Integers
NCERT Exercise 1.2
Write down a pair of integers whose:
- Sum is – 7
Answer: - 14 and 7
- Difference is – 10
Answer: 55 and 65
- Sum is 0
Answer: 24 and – 24
Answer the following:
- Write a pair of integers whose difference gives 8.
Answer: 15 and 7
- Write a negative integer and a positive integer whose sum is – 5.
Answer: - 15 and 10
- 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:
- `( - 6) + ( - 8) = - 8 + (………..)`
Answer: - 6
- `– 53 + ……….. = - 53`
Answer: 0
- `17 + …………… = 0`
Answer: - 17
- `[13 + ( - 12)] + (…………) = 13 + [( - 12) + ( - 7)]`
Answer: - 7
- `( - 4) + [15 + ( - 3)] = [ - 4 + 15] + …………….`
Answer: - 3
Exercise 1.3
Find each of the following products:
- `3 xx ( - 1)`
Answer: `3 xx ( - 1) = - 3`
- `( - 1) xx 225`
Answer: `- 225`
- `( - 21) xx ( - 30)`
Answer: `( - 21) xx ( - 30) = 630`
- `( - 316) xx ( - 1)`
Answer: 316
- `( - 15) xx 0 xx ( - 18)`
Answer: Zero because when any number is multiplied by zero the result is zero.
- `( - 12) xx ( - 11) xx (10)`
Answer: `( - 12) xx ( - 11) xx (10) = 132 xx 10 = 1320`
- `9 xx ( - 3) xx ( - 6)`
Answer: `9 xx ( - 3) xx ( - 6) = 9 xx 18 = 162`
- `( - 18) xx ( - 5) xx ( - 4)`
Answer: `( - 18) xx ( - 5) xx ( - 4) = 90 xx ( - 4) = - 360`
- `( - 1) xx ( -2) xx ( - 3) xx 4`
Answer: `( - 1) xx ( - 2) xx ( - 3) xx 4 = 2 xx ( - 12) = - 24`
- `( - 3) xx ( - 6) xx ( - 2) xx ( - 1)`
Answer: `( - 3) xx ( - 6) xx ( - 2) xx ( - 1) = 18 xx 2 = 36`
Verify the following:
- `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
- `( - 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
- – 22
Answer: 22
- 37
Answer: - 37
- 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:
- `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.
- `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`
- `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`
- `( - 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.
- `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.
- `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.
- `( - 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.
- `( - 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.
- 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`
- 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`
- 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.
- 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
- 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
- `( - 3) xx ………… = 27`
Answer: 9
- `5 xx ………… = - 35`
Answer: - 7
- `……… xx ( - 8) = - 56`
Answer: 7
- `……… xx ( - 12) = 132`
Answer: - 11