Integers

NCERT Exercise 1.2

Write down a pair of integers whose:

  1. Sum is – 7

    Answer: - 14 and 7
  2. Difference is – 10

    Answer: 55 and 65
  3. Sum is 0

    Answer: 24 and – 24

Answer the following:

  1. Write a pair of integers whose difference gives 8.

    Answer: 15 and 7
  2. Write a negative integer and a positive integer whose sum is – 5.

    Answer: - 15 and 10
  3. Write a negative integer and a positive integer whose difference is – 3.

    Answer: Such pair is not possible.

In a quiz, team A scored – 40, 10, 0 and team B scored 10, 0 – 40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order?

Answer: Score of team `A = - 10 + 10 + 0 = - 30`
Score of team `B = 10 + 0 – 40 = - 30`
Both teams have same score. This shows that we can add integers in any order.

Fill in the blanks to make the following statements true:

  1. `( - 6) + ( - 8) = - 8 + (………..)`

    Answer: - 6
  2. `– 53 + ……….. = - 53`

    Answer: 0
  3. `17 + …………… = 0`

    Answer: - 17
  4. `[13 + ( - 12)] + (…………) = 13 + [( - 12) + ( - 7)]`

    Answer: - 7
  5. `( - 4) + [15 + ( - 3)] = [ - 4 + 15] + …………….`

    Answer: - 3

Exercise 1.3

Find each of the following products:

  1. `3 xx ( - 1)`

    Answer: `3 xx ( - 1) = - 3`
  2. `( - 1) xx 225`

    Answer: `- 225`
  3. `( - 21) xx ( - 30)`

    Answer: `( - 21) xx ( - 30) = 630`
  4. `( - 316) xx ( - 1)`

    Answer: 316
  5. `( - 15) xx 0 xx ( - 18)`

    Answer: Zero because when any number is multiplied by zero the result is zero.
  6. `( - 12) xx ( - 11) xx (10)`

    Answer: `( - 12) xx ( - 11) xx (10) = 132 xx 10 = 1320`
  7. `9 xx ( - 3) xx ( - 6)`

    Answer: `9 xx ( - 3) xx ( - 6) = 9 xx 18 = 162`
  8. `( - 18) xx ( - 5) xx ( - 4)`

    Answer: `( - 18) xx ( - 5) xx ( - 4) = 90 xx ( - 4) = - 360`
  9. `( - 1) xx ( -2) xx ( - 3) xx 4`

    Answer: `( - 1) xx ( - 2) xx ( - 3) xx 4 = 2 xx ( - 12) = - 24`
  10. `( - 3) xx ( - 6) xx ( - 2) xx ( - 1)`

    Answer: `( - 3) xx ( - 6) xx ( - 2) xx ( - 1) = 18 xx 2 = 36`

Verify the following:

  1. `18 xx [7 + ( - 3)] ``= [18 xx 7] + [18 xx ( - 3)]`

    Answer: LHS `= 18 xx [7 + ( - 3)] ``= 18 xx [ - 4] = - 72`
    RHS `= [18 xx 7] + [18 xx ( - 3)] ``= 126 + [ - 54] = - 72`
    Here; LHS = RHS
  2. `( - 21) xx [( - 4) + ( - 6)] ``= [( - 21) xx ( -4)] + [( - 21) xx ( - 6)]`

    Answer: LHS `= ( - 21) xx [( - 4) + ( - 6)] ``= - 21 xx [- 10] = 210`
    RHS `= [( - 21) xx ( - 4) + [( - 21) xx ( - 6)] ``= 84 + 126 = 210`
    Here; LHS = RHS

For any integer a, what is ( - 1) x a equal to?

Answer: ( - 1) x a = - a

Determine the integer whose product with ( - 1) is

  1. – 22

    Answer: 22
  2. 37

    Answer: - 37
  3. 0

    Answer: 0

Starting from `( - 1) xx 5`, write various products showing some pattern to show `( - 1) xx ( - 1) = 1`.

Answer: `( - 1) xx 5 = - 5`
`( - 1) xx ( - 3) = 3`
`( - 1) xx 1 = - 1`
`( - 1) xx ( - 1) = 1`

Find the product, using suitable properties:

  1. `26 xx ( - 48) + ( - 48) xx ( - 36)`

    Answer: `26 xx ( - 48) + ( - 48) xx ( - 36)`
    `= ( - 48) xx[26 + ( - 36)]`
    `= ( - 48) xx [ 26 – 36]`
    `= - 48 xx [ - 10] = 480`
    This is showing distributive property of multiplication.
  2. `8 xx 53 xx ( -125)`

    Answer: `8 xx 53 xx ( - 125)` can also be written as `8 xx ( - 125) xx 53`
    This is showing associative property of multiplication.
    `8 xx 53 xx ( - 125) =8 xx ( - 125) xx 53`
    `= ( - 1000) xx 53 = - 53000`
  3. `15 xx ( -25) xx( – 4) xx ( - 10)`

    Answer: This will also show associative property of multiplication.
    `15 xx ( - 25) xx ( - 4) xx ( - 10)`
    `= 15 xx ( - 10) xx ( - 25) xx ( - 4)`
    `=( - 150) xx ( - 100) = 15000`
  4. `( - 41) xx 102`

    Answer: `( - 41) xx 102`
    `= [( - 41) xx 100] + [( - 41) xx 2]`
    `= ( - 4100) + ( - 82) = - 4182`
    This has been solved using distributive property of multiplication.
  5. `625 xx ( - 35) + (- 625) xx 65`

    Answer: `625 xx ( - 35) + ( - 625) xx 65`
    `= ( - 625) xx (35 + 65) ``= ( - 625) xx 100 = - 62500`
    This has been solved using distributive property of multiplication.
  6. `7 xx ( 50 – 2)`

    Answer: `7 xx ( 50 – 2)`
    `= 7 xx 50 + 7 xx ( - 2) = 350 – 14 = 336`
    This has been solved using distributive property of multiplication.
  7. `( - 17) xx ( - 29)`

    Answer: `( - 17) xx ( - 29)`
    `= 17 xx 29 = 17 (30 – 1)`
    `= (17 xx 30) – (17 xx 1)`
    `= 510 – 17 = 493`
    This has been solved using distributive property of multiplication.
  8. `( - 57) xx ( - 19) + 57`

    Answer: `( - 57) xx ( -19) + 57`
    `= 57 xx 19 + 57`
    `= 57 xx (19 + 1)`
    `= 57 xx 20 = 1140`
    This has been solved using distributive property of multiplication.

A certain freezing process required that room temperature be lowered from 40°C at the rate of 5°C every hour. What will be the room temperature 10 hours after the process begins?

Answer: Change in room temperature in 1 hour = - 5° C
So, change in room temperature in 10 hours = ( - 5) x 10 = - 50°C
Room temperature after 10 hours = 40°C - 50°C = - 10°C

In a class test containing 10 questions, 5 marks are awarded for every correct answer and ( - 2) marks are awarded for every incorrect answer and 0 for questions not attempted.

  1. Mohan gets four correct and six incorrect answers. What is his score?

    Answer: Mohan’s score
    `= (4 xx 5) + 6 xx ( - 2) = 20 – 12 = 8`
  2. Reshma gets two correct answers and five incorrect answers, what is her score?

    Answer: Reshma’s score
    `= 2 xx 5 + 5 xx ( - 2) = 10 – 10 = 0`
  3. Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?

    Answer: Heena’s score
    `= 2 xx 5 + 5 xx ( - 2) = 10 – 10 = 0`

A cement company earns a profit of Rs. 8 per bag of white cement sold and a loss of Rs. 5 per bag of grey cement sold.

  1. 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?

    Answer: Company’s profit or loss
    `= 3000 xx 8 + 5000 xx ( - 5)`
    `= 24000 – 25000 = - 1000`
    It is a loss of Rs. 1000
  2. What is the number of white cement bags it must sell to have neither profit nor loss, if the number of grey bags sold is 6,400 bags.

    Answer: Loss in selling grey bags
    `= 6400 xx ( - 5) = - 32000`
    Number of white cement bags required to be sold
    `= 32000 ÷ 8 = 4000`

Replace the blank with an integer to make it a true statement

  1. `( - 3) xx ………… = 27`

    Answer: 9
  2. `5 xx ………… = - 35`

    Answer: - 7
  3. `……… xx ( - 8) = - 56`

    Answer: 7
  4. `……… xx ( - 12) = 132`

    Answer: - 11


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