Ch 01 Integers – Class 7 Maths: Complete Solution

Dive into the fascinating world of integers for NCERT Class 7! This comprehensive post covers all key concepts, properties, and operations on integers, with very simple explanations and classroom-style tips. Explore solved examples, ‘Try It Out’ tasks, and complete step-by-step solutions to Exercises 1.1, 1.2, and 1.3 to master addition, subtraction, multiplication, and division of integers. Perfect for quick revision, exam prep, and building strong number sense!

What are integers?

Integers are all whole numbers and their negatives, like …, −3, −2, −1, 0, 1, 2, 3, ….

Number line basics

• Moving right makes numbers larger, moving left makes numbers smaller on the number line.
• Adding a positive means move right; adding a negative means move left; subtracting a positive means move left; subtracting a negative means move right.

Try These (Page: Recall)

1. A number line representing integers is given below. “−3 and −2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O?”
   Solution: Read each labeled point by counting positions relative to 0 using equal spacing; assign the integer at each label consistently with E = −3 and F = −2.
   Explanation: Each tick is one integer step; labels correspond to the integer directly under the tick when aligned with E and F positions as references.
2. “Arrange 7, −5, 4, 0 and −4 in ascending order and then mark them on a number line to check your answer.”
   Solution: Ascending order: −5, −4, 0, 4, 7.
   Explanation: On a number line, numbers more to the left are smaller; compare by position to sort from left (smallest) to right (largest).
3. “State whether the following statements are correct or incorrect. Correct those which are wrong:”
   (i) “When two positive integers are added we get a positive integer.”
   Solution: Correct.
   Explanation: Sum of positives stays positive, e.g., 2 + 3 = 5.

(ii) “When two negative integers are added we get a positive integer.”
Solution: Incorrect; correct statement: When two negative integers are added, the result is negative.
Explanation: Adding negatives combines their magnitudes and keeps negative sign, e.g., −6 + (−7) = −13.

(iii) “When a positive integer and a negative integer are added, we always get a negative integer.”
Solution: Incorrect; correct statement: Take the difference of their absolute values and keep the sign of the number with the larger absolute value.
Explanation: Example: −9 + 16 = 7 (positive), but 16 + (−23) = −7 (negative) depending on which absolute value is larger.

(iv) “Additive inverse of an integer 8 is (−8) and additive inverse of (−8) is 8.”
Solution: Correct.
Explanation: A number plus its additive inverse equals 0, i.e., 8 + (−8) = 0.

(v) “For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer.”
Solution: Correct.
Explanation: a − b = a + (−b), so subtracting is adding the opposite.

(vi) “(−10) + 3 = 10 − 3”
Solution: Incorrect; left side equals −7 while right side equals 7.
Explanation: The expressions evaluate to different results, so they are not equal.

(vii) “8 + (−7) − (−4) = 8 + 7 − 4”
Solution: Incorrect; left side equals 5 and right side equals 11.
Explanation: Carefully apply signs: subtracting a negative adds, but the right expression changes the second term’s sign incorrectly.

4. “Can you find a pattern for each of the following? If yes, complete them:”
   (a) 7, 3, −1, −5, _____, _____, _____
   Solution: −9, −13, −17.
   Explanation: The pattern decreases by 4 each time: 7 − 4 = 3, 3 − 4 = −1, etc.

(b) −2, −4, −6, −8, _____, _____, _____​
Solution: −10, −12, −14.
Explanation: The pattern decreases by 2 each step.

(c) 15, 10, 5, 0, _____, _____, _____
Solution: −5, −10, −15.
Explanation: The pattern decreases by 5 each step.

(d) −11, −8, −5, −2, _____, _____, _____
Solution: 1, 4, 7.
Explanation: The pattern increases by 3 each step.

Exercise 1.1

1. Following number line shows the temperature in degree celsius (°C) at different places on a particular day.
   (a) Observe this number line and write the temperature of the places marked on it.
   Solution: Read each place’s marked integer directly from the number line ticks (e.g., if Lahulspiti is at −8, write −8°C; list all as seen).
   Explanation: Each point aligns with an integer value on the scale; write those values with °C.

(b) What is the temperature difference between the hottest and the coldest places among the above?
Solution: Difference = hottest − coldest; if hottest = a and coldest = b, then temperature difference is |a − b|
Explanation: Temperature difference is the absolute distance on number line between the two values.

(c) What is the temperature difference between Lahulspiti and Srinagar?
Solution: If Lahulspiti = L and Srinagar = S, difference = |L − S|
Explanation: Distance on number line between two points is the absolute difference.

(d) Can we say temperature of Srinagar and Shimla taken together is less than the temperature at Shimla? Is it also less than the temperature at Srinagar?
Solution: Compute sum S + H and compare with H and S using number line order; typically S + H < H when S is negative; similarly check S + H < S when H is negative.
Explanation: Adding a negative lowers the value; compare using inequality on the number line.

2. In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jack’s scores in five successive rounds were 25, −5, −10, 15 and 10, what was his total at the end?
   Solution: Total = 25 + (−5) + (−10) + 15 + 10 = $25 – 5 – 10 + 15 + 10 = 35$.
   Explanation: Combine positives and negatives: group as $(25+15+10) – (5+10) = 50 – 15 = 35$.
3. At Srinagar temperature was −5°C on Monday and then it dropped by 2°C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by 4°C. What was the temperature on this day?
   Solution: Tuesday: $-5 – 2 = -7$ °C; Wednesday: $-7 + 4 = -3$ °C.
   Explanation: Drop means subtract; rise means add on the number line.
4. A plane is flying at the height of 5000 m above the sea level. At a particular point, it is exactly above a submarine floating 1200 m below the sea level. What is the vertical distance between them?
   Solution: Distance = $5000 – (-1200) = 6200$ m.
   Explanation: Below sea level is negative height; distance is difference between +5000 and −1200 which equals $|5000 – (-1200)|$
5. Mohan deposits ₹ 2,000 in his bank account and withdraws ₹ 1,642 from it, the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in Mohan’s account after the withdrawal.
   Solution: Deposit is positive +2000; withdrawal is −1642; balance = $2000 + (-1642) = 358$.
   Explanation: Deposits increase balance (positive), withdrawals decrease balance (negative).
6. Rita goes 20 km towards east from a point A to the point B. From B, she moves 30 km towards west along the same road. If the distance towards east is represented by a positive integer then, how will you represent the distance travelled towards west? By which integer will you represent her final position from A?
   Solution: West is negative; net = $+20 + (-30) = -10$ km from A.
   Explanation: Opposite directions get opposite signs; the result is 10 km west of A.
7. In a magic square each row, column and diagonal have the same sum. Check which of the following is a magic square.” (two 3×3 squares given)
   Solution: Add each row, column, diagonal for both arrays; the one with equal sums throughout is magic.
   Explanation: A magic square has constant sum in all rows, columns, diagonals; verify by straightforward addition.
8. Verify a − (− b) = a + b for the following values of a and b. (i) a = 21, b = 18 (ii) a = 118, b = 125 (iii) a = 75, b = 84 (iv) a = 28, b = 11
   Solution: Example (i): $21 – (-18) = 21 + 18 = 39$ equals $21 + 18 = 39$; similarly verify others equal.
   Explanation: Subtracting a negative is the same as adding the positive; holds numerically for each pair.
9. Use the sign of >, < or = in the box to make the statements true.
   (a) “(−8) + (−4) ☐ (−8) − (−4)”
   Solution: Left: $-8 + (-4) = -12$; Right: $-8 – (-4) = -8 + 4 = -4$; so −12 < −4, use “<”.
   Explanation: Subtracting a negative increases the value; compare −12 and −4.

(b) (−3) + 7 − (19) ☐ 15 − 8 + (−9)
Solution: Left: $-3 + 7 – 19 = 4 – 19 = -15$; Right: $15 – 8 – 9 = 7 – 9 = -2$; so −15 < −2, use “<”.
Explanation: Evaluate each side step-by-step, then compare.

(c) 23 − 41 + 11 ☐ 23 − 41 − 11
Solution: Left: $23 – 41 + 11 = -18 + 11 = -7$; Right: $23 – 41 – 11 = -18 – 11 = -29$; so −7 > −29, use “>”.
Explanation: A larger (less negative) value is greater on number line.

(d) 39 + (−24) − (15) ☐ 36 + (−52) − (−36)
Solution: Left: $39 – 24 – 15 = 15 – 15 = 0$; Right: $36 – 52 + 36 = -16 + 36 = 20$; so 0 < 20, use “<”.
Explanation: Carefully apply subtracting negative as plus on the right.

(e) −231 + 79 + 51 ☐ −399 + 159 + 81
Solution: Left: $-231 + 79 + 51 = -231 + 130 = -101$; Right: $-399 + 159 + 81 = -399 + 240 = -159$; so −101 > −159, use “>”.
Explanation: Compare by bringing to single totals on each side.

10. A water tank has steps inside it. A monkey is sitting on the topmost step (i.e., the first step). The water level is at the ninth step.
    (i) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level?
    (ii) After drinking water, he wants to go back. For this, he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step?
    (iii) If the number of steps moved down is represented by negative integers and the number of steps moved up by positive integers, represent his moves in part (i) and (ii) by completing the following; (a) −3 + 2 − … = −8 (b) 4 − 2 + … = 8. In (a) the sum (−8) represents going down by eight steps. So, what will the sum 8 in (b) represent?
    Solution:
    (i) Net per move = $-3 + 2 = -1$ step; from step 1 to step 9 is 8 steps down, so needs 8 moves.
    (ii) Net per move = $+4 – 2 = +2$ steps up; from step 9 to step 1 is 8 steps up, so needs 4 moves.
    (iii) (a) Sequence of 8 moves: $-3 + 2 – 3 + 2 – 3 + 2 – 3 + 2 = -8$; (b) Sequence of 4 moves: $4 – 2 + 4 – 2 + 4 – 2 + 4 – 2 = 8$; the sum 8 represents going up by eight steps.
    Explanation: Each “move” is a pair forming a net step; total required net movement dictates number of moves; algebraically the partial sums match target displacement.

Properties of addition/subtraction

• Closure: For integers a, b, both a + b and a − b are integers.
• Commutative: a + b = b + a; but subtraction is not commutative.
• Associative: a + (b + c) = (a + b) + c; subtraction is not associative.
• Additive identity: a + 0 = a.

Exercise 1.2

1. Write down a pair of integers whose: (a) sum is −7 (b) difference is −10 (c) sum is 0.
   Solution:
   (a) One possible: $-3 + (-4) = -7$.
   (b) One possible: $(-7) – 3 = -10$.
   (c) One possible: $5 + (-5) = 0$.
   Explanation: Create examples by adjusting signs and magnitudes to meet the required sum or difference.
2. (a) Write a pair of negative integers whose difference gives 8.
   (b) Write a negative integer and a positive integer whose sum is −5.
   (c) Write a negative integer and a positive integer whose difference is −3.
   Solution:
   (a) $-2 – (-10) = 8$.
   (b) $-8 + 3 = -5$.
   (c) $-1 – 2 = -3$.
   Explanation: For differences, remember a − b = a + (−b); choose values accordingly.
3. 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?
   Solution: Team A total = $-40 + 10 + 0 = -30$; Team B total = $10 + 0 + (-40) = -30$; both equal; yes, addition is commutative and associative so order doesn’t change the sum.
   Explanation: Reordering terms does not affect the total for addition; both totals match.
4. Fill in the blanks to make the following statements true:
   (i) (−5) + (−8) = (−8) + (…………) → (−5)
   (ii) −53 + ………… = −53 → 0
   (iii) 17 + ………… = 0 → −17
   (iv) [13 + (−12)] + (…………) = 13 + [(−12) + (−7)] → −7
   (v) (−4) + [15 + (−3)] = [−4 + 15] + ………… → −3
   Explanation: Use commutative, identity, inverse, and associative properties for correct fillers.

Multiplication of integers

• Positive × Negative = Negative: $a \times (-b) = -(a\times b)$.
• Negative × Negative = Positive: $(-a)\times(-b)=a\times b$.
• Even number of negatives → positive product; odd number of negatives → negative product.

Try These (Multiplication)

1. Find: 4 × (−8), 8 × (−2), 3 × (−7), 10 × (−1) using number line.​
   Solution: $4\times(-8)=-32$, $8\times(-2)=-16$, $3\times(-7)=-21$, $10\times(-1)=-10$.
   Explanation: Repeated jumps left on number line equal repeated addition of a negative.
2. Find: (i) 6 × (−19) (ii) 12 × (−32) (iii) 7 × (−22)
   Solution: (i) $-114$, (ii) $-384$, (iii) $-154$.
   Explanation: Multiply magnitudes, add a single negative sign for one negative factor.
3. Using patterns, find (−4) × 8, (−3) × 7, (−6) × 5 and (−2) × 9. Check whether (−4) × 8 = 4 × (−8), etc.
   Solution: $(−4)\times 8 = -32 = 4\times(-8)$; $(−3)\times 7 = -21 = 3\times(-7)$; $(−6)\times 5=-30=6\times(-5)$; $(−2)\times 9=-18=2\times(-9)$.
   Explanation: One negative factor makes the product negative, and commutativity holds.
4. Find: (−31) × (−100), (−25) × (−72), (−83) × (−28)
   Solution: $3100$, $1800$, $2324$.
   Explanation: Two negatives give a positive; multiply magnitudes.

Properties of multiplication

• Closure, Commutative, Associative, Multiplicative identity 1: $a\times b$ is integer; $a\times b=b\times a$; $(a\times b)\times c=a\times(b\times c)$; $a\times 1= a$.
• Distributive over addition and subtraction: $a\times(b+c)=ab+ac$, $a\times(b-c)=ab-ac$.
• Multiplication by zero gives zero: $a\times 0=0$.

Making multiplication easier

Use properties to simplify: e.g., $(−25)\times 37\times 4 = (−100)\times 37 = −3700$; $16\times 12=16\times(10+2)=160+32=192$.

Exercise 1.3

1. Find each of the following products:
   (a) $3 \times (−1)$ → $-3$
   (b) $(−1) \times 225$ → $-225$
   (c) $(−21) \times (−30)$ → $630$
   (d) $(−316) \times (−1)$ → $316$
   (e) $(−15) \times 0 \times (−18)$ → $0$
   (f) $(−12) \times (−11) \times (10)$ → $1320$
   (g) $9 \times (−3) \times (−6)$ → $162$​
   (h) $(−18) \times (−5) \times (−4)$ → $-360$
   (i) $(−1) \times (−2) \times (−3) \times 4$ → $-24$
   (j) $(−3) \times (−6) \times (−2) \times (−1)$ → $36$
   Explanation: Count negatives to decide sign; multiply magnitudes accordingly.
2. Verify the following:
   (a) $18 \times [7 + (−3)] = [18 \times 7] + [18 \times (−3)]$
   Solution: LHS $=18\times 4=72$; RHS $=126 + (-54) = 72$; equal.
   Explanation: Shows distributive property over addition.
   (b) $(−21) \times [(−4) + (−6)] = [(−21) \times (−4)] + [(−21) \times (−6)]$
   Solution: LHS $=(-21)\times(-10)=210$; RHS $=84+126=210$; equal.
   Explanation: Distributive property holds for negatives too.
3. (i) For any integer a, what is (−1) × a equal to?
   (ii) Determine the integer whose product with (−1) is (a) −22 (b) 37 (c) 0
   Solution:
   (i) $(−1)\times a = -a$.
   (ii) (a) $a=22$; (b) $a=-37$; (c) $a=0$.
   Explanation: Multiplying by −1 gives the additive inverse.
4. Starting from (−1) × 5, write various products showing some pattern to show (−1) × (−1) = 1.
   Solution: $(-1)\times 5=-5$, $(-1)\times 4=-4$, $(-1)\times 3=-3$, $(-1)\times 2=-2$, $(-1)\times 1=-1$, $(-1)\times 0=0$, then decrease the second factor: $(-1)\times (-1)=1$ by pattern of adding 1 each step past zero.​
   Explanation: Observing the pattern as the second factor decreases by 1 increases the product by 1, leading to $(-1)\times(-1)=1$.
5. Find the product, using suitable properties:
   (a) $26 \times (−48) + (−48) \times (−36)$
   Solution: Factor $(−48)$: $(−48)\times(26 – (−36)) = (−48)\times 62 = -2976$.
   Explanation: Use distributive property in reverse (factorisation).

(b) $8 \times 53 \times (−125)$
Solution: Combine $8 \times (−125) = -1000$; then $-1000 \times 53 = -53000$.
Explanation: Associativity and grouping ease computation.

(c) $15 \times (−25) \times (−4) \times (−10)$
Solution: First $15 \times (−25) = -375$; then $\times (−4) = 1500$; then $\times (−10) = -15000$.
Explanation: Three negatives overall give a negative product.

(d) $(−41) \times 102$
Solution: $102 = 100 + 2$; $(−41)\times 100 + (−41)\times 2 = -4100 – 82 = -4182$.
Explanation: Distributive over addition.

(e) $625 \times (−35) + (−625) \times 65$
Solution: Factor $625$: $625\times[(-35) + (-65)] = 625\times(-100) = -62500$.
Explanation: Factor common term, then add inside bracket.

(f) $7 \times (50 − 2)$
Solution: $7\times 48 = 336$.
Explanation: Distribute or compute directly by difference.

(g) $(−17) \times (−29)$
Solution: Positive product: $17\times 29 = 17\times(30-1)=510-17=493$.
Explanation: Two negatives make positive; use distributive trick for 29.

(h) $(−57) \times (−19) + 57$
Solution: $(−57)\times(−19) = 57\times 19 = 57\times(20-1)=1140-57=1083$; then $1083 + 57 = 1140$.
Explanation: Convert to positive multiplication and use distribution over (20−1).

6. A certain freezing process requires 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?
   Solution: Change in 10 hours = $10\times (-5) = -50$;
   final temperature = $40 + (-50) = -10$ °C.
   Explanation: Repeated decrease is multiplication by a negative rate.
7. 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.
   (i) Mohan gets four correct and six incorrect answers. What is his score?
   Solution: $4\times 5 + 6\times(-2) = 20 – 12 = 8$.
   Explanation: Add marks for corrects and subtract for incorrects.

(ii) Reshma gets five correct answers and five incorrect answers, what is her score?
Solution: $5\times 5 + 5\times(-2) = 25 – 10 = 15$.
Explanation: Same marking scheme applied.

(iii) Heena gets two correct and five incorrect answers out of seven questions she attempts. What is her score?
Solution: $2\times 5 + 5\times(-2) = 10 – 10 = 0$.
Explanation: Net effect cancels to zero.

8. 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.
   (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?
   Solution: Profit part: $3000\times 8=24000$; Loss part: $5000\times 5=25000$; Net = $24000 + (-25000) = -1000$ → ₹ 1,000 loss.
   Explanation: Treat loss as negative and add to find net.

(b) 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.
Solution: Let white bags = x; equation: $8x + (-5)\times 6400 = 0$; so $8x = 32000$; $x = \frac{32000}{8} = 4000$.​
Explanation: Break-even means total profit + total loss = 0; solve for x.

9. Replace the blank with an integer to make it a true statement.
   (a) (−3) × _____ = 27 → $-9$
   (b) 5 × _____ = −35 → $-7$
   (c) _____ × (−8) = −56 → $7$
   (d) _____ × (−12) = 132 → $-11$
   Explanation: Solve by dividing target by the given factor, keeping sign rules in mind.

Division of integers

• Positive ÷ Negative = Negative; Negative ÷ Positive = Negative; Negative ÷ Negative = Positive.
• Division by zero is not defined; dividing zero by a nonzero integer gives zero.

Try These (Division)

1. Find: (a) (−100) ÷ 5 (b) (−81) ÷ 9 (c) (−75) ÷ 5 (d) (−32) ÷ 2
   Solution: (a) $-20$; (b) $-9$; (c) $-15$; (d) $-16$.
   Explanation: Divide magnitudes and apply one negative sign for single negative in division.
2. Find: (a) 125 ÷ (−25) (b) 80 ÷ (−5) (c) 64 ÷ (−16)
   Solution: (a) $-5$; (b) $-16$; (c) $-4$.
   Explanation: Positive divided by negative is negative.
3. Find: (a) (−36) ÷ (−4) (b) (−201) ÷ (−3) (c) (−325) ÷ (−13)
   Solution: (a) $9$; (b) $67$; (c) $25$.
   Explanation: Negative divided by negative is positive.

Key Points

• Addition/subtraction of integers stays within integers; 0 is the additive identity.
• Multiplication is closed, commutative, associative; 1 is multiplicative identity; distributive laws help simplify.
• Sign rules: product/quotient of even number of negatives is positive; odd number is negative.
• Division by zero is not defined; a ÷ 1 = a.

If a fully worked “magic square” check or the temperatures’ exact labeled values are needed, state the exact grid or the precise number line marks, and each sum/value will be computed line by line accordingly.

Interactive Quiz on Integers – Chapter 01 NCERT Maths