tokens, argue out the following statements: (a) 7 − 18 = 7 + (−18) (additive inverse of 18 is −18) (b) 4 − (−12) = 4 + 12 (additive inverse of −12 is 12)

Using the token interpretation, find the values of: (a) 3 × (−2) (b) (−5) × (−2) (c) (−4) × (−1) (d) (−7) × 3

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

Try to frame a simple rule to multiply two integers.

Find the following products: (a) 4 × (−3) (b) (−6) × (−3) (c) (−5) × (−1) (d) (−8) × 4 (e) (−9) × 10 (f) 10 × (−17)

Observe the following pairs of multiplications (fill in the blanks where needed): | 3 × −4 = −12 | −$4 \times 3$ = −12 | | −$30 \times 12$ = ___ | 12 × −30 = ___ | | −15 × −8 = 120 | −8 × −15 = 120 | | 14 × −5 = −70 | −5 × ___ = −70 |

Find the values of: (a) 14 × (−15) (b) −16 × (−5) (c) 36 ÷ (−18) (d) (−46) ÷ (−23)

A freezing process requires that the room temperature be lowered from 32°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?

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 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.

Replace the blank with an integer to make a true statement. (a) (−3) × ___ = 27 (b) 5 × ___ = (−35) (c) ___ × (−8) = (−56) (d) ___ × (−12) = 132 (e) ___ ÷ (−8) = 7 (f) ___ ÷ 12 = −11

Find the values of the following expressions: (a) (−5) × (18 + (−3)) (b) (−7) × 4 × (−1) (c) (−2) × (−1) × (−5) × (−3)

Find the values of the following expressions: (a) (−27) ÷ 9 (b) 84 ÷ (−4) (c) (−56) ÷ (−2)

Find the integer whose product with (−1) is: (a) 27 (b) −31 (c) −1 (d) 1 (e) 0

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.

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

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?

Imagine you're in a place where the temperature drops by 5°C each hour. If the temperature is currently at 8°C, write an expression which denotes the temperature after 4 hours.