Integers

Q1: Write the opposite of each of the following statements:

i. Gain of ₹ 90
→ Gain means profit, which is positive.
→ So, gain of ₹ 90 = +90
→ Opposite of gain = loss
Answer: –90 (Loss of ₹ 90)

ii. 70 m above sea-level
→ Above sea-level is positive
→ So, +70 m
→ Opposite = below sea-level
Answer: –70 m (Below sea-level)

iii. 200 m due west
→ West direction considered negative
→ 200 m due west = –200 m
→ Opposite direction = East
Answer: +200 m (Due East)

iv. 25% increase
→ Increase = Positive
→ So, +25%
→ Opposite of increase = decrease
Answer: –25% (25% Decrease)

v. 10°C below freezing point
→ Below freezing = negative
→ –10°C
→ Opposite = above freezing
Answer: +10°C (Above freezing point)

vi. 30 m to the left
→ Left is negative
→ –30 m
→ Opposite direction = Right
Answer: +30 m (To the right)

vii. Below average
→ Below = negative
→ Opposite = above average
Answer: Above average (Positive)

viii. Gaining a weight of 1 kg
→ Gain = positive
→ +1 kg
→ Opposite = losing weight
Answer: –1 kg (Losing 1 kg)

ix. Depositing ₹ 500 in a bank
→ Deposit = positive
→ +500
→ Opposite = Withdrawal
Answer: –₹500 (Withdrawal of ₹ 500)

x. +45
→ Direct number
→ Opposite = negative
Answer: –45

xi. –50
→ Direct number
→ Opposite = positive
Answer: +50

Q2: Indicate the following using negative sign :

i. An increase of 8
Answer: A decrease of -8

ii. Loss of ₹ 300
Answer: Gain of ₹ -300

iii. Spending ₹ 725
Answer: Saving ₹ -725 or Receiving ₹ -725

iv. 53 m below sea-level
Answer: -53 m above sea-level

v. 20°C below freezing point
Answer: -20°C above freezing point

vi. 4 above average
Answer: -4 below average

Q3: Write all integers between:

i. 3 and 10
→ Integers between 3 and 10 (excluding 3 and 10):
= 4, 5, 6, 7, 8, 9
Answer: 4, 5, 6, 7, 8, 9

ii. –3 and 6
→ Integers between –3 and 6:
= –2, –1, 0, 1, 2, 3, 4, 5
Answer: –2, –1, 0, 1, 2, 3, 4, 5

iii. –7 and 0
→ Integers between –7 and 0:
= –6, –5, –4, –3, –2, –1
Answer: –6, –5, –4, –3, –2, –1

iv. –12 and –8
→ Integers between –12 and –8:
= –11, –10, –9
Answer: –11, –10, –9

v. –20 and –14
→ Integers between –20 and –14:
= –19, –18, –17, –16, –15
Answer: –19, –18, –17, –16, –15

vi. –8 and 4
→ Integers between –8 and 4:
= –7, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3
Answer: –7, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3

Q4: Write seven integers, each greater than -6.

Step 1: We are asked to list integers that are greater than –6.
Step 2: Integers greater than –6 are those numbers which lie to the right of –6 on the number line.
So, integers greater than –6 are:
= –5, –4, –3, –2, –1, 0, 1, 2, 3, …
Step 3: From the above list, select any **seven** integers greater than –6:
→ Let us choose the first seven:
Answer: –5, –4, –3, –2, –1, 0, 1

Q5: Fill in the blanks by using > or < or = in each of the following:

i. –5 ___ 3
→ On the number line, –5 is to the left of 3.
→ Hence, –5 is less than 3.
Answer: –5 < 3

ii. 0 ___ –4
→ 0 lies to the right of –4 on the number line.
→ So, 0 is greater than –4.
Answer: 0 > –4

iii. –8 ___ –14
→ –8 is to the right of –14 on the number line.
→ So, –8 is greater than –14.
Answer: –8 > –14

iv. –23 ___ –16
→ –23 is further left than –16 on the number line.
→ So, –23 is less than –16.
Answer: –23 < –16

v. 15 ___ –15
→ 15 is a positive number and –15 is a negative number.
→ Positive numbers are always greater than negative numbers.
Answer: 15 > –15

vi. –1 ___ –12
→ –1 is closer to zero than –12.
→ So, –1 is greater than –12.
Answer: –1 > –12

vii. 9 ___ (–1)
→ 9 is a positive number, and –1 is a negative number.
→ So, 9 is greater than –1.
Answer: 9 > –1

viii. (–9) ___ 1
→ –9 is to the left of 1 on the number line.
→ So, –9 is less than 1.
Answer: –9 < 1

ix. (–9) ___ (–1)
→ –9 is further left than –1 on the number line.
→ So, –9 is less than –1.
Answer: –9 < –1

Q6: Evaluate each of the following :

i. |–17| – |8|
→ The absolute value of –17 is 17, and the absolute value of 8 is 8.
→ Now, evaluate the expression:
= |–17| – |8| = 17 – 8 = 9
Answer: 9

ii. |–14| + |5|
→ The absolute value of –14 is 14, and the absolute value of 5 is 5.
→ Now, evaluate the expression:
= |–14| + |5| = 14 + 5 = 19
Answer: 19

iii. |–9| + |–6|
→ The absolute value of –9 is 9, and the absolute value of –6 is 6.
→ Now, evaluate the expression:
= |–9| + |–6| = 9 + 6 = 15
Answer: 15

iv. |–23| – |–12|
→ The absolute value of –23 is 23, and the absolute value of –12 is 12.
→ Now, evaluate the expression:
= |–23| – |–12| = 23 – 12 = 11
Answer: 11

v. |–8| + |–6| – |–10|
→ The absolute value of –8 is 8, the absolute value of –6 is 6, and the absolute value of –10 is 10.
→ Now, evaluate the expression:
= |–8| + |–6| – |–10| = 8 + 6 – 10 = 4
Answer: 4

vi. |–6| + |–1| – |–5|
→ The absolute value of –6 is 6, the absolute value of –1 is 1, and the absolute value of –5 is 5.
→ Now, evaluate the expression:
= |–6| + |–1| – |–5| = 6 + 1 – 5 = 2
Answer: 2

Q7: Write the following integers in ascending order:

i. 14, –7, –13, 0, 6, –2, 30
→ To arrange the integers in ascending order, we begin by placing the smallest number first.
→ The integers from smallest to largest are:
= –13, –7, –2, 0, 6, 14, 30
Answer: –13, –7, –2, 0, 6, 14, 30

ii. –18, 14, –25, 32, –5, 2
→ To arrange the integers in ascending order:
= –25, –18, –5, 2, 14, 32
Answer: –25, –18, –5, 2, 14, 32

iii. –9, –24, 26, –2, –11, 18, 9
→ To arrange the integers in ascending order:
= –24, –11, –9, -2, 9, 18, 26
Answer: –24, –11, –9, -2, 9, 18, 26

iv. 37, –28, –15, 7, 23, –19, –6
→ To arrange the integers in ascending order:
= –28, –19, –15, –6, 7, 23, 37
Answer: –28, –19, –15, –6, 7, 23, 37

Q8: Write the following integers in descending order:

i. –16, –21, 8, 0, –7, 14
→ To arrange the integers in descending order, we begin by placing the largest number first.
→ The integers from largest to smallest are:
= 14, 8, 0, –7, –16, –21
Answer: 14, 8, 0, –7, –16, –21

ii. –4, 31, –41, –23, 6, 0, –11
→ To arrange the integers in descending order:
= 31, 6, 0, –4, –11, –23, –41
Answer: 31, 6, 0, –4, –11, –23, –41

iii. –60, –9, –36, –8, 5, 15, 0
→ To arrange the integers in descending order:
= 15, 5, 0, –8, –9, –36, –60
Answer: 15, 5, 0, –8, –9, –36, –60

iv. –10, –43, –2, 1, –14, 6, 22
→ To arrange the integers in descending order:
= 22, 6, 1, –2, –10, –14, –43
Answer: 22, 6, 1, –2, –10, –14, –43

v. –8, –53, 53, –100, 62, 16, –82
→ To arrange the integers in descending order:
= 62, 53, 16, –8, –53, –82, –100
Answer: 62, 53, 16, –8, –53, –82, –100

Q9: Represent each of the following on the number line:

i. All integers from –5 to 3.
→ The integers from –5 to 3 are:
= –5, –4, –3, –2, –1, 0, 1, 2, 3

←───|───|───|───|───|───|───|───|───|────→
   -5  -4  -3  -2  -1   0   1   2   3

Answer: All integers from –5 to 3 are represented on the number line.

ii. All negative integers greater than –6.
→ The negative integers greater than –6 are:
= –5, –4, –3, –2, –1

←───|───|───|───|───|────→
   -5  -4  -3  -2  -1

Answer: All negative integers greater than –6 are represented on the number line.

iii. All non-negative integers less than 6.
→ The non-negative integers less than 6 are:
= 0, 1, 2, 3, 4, 5

←───|───|───|───|───|───|───→
    0   1   2   3   4   5

Answer: All non-negative integers less than 6 are represented on the number line.

iv. All even positive integers less than 12.
→ The even positive integers less than 12 are:
= 2, 4, 6, 8, 10

←───|───|───|───|───|─────→
    2   4   6   8   10

Answer: All even positive integers less than 12 are represented on the number line.

v. All odd negative integers greater than –8.
→ The odd negative integers greater than –8 are:
= –7, –5, –3, –1

←───|───|───|───|────→
   -7  -5  -3  -1

Answer: All odd negative integers greater than –8 are represented on the number line.