ML Aggarwal Class 7 Maths Solutions Chapter 01 Integers

Exercise 1.1

Question 1. Some integers are marked on the following number line:
(i) Write these integers in ascending order.
(ii) Write these integers in descending order.
(iii) Few dots have been marked on the above number line. Write appropriate integer at each dot.
Answer: The integers shown on the number line are: -18, -9, -4, 0, 3, 8, 12.
(i) In ascending order: -18, -9, -4, 0, 3, 8, 12
(ii) In descending order: 12, 8, 3, 0, -4, -9, -18
(iii) The dots marked on the number line represent the following integers: -17, -14, -11, -7, -5, -3, 1, 4, 7, 9, 11
In simple words: Ascending order means going from smallest (most negative) to largest. Descending order means going from largest to smallest (most negative).

Exam Tip: Always remember that on a number line, smaller numbers are to the left and larger numbers are to the right. Use this rule to arrange integers correctly every time.

Question 2. Arrange 7, -5, 4, 0 and -4 in ascending order and mark them on a number line to check your answer.
Answer: The negative integers given are -5 and -4. In ascending order: -5 < -4.
The positive integers given are 7 and 4. In ascending order: 4 < 7.
Since 0 is between negative and positive integers, the complete ascending order is:
-5 < -4 < 0 < 4 < 7, or written as: -5, -4, 0, 4, 7.
On the number line, -5 lies to the left of -4, which lies to the left of 0, which lies to the left of 4, which lies to the left of 7. This confirms our ascending arrangement.
In simple words: First list all negative numbers from smallest to biggest, then put zero, then put all positive numbers from smallest to biggest.

Exam Tip: Always group negative integers together and positive integers separately before ordering them. This makes the process clearer and helps avoid mistakes.

Question 3. In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Rohit's scores in five successive rounds were 15, -3, -7, 12 and 8, what was his total at the end?
Answer: Rohit's scores across five rounds were 15, -3, -7, 12, and 8.
Total score = 15 + (-3) + (-7) + 12 + 8
Grouping positive and negative values: = (15 + 12 + 8) + ((-3) + (-7))
= 35 + (-10)
= 35 - 10
= 25
Therefore, Rohit's total score at the end was 25.
In simple words: Add all the positive marks together, add all the negative marks together, then combine both results to find the final score.

Exam Tip: Grouping positive and negative integers before adding them makes calculations faster and reduces the chance of sign errors.

Question 4. Ruchi deposited Rs. 4370 in her account on Monday and then withdrew Rs. 2875 on Tuesday. Next day she deposited Rs. 1550. What was her balance on Thursday?
Answer: Amount deposited on Monday = Rs. 4370
Amount withdrawn on Tuesday = Rs. 2875 (shown as -2875)
Amount deposited on Wednesday = Rs. 1550
Balance on Thursday = Rs. 4370 + Rs. (-2875) + Rs. 1550
= Rs. 4370 - Rs. 2875 + Rs. 1550
= Rs. 1495 + Rs. 1550
= Rs. 3045
Therefore, Ruchi's balance on Thursday was Rs. 3045.
In simple words: Start with the money put in, subtract the money taken out, then add any new money deposited to find what remains in the account.

Exam Tip: Deposits are positive and withdrawals are negative. Write them with their correct signs before adding to find the final balance.

Question 5. Evaluate the following:
(i) |-13| - |9|
(ii) |13 - 5| - |-9|
(iii) |35 - 21| - |8 - 3|
Answer:
(i) |-13| - |9| = 13 - 9 [since |-13| = 13 and |9| = 9] = 4
(ii) |13 - 5| - |-9| = |8| - |-9| = 8 - 9 [since |8| = 8 and |-9| = 9] = -1
(iii) |35 - 21| - |8 - 3| = |14| - |5| = 14 - 5 [since |14| = 14 and |5| = 5] = 9
In simple words: The absolute value symbol removes the negative sign. Solve what is inside the bars first, then take the absolute value, and finally perform the subtraction.

Exam Tip: Simplify expressions inside absolute value bars before finding the absolute value. This prevents careless errors and makes the solution clearer.

Question 6. Arrange the following integers in ascending order: -39, 35, -102, 0, -51, -5, -6, 7
Answer: The negative integers given are -39, -102, -51, -5, -6. In ascending order: -102 < -51 < -39 < -6 < -5.
The positive integers given are 35 and 7. In ascending order: 7 < 35.
Since 0 lies between negative and positive integers, all integers in ascending order are:
-102 < -51 < -39 < -6 < -5 < 0 < 7 < 35
Written as: -102, -51, -39, -6, -5, 0, 7, 35.
In simple words: Start with the most negative number and work your way up to the most positive number. Remember that numbers farther to the left on a number line are always smaller.

Exam Tip: For negative integers, the one with the larger absolute value is actually smaller. For example, -102 is smaller than -51 even though 102 is larger than 51.

Question 7. Arrange the following integers in descending order: -31, 139, -203, -97, 0, 4208
Answer: The positive integers given are 139 and 4208. In descending order: 4208 > 139.
The negative integers given are -31, -203, -97. In descending order: -31 > -97 > -203.
Since 0 is less than positive integers but greater than negative integers, all integers in descending order are:
4208 > 139 > 0 > -31 > -97 > -203
Written as: 4208, 139, 0, -31, -97, -203.
In simple words: Start with the largest number and go down to the smallest number. For negative numbers, the one closest to zero is actually the largest.

Exam Tip: Remember that in descending order, all positive numbers come before zero, and zero comes before all negative numbers. Among negatives, those closest to zero are largest.

Question 8. State whether each of the following statement is true or false:
(i) 0 is the successor of -1 in integers
(ii) 0 has no predecessor in integers
(iii) -2 is the predecessor of -1
(iv) 0 is greater than every negative integer.
Answer:
(i) True - The successor of any integer is found by adding 1. Since -1 + 1 = 0, the value 0 is indeed the successor of -1.
(ii) False - Every integer has a predecessor, not just some. The predecessor of 0 is found by subtracting 1, giving -1 (calculated as 0 - 1).
(iii) True - A number's predecessor is the integer positioned immediately to its left on the number line, obtained by subtracting 1. Since -1 - 1 = -2, the value -2 is the predecessor of -1.
(iv) True - On a standard number line drawn horizontally, 0 sits to the right of every negative integer, making it greater than any negative value.
In simple words: A successor is found by adding 1, and a predecessor is found by subtracting 1. Every integer except those at the boundaries has both a successor and a predecessor.

Exam Tip: Know the definitions clearly: successor = adding 1, predecessor = subtracting 1. Test your answer by checking on a number line if you are unsure.

Question 9. Use the sign >, < or = in the box to make the following statements true:
(i) (-11) + (-7) ___ (-11) - (-7)
(ii) 23 - 41 + 11 ___ 23 - 41 - 11
(iii) 40 - (-39) + (-5) ___ 40 + (-39) - (-5)
(iv) (-3) + 13 - (15) ___ 25 - (-2) + (-33)
Answer:
(i) LHS = (-11) + (-7) = -(11 + 7) = -18
RHS = (-11) - (-7) = -11 + 7 = -4
Since -18 < -4, the sign is:
(-11) + (-7) < (-11) - (-7)

(ii) LHS = 23 - 41 + 11 = 34 - 41 = -7
RHS = 23 - 41 - 11 = 23 - 52 = -29
Since -7 > -29, the sign is:
23 - 41 + 11 > 23 - 41 - 11

(iii) LHS = 40 - (-39) + (-5) = 40 + 39 - 5 = 74
RHS = 40 + (-39) - (-5) = 40 - 39 + 5 = 6
Since 74 > 6, the sign is:
40 - (-39) + (-5) > 40 + (-39) - (-5)

(iv) LHS = (-3) + 13 - 15 = 10 - 15 = -5
RHS = 25 - (-2) + (-33) = 25 + 2 - 33 = -6
Since -5 > -6, the sign is:
(-3) + 13 - (15) > 25 - (-2) + (-33)
In simple words: Calculate both sides separately, then compare the results to choose the correct comparison sign.

Exam Tip: Always work out the left side completely and the right side completely before comparing. Careless sign errors are the most common mistakes here.

Exercise 1.2

Question 1. Write a pair of integers whose:
(i) sum is -3
(ii) difference is -5
(iii) difference is 4
Answer:
(i) Sum is -3: One such pair is -5 and 2, because -5 + 2 = -3.
(ii) Difference is -5: One such pair is -2 and 3, because (-2) - 3 = -2 - 3 = -5.
(iii) Difference is 4: One such pair is -7 and -11, because (-7) - (-11) = -7 + 11 = 4.
In simple words: To find a pair with a given sum, think of numbers that add to that sum. To find a pair with a given difference, think of numbers that when subtracted give that result.

Exam Tip: Many pairs work for each condition - your answer just needs to satisfy the given condition. Always verify your answer by computing the sum or difference to confirm.

Question 2. In a quiz, team A scored -30, 20, 0 and team B scored 20, 0, -30 in three successive rounds. Which team scored more? Can we say that we can add integers in any order?
Answer: Total score of Team A = (-30) + 20 + 0 = -10 + 0 = -10
Total score of Team B = 20 + 0 + (-30) = 20 - 30 = -10
Since both teams scored -10, both teams scored equally.
Yes, we can add integers in any order. This is because integer addition follows both the commutative property (order does not matter) and the associative property (grouping does not matter).
In simple words: Adding integers gives the same total no matter what order you add them in. The answer stays the same whether you add left to right or rearrange the numbers.

Exam Tip: This demonstrates the commutative property of addition. Always remember that a + b + c = c + b + a for any integers a, b, and c.

Question 3. Find the sum of integers -72, 237, 84, 72, -184, -37
Answer: Sum = (-72) + 237 + 84 + 72 + (-184) + (-37)
Grouping convenient integers together:
= [(-72) + 72] + [237 + 84] + [(-184) + (-37)]
= 0 + 321 + (-221)
= 321 - 221
= 100
Therefore, the required sum is 100.
In simple words: Look for pairs of numbers that add to zero first, then add the remaining numbers. This approach makes the calculation much quicker.

Exam Tip: Group integers strategically - pair opposites (like -72 and 72) to make them zero, then work with the rest. This reduces the work and minimizes errors.

Question 4. Write two integers which are smaller than -3, but their difference is greater than -3
Answer: Let the two integers be -5 and -4.
Both -5 and -4 are smaller than -3 (since they are farther to the left on the number line).
Their difference: (-5) - (-4) = -5 + 4 = -1
Since -1 > -3, the condition is satisfied.
Therefore, the two integers are -5 and -4.
In simple words: Find two negative numbers that are both smaller than -3. Then subtract one from the other and check that the result is bigger than -3.

Exam Tip: When subtracting a negative number, remember to flip the sign (subtracting a negative becomes adding). This is where most mistakes happen - be extra careful with the signs.

Exercise 1.3

Question 1. Find the following products:
(i) 7 × (-35)
(ii) (-13) × (-15)
(iii) (-12) × (-11) × (-10)
(iv) (-13) × 0 × (-24)
(v) (-1) × (-2) × (-3) × 4
(vi) (-3) × (-6) × (-2) × (-1)
Answer:
(i) 7 × (-35) = -(7 × 35) = -245 [Positive × Negative = Negative]

(ii) (-13) × (-15) = +(13 × 15) = 195 [Negative × Negative = Positive]

(iii) (-12) × (-11) × (-10) = [(-12) × (-11)] × (-10) = 132 × (-10) = -1320 [Positive × Negative = Negative]

(iv) (-13) × 0 × (-24) = 0 [Any number multiplied by 0 equals 0]

(v) (-1) × (-2) × (-3) × 4 = [(-1) × (-2)] × [(-3) × 4] = 2 × (-12) = -24 [Positive × Negative = Negative]

(vi) (-3) × (-6) × (-2) × (-1) = [(-3) × (-6)] × [(-2) × (-1)] = 18 × 2 = 36 [Negative × Negative = Positive]
In simple words: When multiplying, count how many negative numbers you have. If the count is odd, the answer is negative. If the count is even, the answer is positive.

Exam Tip: The sign rule for multiplication is crucial: odd number of negatives gives negative result, even number of negatives gives positive result. Use this shortcut to check your work quickly.

Question 2. Verify the following:
(i) 37 × [6 + (-3)] = 37 × 6 + 37 × (-3)
(ii) (-21) × [(-6) + (-4)] = (-21) × (-6) + (-21) × (-4)
Answer:
(i) 37 × [6 + (-3)] = 37 × 6 + 37 × (-3)
LHS = 37 × [6 + (-3)] = 37 × 3 = 111
RHS = 37 × 6 + 37 × (-3) = 222 + (-111) = 222 - 111 = 111
Since LHS = RHS, the statement is verified. [Distributive Property]

(ii) (-21) × [(-6) + (-4)] = (-21) × (-6) + (-21) × (-4)
LHS = (-21) × [(-6) + (-4)] = (-21) × (-10) = 210
RHS = (-21) × (-6) + (-21) × (-4) = 126 + 84 = 210
Since LHS = RHS, the statement is verified. [Distributive Property]
In simple words: Multiplying a number by a sum is the same as multiplying the number by each part separately and then adding the results together.

Exam Tip: This is the distributive property in action. Always verify both sides separately and confirm they are equal. This property is very useful for making hard multiplications easier.

Question 3. Using suitable properties, evaluate the following:
(i) 8 × 53 × (-125)
(ii) (-8) × (-2) × 3 × (-5)
(iii) (-6) × 2 × (-8) × 5
(iv) 15 × (-25) × (-4) × (-10)
(v) 26 × (-48) + (-48) × (-36)
(vi) 724 × (-56) + (-724) × 44
(vii) (-47) × 102
(viii) (-39) × (-97)
Answer:
(i) 8 × 53 × (-125) = 53 × [8 × (-125)] [Commutative and Associative Properties] = 53 × (-1000) = -53000

(ii) (-8) × (-2) × 3 × (-5) = [(-8) × (-2)] × [3 × (-5)] [Associative Property] = 16 × (-15) = -240

(iii) (-6) × 2 × (-8) × 5 = [(-6) × (-8)] × [2 × 5] [Rearranging] = 48 × 10 = 480

(iv) 15 × (-25) × (-4) × (-10) = [15 × (-4)] × [(-25) × (-10)] [Rearranging] = (-60) × 250 = -15000

(v) 26 × (-48) + (-48) × (-36) = (-48) × [26 + (-36)] [Distributive Property] = (-48) × (-10) = 480

(vi) 724 × (-56) + (-724) × 44 = 724 × (-56) + (-724) × 44 = 724 × [(-56) + (-44)] [Distributive Property, noting (-724) = -(724)] = 724 × (-100) = -72400

(vii) (-47) × 102 = (-47) × (100 + 2) [Breaking 102 into parts] = (-47) × 100 + (-47) × 2 [Distributive Property] = -4700 + (-94) = -4794

(viii) (-39) × (-97) = (-39) × (-100 + 3) [Breaking -97 into parts] = (-39) × (-100) + (-39) × 3 [Distributive Property] = 3900 + (-117) = 3783
In simple words: Break larger numbers into smaller, easier parts. Rearrange and group terms to make calculations simpler. Use properties to avoid lengthy calculations.

Exam Tip: Choose groupings that create round numbers or that allow factoring to reduce work. For example, 8 × 125 = 1000 and 100 is easy to work with, so use these whenever you see them.

Question 1. Evaluate the following:
(i) \( (-36) \div (-9) \)
(ii) \( 150 \div (-25) \)
(iii) \( (-270) \div 27 \)
(iv) \( (-59) \div 59 \)
(v) \( 0 \div (-17) \)
(vi) \( (-784) \div (-56) \)
Answer:
(i) \( (-36) \div (-9) = \frac{-36}{-9} = 4 \)
(ii) \( 150 \div (-25) = \frac{150}{-25} = -6 \)
(iii) \( (-270) \div 27 = \frac{-270}{27} = -10 \)
(iv) \( (-59) \div 59 = \frac{-59}{59} = -1 \)
(v) \( 0 \div (-17) = \frac{0}{-17} = 0 \) [Since zero divided by any non-zero integer is zero]
(vi) \( (-784) \div (-56) = \frac{-784}{-56} = 14 \)
In simple words: When you divide two numbers, write them as a fraction. A negative divided by a negative gives a positive. A negative divided by a positive (or positive divided by negative) gives a negative. Zero divided by anything is always zero.

Exam Tip: Remember the sign rule for division: same signs give positive, different signs give negative. Always simplify the fraction fully before stating your final answer.

Question 2. Evaluate the following:
(i) \( 13 \div [(-2) + 1] \)
(ii) \( (-47) \div [(-45) + (-2)] \)
(iii) \( [(-6) + 5] \div [(-2) + 1] \)
(iv) \( [(-48) \div (-6)] \div (-2) \)
Answer:
(i) \( 13 \div [(-2) + 1] = 13 \div (-1) = -13 \)
(ii) \( (-47) \div [(-45) + (-2)] = (-47) \div (-47) = 1 \)
(iii) \( [(-6) + 5] \div [(-2) + 1] = (-1) \div (-1) = 1 \)
(iv) \( [(-48) \div (-6)] \div (-2) = 8 \div (-2) = -4 \)
In simple words: Work out what is inside the brackets first. Then perform the division using the sign rules. A number divided by itself (when not zero) equals 1.

Exam Tip: Always solve brackets before anything else. This is the order of operations - BODMAS/PEMDAS - and ignoring it is the most common source of wrong answers.

Question 3. Verify that \( (a \div b) \div c \neq a \div (b \div c) \) for \( a = -225 \), \( b = 15 \) and \( c = -3 \)
Answer: Given: \( a = -225 \), \( b = 15 \) and \( c = -3 \)

LHS \( = (a \div b) \div c \)
\( = [(-225) \div 15] \div (-3) \)
\( = (-15) \div (-3) \)
\( = 5 \)

RHS \( = a \div (b \div c) \)
\( = (-225) \div [15 \div (-3)] \)
\( = (-225) \div (-5) \)
\( = 45 \)

Since \( 5 \neq 45 \), we have \( (a \div b) \div c \neq a \div (b \div c) \) for \( a = -225 \), \( b = 15 \) and \( c = -3 \).

This shows that division is not associative for integers.
In simple words: When you change where you put the brackets in a division problem, you get a different answer. This means division is not associative - the order in which you divide matters a lot.

Exam Tip: Write out both sides (LHS and RHS) separately and show all working steps. The key is to demonstrate the property clearly by proving the two sides are unequal.

Question 4. Verify that \( a \div (b + c) \neq (a \div b) + (a \div c) \) for (i) \( a = -10 \), \( b = 1 \) and \( c = 1 \) (ii) \( a = 12 \), \( b = 1 \) and \( c = -2 \)
Answer:
(i) Given: \( a = -10 \), \( b = 1 \) and \( c = 1 \)

LHS \( = a \div (b + c) \)
\( = (-10) \div (1 + 1) \)
\( = (-10) \div 2 \)
\( = -5 \)

RHS \( = (a \div b) + (a \div c) \)
\( = [(-10) \div 1] + [(-10) \div 1] \)
\( = (-10) + (-10) \)
\( = -20 \)

Since \( -5 \neq -20 \), we have \( a \div (b + c) \neq (a \div b) + (a \div c) \) for \( a = -10 \), \( b = 1 \) and \( c = 1 \).

(ii) Given: \( a = 12 \), \( b = 1 \) and \( c = -2 \)

LHS \( = a \div (b + c) \)
\( = 12 \div [1 + (-2)] \)
\( = 12 \div (-1) \)
\( = -12 \)

RHS \( = (a \div b) + (a \div c) \)
\( = (12 \div 1) + [12 \div (-2)] \)
\( = 12 + (-6) \)
\( = 12 - 6 \)
\( = 6 \)

Since \( -12 \neq 6 \), we have \( a \div (b + c) \neq (a \div b) + (a \div c) \) for \( a = 12 \), \( b = 1 \) and \( c = -2 \).
In simple words: Division does not distribute over addition. Dividing a sum gives a different result than dividing each part and adding. Always do what is in the brackets first.

Exam Tip: This is a key concept: division is not distributive. Show clear, step-by-step working for both sides to earn full marks. Write the conclusion clearly stating why the two sides are not equal.

Question 5. Fill in the blanks to make the following statements true:
(i) \( 239 \div \ldots = 1 \)
(ii) \( (-85) \div \ldots = -1 \)
(iii) \( (-213) \div \ldots = 1 \)
(iv) \( (-43) \div \ldots = 43 \)
(v) \( \ldots \div (-21) = 4 \)
(vi) \( (-66) \div \ldots = -3 \)
Answer:
(i) \( 239 \div 239 = 1 \)
(ii) \( (-85) \div 85 = -1 \)
(iii) \( (-213) \div (-213) = 1 \)
(iv) \( (-43) \div (-1) = 43 \)
(v) \( (-84) \div (-21) = 4 \)
(vi) \( (-66) \div 22 = -3 \)

Explanation:
(i) Any non-zero number divided by itself results in 1.
(ii) For the quotient to be negative, you must divide a negative number by a positive number.
(iii) When a negative number is divided by itself, the result is 1.
(iv) Dividing a negative number by - 1 reverses its sign, making it positive.
(v) Multiply the quotient and divisor: \( 4 \times (-21) = -84 \).
(vi) Divide the dividend by the quotient: \( (-66) \div (-3) = 22 \).
In simple words: A number divided by itself is 1. A negative divided by negative is positive. A negative divided by positive is negative. Dividing by - 1 flips the sign of a number.

Exam Tip: Use the inverse operation to check your answer - if you found the divisor, multiply it by the quotient to see if you get back the dividend. This verification takes just a few seconds and catches errors.

Question 6. In a competition 3 marks are given for every correct answer and (-2) marks are given for every incorrect answer and no marks for not attempting any question.
(i) Sachin scored 24 marks. If he got 14 correct answers, how many questions has he attempted incorrectly?
(ii) Nalini scores (-7) marks in this competition, though she has got 9 correct answers. How many questions she has attempted incorrectly?
Answer: Marks given for each correct answer \( = 3 \)
Marks given for each incorrect answer \( = -2 \)

(i) Sachin's total score \( = 24 \)
Sachin's correct answers \( = 14 \)
Marks for correct answers \( = 14 \times 3 = 42 \)
Marks obtained for incorrect answers \( = \) Total score - Marks for correct answers
\( = 24 - 42 = -18 \)
Number of incorrect answers \( = \) Marks obtained for incorrect answers \( \div \) Marks given for each incorrect answer
\( = (-18) \div (-2) = 18 \div 2 = 9 \)

Therefore, Sachin attempted 9 questions incorrectly.

(ii) Nalini's total score \( = -7 \)
Marks scored for 9 correct answers \( = 9 \times 3 = 27 \)
Marks obtained for incorrect answers \( = \) Total score - Marks for correct answers
\( = -7 - 27 = -34 \)
Number of incorrect answers \( = \) Marks obtained for incorrect answers \( \div \) Marks given for each incorrect answer
\( = (-34) \div (-2) = 34 \div 2 = 17 \)

Therefore, Nalini attempted 17 questions incorrectly.
In simple words: Start by finding marks from correct answers. Subtract this from the total score to find marks from wrong answers. Then divide by the marks per wrong answer to get the number of incorrect attempts.

Exam Tip: Set up the problem clearly showing each step: marks from correct, marks from incorrect, then use division to find the count. Always check: (correct × 3) + (incorrect × -2) should equal the total score.

Question 7. An elevator descends into a mine shaft at the rate of 6 m/min. If the descend starts from 10 m above the ground level, how long will it take to reach the shaft 350 m below the ground level?
Answer: The elevator's starting location is 10 m above the ground, which equals +10 m. Its final location is 350 m below ground, which equals -350 m. The total distance it must descend is calculated as 10 - (-350) = 10 + 350 = 360 m. The descent rate is 6 m/min. Therefore, the time required is 360 m divided by 6 m/min, which equals 60 minutes or 1 hour.
In simple words: The elevator goes down from 10 m above ground to 350 m below ground. The total drop is 360 m. At 6 m every minute, it takes 60 minutes or 1 hour.

Exam Tip: Always treat above-ground positions as positive and below-ground positions as negative. Set up the distance equation carefully before dividing by the rate.

Exercise 1.5

Question 1. 7 - 8 ÷ (-2) + 3 × (-4)
Answer: Working with the given expression 7 - 8 ÷ (-2) + 3 × (-4), we start by performing division and multiplication from left to right: 7 - (-4) + (-12). Next, we handle the brackets to get 7 + 4 - 12. Combining these: 11 - 12 = -1. Therefore, 7 - 8 ÷ (-2) + 3 × (-4) = -1.
In simple words: Do division and multiplication first, then add and subtract from left to right. You should get -1.

Exam Tip: Always follow BODMAS - Division and Multiplication before Addition and Subtraction. Track negative signs carefully when removing brackets.

Question 2. 9 - {7 - 24 ÷ (8 + 6 × 2 - 16)}
Answer: Starting with the expression 9 - {7 - 24 ÷ (8 + 6 × 2 - 16)}, we simplify the innermost brackets first. Inside the parentheses: 8 + 12 - 16 = 4. Then 24 ÷ 4 = 6. Next: 7 - 6 = 1. Finally: 9 - 1 = 8. Therefore, 9 - {7 - 24 ÷ (8 + 6 × 2 - 16)} = 8.
In simple words: Work from the innermost brackets outward. Start with multiplication, then division, then subtraction.

Exam Tip: Use different bracket types systematically - parentheses first, then square brackets, then curly brackets. Show each step clearly for full marks.

Question 3. -11 - [-6 - {3 - 5(8 ÷ 4 - 1)}]
Answer: We simplify the expression -11 - [-6 - {3 - 5(8 ÷ 4 - 1)}] step by step. Inside the smallest brackets: 8 ÷ 4 = 2, so 2 - 1 = 1. Then 5(1) = 5. Next: 3 - 5 = -2. Moving outward: -6 - (-2) = -6 + 2 = -4. Finally: -11 - (-4) = -11 + 4 = -7. Therefore, -11 - [-6 - {3 - 5(8 ÷ 4 - 1)}] = -7.
In simple words: Start with the smallest brackets and work your way out. When you remove a minus sign before a bracket, flip all the signs inside.

Exam Tip: When a negative sign appears before brackets, every term inside reverses its sign. Double-check this crucial step to avoid errors.

Question 4. (-3) × (-12) ÷ (-4) + 3 × 6
Answer: Starting with (-3) × (-12) ÷ (-4) + 3 × 6, we perform multiplication and division from left to right. First: (-3) × (-12) = 36. Then: 36 ÷ (-4) = -9. Next: 3 × 6 = 18. Finally: -9 + 18 = 9. Therefore, (-3) × (-12) ÷ (-4) + 3 × 6 = 9.
In simple words: Multiply and divide from left to right first. Then add or subtract the results at the end.

Exam Tip: Remember that negative times negative equals positive, but negative times positive (or negative divided by positive) equals negative. Keep track of signs throughout.

Question 5. 14 ÷ (3 of 2 - 3 + 4) - 9(5 - 3)
Answer: We simplify 14 ÷ (3 of 2 - 3 + 4) - 9(5 - 3) by starting with the brackets. In the first bracket, 3 of 2 means 3 × 2 = 6, so 6 - 3 + 4 = 7. The second bracket: 5 - 3 = 2. Now we have 14 ÷ 7 - 9 × 2. Division and multiplication give: 2 - 18 = -16. Therefore, 14 ÷ (3 of 2 - 3 + 4) - 9(5 - 3) = -16.
In simple words: The word "of" means multiply. Work out both brackets, then do division and multiplication, then subtract.

Exam Tip: "Of" is an older notation meaning multiplication. Treat it with the same priority as × and ÷ in the order of operations.

Objective Type Questions - Mental Maths

Question 1. Fill in the blanks:
(i) ... is the greatest negative integer.
(ii) ((-10) + 3) + (-12) = (-10) + (3 + ....))
(iii) The product of three negative integers and the product of two positive integers is a .... integer.
(iv) The division of any integer by zero is .....
(v) The integer whose product with (-1) is 22 is ....
(vi) (-15) × ... = 120
(vii) .... ÷ (-6) = -12
(viii) (-10) × ((-15) + 33) = .... × (-15) + (-10) × 33
(ix) .... ÷ (-25) = 0
(x) ((-8) × (-13)) × 27 = (-8) × ((.....) × 27)
(xi) 13 × (-6) = - (..... × .....) = .....
(xii) (-a) + b = b + additive inverse of .....
(xiii) When -25 is divided by ..... the quotient is 5
(xiv) There are .... pairs of integers satisfying a + b = -1
(xv) The value of the expression ((-60) ÷ 12) ÷ (-5) is ....
Answer:
(i) -1 is the greatest negative integer.
(ii) ((-10) + 3) + (-12) = (-10) + (3 + -12)
(iii) The product of three negative integers and the product of two positive integers is a negative integer.
(iv) The division of any integer by zero is not defined
(v) The integer whose product with (-1) is 22 is -22
(vi) (-15) × -8 = 120
(vii) 72 ÷ (-6) = -12
(viii) (-10) × ((-15) + 33) = -10 × (-15) + (-10) × 33
(ix) 0 ÷ (-25) = 0
(x) ((-8) × (-13)) × 27 = (-8) × ((-13) × 27)
(xi) 13 × (-6) = - (13 × 6) = -78
(xii) (-a) + b = b + additive inverse of a
(xiii) When -25 is divided by -5 the quotient is 5
(xiv) There are infinitely many pairs of integers satisfying a + b = -1
(xv) The value of the expression ((-60) ÷ 12) ÷ (-5) is 1
In simple words: -1 is the largest of all negative numbers. When you divide by zero, there is no answer. Negative times negative gives positive. The additive inverse of any number a is -a.

Exam Tip: Study key concepts: -1 is the greatest negative integer, division by zero is undefined, additive inverses reverse the sign, and properties like associativity and distributivity apply to specific operations.

Question 2. State whether the following statements are true (T) or false (F):
(i) For every integer a, |a| is either positive or zero.
(ii) The difference of two negative integers cannot be a positive integer.
(iii) We can write a pair of integers whose sum is not an integer.
(iv) If we divide an integer by (-1), then the result is the additive inverse of the integer.
(v) 1 is the additive identity of integers.
(vi) (-17) × 6 is a whole number.
(vii) (-5) × (-8) × 0 is a positive integer.
(viii) (-237) × 0 is same as 0 × (-89)
(ix) Closure property holds for subtraction of integers.
(x) Commutative property does not hold for subtraction of integers.
(xi) Associative property holds for subtraction of integers.
(xii) Closure property holds for division of integers.
(xiii) Commutative property does not hold for division of integers.
(xiv) Multiplication fact (-8) × (-12) = 96 is same as division fact 96 ÷ (-12) = -8
(xv) [(-32) ÷ 8] ÷ 2 = (-32) ÷ (8 ÷ 2)
(xvi) For every integer a, a ÷ a = 1
(xvii) The successor of 0 × (-10) is 1 × (-10)
Answer:
(i) True
Reason - The absolute value of any integer is always non-negative, meaning it is either positive or zero, never negative.
(ii) False
Reason - Take the example (-2) - (-5) = -2 + 5 = 3. Here, subtracting one negative from another gives a positive result.
(iii) False
Reason - When you add any two integers, the answer is always an integer. This is the closure property that applies to addition of integers.
(iv) True
Reason - For any integer a, dividing by (-1) gives -a, which is the additive inverse (opposite) of a.
(v) False
Reason - Zero is the additive identity for integers (a + 0 = a). One is the multiplicative identity (a × 1 = a).
(vi) False
Reason - (-17) × 6 = -102, a negative integer. Whole numbers include only zero and positive integers, not negatives.
(vii) False
Reason - (-5) × (-8) × 0 = 40 × 0 = 0. Zero is neither positive nor negative.
(viii) True
Reason - Both expressions equal zero because any integer multiplied by zero is zero.
(ix) True
Reason - Subtracting any two integers always produces an integer, so closure holds for subtraction.
(x) True
Reason - Subtraction is not commutative. For example, 5 - 3 = 2 but 3 - 5 = -2. The order matters and changing it gives different answers.
(xi) False
Reason - Subtraction is not associative. For example, (5 - 3) - 2 = 0 but 5 - (3 - 2) = 4. The placement of brackets changes the outcome.
(xii) False
Reason - Dividing two integers does not always yield an integer. For instance, 1 ÷ 2 is not a whole number.
(xiii) True
Reason - Division is not commutative for integers. For example, 12 ÷ (-3) = -4, but (-3) ÷ 12 is not even an integer.
(xiv) True
Reason - These two statements express the same relationship between multiplication and division. If (-8) × (-12) = 96, then 96 ÷ (-12) must equal -8.
(xv) False
Reason - Left side: [(-32) ÷ 8] ÷ 2 = (-4) ÷ 2 = -2. Right side: (-32) ÷ (8 ÷ 2) = (-32) ÷ 4 = -8. Since -2 ≠ -8, division is not associative.
(xvi) False
Reason - When a = 0, the expression 0 ÷ 0 is undefined. The statement is only true when a is a non-zero integer.
(xvii) False
Reason - First, 0 × (-10) = 0, and the successor (next integer) of 0 is 1. However, 1 × (-10) = -10, not 1. Since 1 ≠ -10, the statement is false.
In simple words: Absolute value is always zero or positive. Not all operations (like subtraction and division) follow commutative and associative rules like addition does. Zero times anything is zero. Division by zero is never allowed.

Exam Tip: Memorize which properties hold for each operation: closure applies to all four operations with integers except division; commutativity and associativity do not apply to subtraction or division. Always verify with concrete examples.

Question 3. State whether the following statements are true or false. Justify your answer.
(i) The sum of a positive integer and a negative integer is always a positive integer.
(ii) The sum of two integers is always greater than their difference.
(iii) For any two integers a and b, the inequality -a < b is always true.
(iv) The product of two integers is always greater than the sum of the integers.
Answer:
(i) False
Reason - The sign of the sum depends on the absolute values of both integers. For instance, 3 + (-5) = -2, which is negative, not positive. The result depends on which number has the larger absolute value.
(ii) False
Reason - Consider a = 5 and b = -3. The sum is 5 + (-3) = 2. The difference is 5 - (-3) = 8. Here the sum (2) is actually less than the difference (8), not greater.
(iii) False
Reason - Take a = -5 and b = -10. Then -a = 5 and b = -10. Since 5 > -10, we have -a > b, so the inequality -a < b does not hold in all cases.
(iv) False
Reason - Let a = 1 and b = 2. The sum equals 1 + 2 = 3. The product equals 1 × 2 = 2. Here the product (2) is less than the sum (3), contradicting the claim.
In simple words: Adding a positive and negative number depends on which is bigger. You can subtract and get a larger answer than adding. Multiplying two small positive numbers gives a smaller result than adding them.

Exam Tip: For "always true" statements, find just one counterexample to prove it false. Use small, simple numbers like 1, 2, -1, -2, -3, -5 to test claims quickly.

Multiple Choice Questions

Question 4. If the integers 10, -7, 5, 3, -4 and 0 are marked on the number line, then the integer which lies on the extreme left is
(a) 10
(b) 0
(c) -7
(d) -4
Answer: (c) -7
The smallest integer will be positioned at the extreme left on the number line. Among 10, -7, 5, 3, -4, and 0, the smallest value is -7. Therefore, -7 is the integer that lies on the extreme left.
In simple words: On a number line, the furthest left always has the smallest (most negative) number. Among all these integers, -7 is the smallest.

Exam Tip: The extreme left of a number line always corresponds to the smallest (most negative) integer. Rank all given numbers from smallest to largest to find the answer quickly.

Question 5. On the number line, the value of (-3) × 3 lies on the right hand side of
(a) -10
(b) -6
(c) 0
(d) 9
Answer: (a) -10
First, we calculate (-3) × 3 = -9. On the number line, -9 lies to the right of -10 (since -9 is greater than -10). We can verify that -9 lies to the left of -6, 0, and 9, so it does not lie to the right of these values.
In simple words: First multiply to get -9. Then place -9 on the number line. Moving left means more negative, moving right means less negative (or more positive). So -9 is to the right of -10.

Exam Tip: Always compute the expression first. Then visualize the result on a number line. Remember: to the right means larger (less negative), to the left means smaller (more negative).

Question 6. The value of 5 ÷ (-1) does not lie between
(a) 0 and -10
(b) 0 and 10
(c) -3 and -10
(d) -7 and 7
Answer: (b) 0 and 10
We first compute 5 ÷ (-1) = -5. Now we check each option to see where -5 lies. -5 does lie between 0 and -10 (between the two values on the number line). -5 does lie between -3 and -10. -5 does lie between -7 and 7. However, -5 does not lie between 0 and 10, since -5 is negative and 0-10 is the range of all non-negative numbers up to 10. Since -5 falls outside this range, option (b) is correct.
In simple words: Calculate -5. Check if it fits between each pair. -5 is negative, so it cannot be between 0 and 10 (all positive numbers).

Exam Tip: For "does not lie between," check each option by seeing if the number falls in that range on the number line. A negative number can never lie between 0 and a positive number.

Question 7. The next number in the pattern -62, -37, -12, ... is
(a) 25
(b) 0
(c) 13
(d) -13
Answer: (c) 13
In simple words: Look at how much each number changes. From -62 to -37 is +25. From -37 to -12 is also +25. So the next number is -12 plus 25, which gives 13.

Exam Tip: Always find the difference between consecutive numbers first - this tells you the pattern rule you need to follow.

Question 8. Multiplication of integers satisfies the property of
(a) closure
(b) commutativity
(c) associativity
(d) all of these
Answer: (d) all of these
In simple words: When you multiply two integers, you always get an integer (closure). The order does not matter: 3 × 5 equals 5 × 3 (commutativity). Grouping does not matter either: (2 × 3) × 4 equals 2 × (3 × 4) (associativity). All three properties work for multiplication.

Exam Tip: Learn the definitions of closure, commutativity, and associativity with simple number examples - this helps you recognise which property is being tested.

Question 9. The number of integers between -20 and -10 are
(a) 8
(b) 9
(c) 10
(d) 11
Answer: (b) 9
In simple words: The integers strictly between -20 and -10 (not including -20 and -10 themselves) are: -19, -18, -17, -16, -15, -14, -13, -12, -11. Count them - there are exactly 9.

Exam Tip: When asked for numbers "between" two values, exclude the endpoints unless the question says "between and including".

Question 10. If the sum of two integers is -10 and one of them is 2, then the other is
(a) 8
(b) -8
(c) 12
(d) -12
Answer: (d) -12
In simple words: If two numbers add up to -10 and one is 2, then the other must be -10 minus 2, which is -12. Check: 2 + (-12) = -10. Correct!

Exam Tip: Always verify your answer by substituting back into the original condition to make sure it works.

Question 11. The integer that must be subtracted from -5 to obtain -12 is
(a) 7
(b) -7
(c) 17
(d) -17
Answer: (a) 7
In simple words: Set up the equation: -5 minus something equals -12. Rearrange: something equals -5 minus (-12), which is -5 plus 12, giving 7. Check: -5 - 7 = -12. Correct!

Exam Tip: When a question asks "what must be subtracted", write it as an equation first - this makes it clearer what you are solving for.

Question 12. Which of the following is not the additive inverse of a?
(a) -(-a)
(b) -a
(c) a ÷ (-1)
(d) a × (-1)
Answer: (a) -(-a)
In simple words: The additive inverse of a number is what you add to it to get zero. For any number a, the additive inverse is -a. Now check each option: -(-a) equals a (not -a), so it is not the additive inverse. The other three all simplify to -a, so they are additive inverses.

Exam Tip: Always simplify each option fully before deciding - do not just glance at the form of the expression.

Question 13. 0 ÷ (-10) is equal to
(a) 0
(b) -1
(c) -10
(d) none of these
Answer: (a) 0
In simple words: Zero divided by any non-zero number always gives zero. It does not matter what the divisor is - if the dividend is zero, the answer is always zero.

Exam Tip: Memorise this rule: 0 divided by any non-zero integer equals 0. This is different from division by zero, which is undefined.

Question 14. (-33) × 102 + (-33) × (-2) is equal to
(a) 3300
(b) -3300
(c) 3432
(d) -3432
Answer: (b) -3300
In simple words: Both terms share the factor -33. Pull it out: -33 × (102 + (-2)) = -33 × (102 - 2) = -33 × 100 = -3300. This method (the distributive property) makes the calculation much simpler than working out each term separately.

Exam Tip: Look for common factors you can pull out - the distributive property often simplifies long multiplication problems dramatically.

Question 15. 101 × (-1) + 0 ÷ (-1) is equal to
(a) -101
(b) 101
(c) -102
(d) 102
Answer: (a) -101
In simple words: Work out each operation. 101 × (-1) gives -101. Then 0 ÷ (-1) gives 0 (because zero divided by anything non-zero is zero). Add them: -101 + 0 = -101.

Exam Tip: When you see a mix of operations, do them step-by-step in order of precedence (multiply and divide before add and subtract).

Question 16. (-3) × 5 is not equal to
(a) 3 × (-5)
(b) -(3 × 5)
(c) (-3) × (-5)
(d) 5 × (-3)
Answer: (c) (-3) × (-5)
In simple words: (-3) × 5 equals -15. Now check each option: 3 × (-5) = -15 (equal), -(3 × 5) = -15 (equal), (-3) × (-5) = 15 (not equal - this is positive), and 5 × (-3) = -15 (equal). Only option (c) gives a different answer.

Exam Tip: Remember: negative times negative equals positive, negative times positive equals negative, and positive times positive equals positive.

Question 17. If a and b are two integers, then which of the following may not be an integer?
(a) a + b
(b) a - b
(c) a × b
(d) a ÷ b
Answer: (d) a ÷ b
In simple words: Integers are closed under addition, subtraction, and multiplication - this means if you add, subtract, or multiply two integers, you always get an integer back. However, division is different. If you divide one integer by another, you may get a fraction that is not an integer. For example, 1 ÷ 2 = 0.5, which is not an integer.

Exam Tip: The key word is "closed" - learn which operations preserve the integer set and which do not. Division always fails the closure test for integers.

Question 18. For a non-zero integer a, which of the following is not defined?
(a) a ÷ 0
(b) 0 ÷ a
(c) a ÷ 1
(d) a ÷ a
Answer: (a) a ÷ 0
In simple words: Division by zero is never allowed in mathematics - it is undefined and has no meaning. All other divisions here are valid: 0 ÷ a = 0, a ÷ 1 = a, and a ÷ a = 1 (for non-zero a).

Exam Tip: Memorise this rule: you can never divide by zero, no matter what the numerator is. This is a fundamental restriction in mathematics.

Question 19. Statement I: 0 is the multiplicative identity for integers. Statement II: The number 0 is an integer.
Answer: Statement I is false. The multiplicative identity for integers is 1, not 0. When you multiply any integer by 1, you get that same integer back (a × 1 = a). When you multiply by 0, you always get 0, which is not the original number.
Statement II is true. The number 0 is indeed an integer. It is neither positive nor negative - it sits at the boundary between positive and negative integers on the number line.
In simple words: The multiplicative identity is the number that does not change other numbers when you multiply by it. That is 1, not 0. And yes, 0 counts as an integer.

Exam Tip: Do not confuse the additive identity (0, the number you add to get no change) with the multiplicative identity (1, the number you multiply by to get no change).

Question 20. Statement I: When we multiply the absolute values of two non-zero integers, we always get a positive integer. Statement II: |-a| = |a|, for all integers a.
Answer: Statement I is true. The absolute value of any non-zero integer is always positive. When you multiply two positive numbers together, you always get a positive result. For instance, |-7| = 7 and |-3| = 3, and 7 × 3 = 21, which is positive.
Statement II is true. By the definition of absolute value, the absolute value of a number and the absolute value of its opposite are always equal. For example, |-5| = 5 and |5| = 5, so they are the same. This holds for every integer a.
In simple words: Absolute value strips away the minus sign and leaves only the distance from zero (a positive number). Multiplying two positive numbers gives a positive result. And a number and its opposite have the same absolute value.

Exam Tip: When evaluating statements about absolute value, think of absolute value as "distance from zero" - this makes the properties much clearer.

Question 21. Statement I: Multiplying two integers with different signs results in a positive integer. Statement II: If a, b and c are integers, then a × (b × c) = (a × b) × (a × c).
Answer: Statement I is false. When you multiply two integers with different signs, the product is always negative. For example, 3 × (-5) = -15, which is negative, not positive.
Statement II is false. This equation is incorrect. The correct associative property of multiplication is a × (b × c) = (a × b) × c. The expression (a × b) × (a × c) is different - it applies the distributive law incorrectly by multiplying a by both b and c separately and then multiplying those products together, which does not equal a × (b × c) in general.
In simple words: Negative times positive always gives negative. And the associative property means you can move brackets around when multiplying three numbers, but you cannot rearrange the numbers themselves or introduce extra copies of them.

Exam Tip: Distinguish between the associative property (moving brackets) and the distributive property (multiplying one factor by a sum). They are different rules.

Question 22. Statement I: If a is any non-zero integer, then 0 ÷ a = 0 and a ÷ a = 1. Statement II: All the four operations - addition, subtraction, multiplication and division follow the property of closure for all integers.
Answer: Statement I is true. For any non-zero integer a, dividing zero by a always gives zero (since 0 = a × 0). Dividing a by itself always gives 1 (since a = a × 1). Both statements hold without exception.
Statement II is false. Integers are closed under addition, subtraction, and multiplication - meaning these operations always produce an integer when applied to two integers. However, division does not satisfy closure. For example, 1 ÷ 2 = 0.5, which is not an integer. So division fails the closure property for the set of integers.
In simple words: Zero divided by anything non-zero is zero, and a number divided by itself is one. But division as a whole is not closed for integers because you can end up with a fraction.

Exam Tip: Closure means "the operation never leaves the set". Learn which operations are closed for integers and which are not - this appears frequently on exams.

Question 1. Evaluate the following:
(i) (-7) × (-9) × (-11)
(ii) (-5) × 7 × (-6) × (-8)
(iii) (-1024) ÷ 32
(iv) (-216) ÷ (-12)
Answer:
(i) (-7) × (-9) × (-11)
\[ = [(-7) \times (-9)] \times (-11) \]
\[ = 63 \times (-11) \]
\[ = -693 \]
(ii) (-5) × 7 × (-6) × (-8)
\[ = [(-5) \times 7] \times [(-6) \times (-8)] \]
\[ = (-35) \times 48 \]
\[ = -1680 \]
(iii) (-1024) ÷ 32
\[ = \frac{-1024}{32} \]
\[ = -(1024 \div 32) \]
\[ = -32 \]
(iv) (-216) ÷ (-12)
\[ = \frac{-216}{-12} \]
\[ = +(216 \div 12) \]
\[ = 18 \]
In simple words: Group the numbers cleverly - multiply pairs together first to make the calculation simpler. When both numbers have the same sign, the answer is positive. When they have opposite signs, the answer is negative.

Exam Tip: Use grouping and the associative property to rearrange calculations in a convenient order. This reduces mistakes and speeds up your work.

Question 2. What will be the sign of the product if we multiply 39 negative integers and 98 positive integers?
Answer: Product of 39 negative integers: Since 39 is an odd number, the product of 39 negative integers will be negative. (An even count of negatives gives a positive; an odd count gives a negative.)
Product of 98 positive integers: The product of 98 positive integers is positive. Positive numbers multiplied together always give a positive result.
Combined product: (negative integer) × (positive integer) = negative integer
In simple words: Count how many negative numbers you are multiplying: if the count is odd, the final answer is negative; if even, the answer is positive. Then multiply that result by a positive number - the sign does not change.

Exam Tip: When multiplying many integers, count the negatives: odd count means negative answer, even count means positive answer. This rule alone can solve the whole problem.

Question 3. Use the sign >, < or = in the box to make the following statements true:
(i) (-15) + 38 ___ 27 + (-50)
(ii) (-13) × 0 × (-5) ___ (-7) × (-6) × 14
(iii) (-18) ÷ (-3) ___ (-10) + (-15) + 31
(iv) (-5) × (-7) × (-10) ___ (-1400) ÷ (-4)
Answer:
(i) LHS = (-15) + 38 = 23
RHS = 27 + (-50) = -23
Since 23 > -23, the sign is >

(ii) LHS = (-13) × 0 × (-5) = 0 (because one factor is zero)
RHS = (-7) × (-6) × 14 = 42 × 14 = 588
Since 0 < 588, the sign is <

(iii) LHS = (-18) ÷ (-3) = 6
RHS = (-10) + (-15) + 31 = -25 + 31 = 6
Since 6 = 6, the sign is =

(iv) LHS = (-5) × (-7) × (-10) = 35 × (-10) = -350
RHS = (-1400) ÷ (-4) = 350
Since -350 < 350, the sign is <
In simple words: Work out the left side, then the right side. Compare the results and choose the correct symbol. Be careful with negative numbers and with any zeros - they change the result completely.

Exam Tip: Always calculate both sides fully before comparing - do not try to predict the relationship without doing the arithmetic. One calculation mistake gives a wrong symbol.

Question 3. Compare the following pairs of numbers using the appropriate symbol (<, =, or >).
(ii) (−13) × 0 × (−5) and (−7) × (−6) × 14
(iii) (−18) ÷ (−3) and (−10) + (−15) + 31
(iv) (−5) × (−7) × (−10) and (−1400) ÷ (−4)
Answer:
(ii) Work out the left side: (−13) × 0 × (−5) = 0. Work out the right side: (−7) × (−6) × 14 = 42 × 14 = 588. Since 0 is less than 588, we have (−13) × 0 × (−5) < (−7) × (−6) × 14.

(iii) Work out the left side: (−18) ÷ (−3) = 6. Work out the right side: (−10) + (−15) + 31 = −25 + 31 = 6. Since both sides equal 6, we have (−18) ÷ (−3) = (−10) + (−15) + 31.

(iv) Work out the left side: (−5) × (−7) × (−10) = 35 × (−10) = −350. Work out the right side: (−1400) ÷ (−4) = 350. Since −350 is less than 350, we have (−5) × (−7) × (−10) < (−1400) ÷ (−4).
In simple words: To compare two expressions, work them out one at a time, then use the correct symbol to show which is bigger, smaller, or equal.

Exam Tip: Always evaluate both sides completely before comparing. A common mistake is forgetting the order of operations or making a sign error with negative numbers.

Question 4. Is ((−45) ÷ (−15)) ÷ (−3) = (−45) ÷ ((−15) ÷ (−3))?
Answer: Start with the left side: ((−45) ÷ (−15)) ÷ (−3) = 3 ÷ (−3) = −1. Now work out the right side: (−45) ÷ ((−15) ÷ (−3)) = (−45) ÷ 5 = −9. Since −1 ≠ −9, the two sides are not equal. Therefore, ((−45) ÷ (−15)) ÷ (−3) ≠ (−45) ÷ ((−15) ÷ (−3)). This shows that division is not associative for integers - the order of grouping changes the result.
In simple words: When you divide, it matters which division you do first. If you change the grouping, you get a different answer, so division is not associative.

Exam Tip: To prove something is not associative, find one counterexample where changing the grouping gives different results. Always show both sides of your calculation clearly.

Question 5. Simplify the following:
(i) (−7) + (−6) ÷ 2 - {(−5) × (−4) - (3 - 5)}
(ii) 11 - [7 - {5 - 3(9 - 3 - 6)}]
Answer:
(i) Start with the expression (−7) + (−6) ÷ 2 − {(−5) × (−4) − (3 − 5)}. First, simplify division, multiplication, and innermost brackets: (−7) + (−3) − {20 − (−2)}. Next, remove the round brackets: (−7) + (−3) − {20 + 2}. Then simplify the curly brackets: (−7) + (−3) − 22. Finally, combine from left to right: −7 − 3 − 22 = −32.

(ii) Start with the expression 11 - [7 - {5 - 3(9 - 3 - 6)}]. First, work out what is under the overline: 9 - 3 - 6 becomes 9 - (−3) = 9 + 3. Now simplify inside the round brackets: 5 - 3(12). Then simplify the multiplication: 5 - 36 = −31. Next, remove the curly brackets: 7 - (−31) = 7 + 31. Now simplify the square brackets: 11 - 38. Finally, compute: 11 - 38 = −27.
In simple words: Always work from the inside out, removing the smallest brackets first. Handle multiplication and division before addition and subtraction, and remember that removing a minus sign in front of brackets flips all the signs inside.

Exam Tip: Set out your working step by step and label each step (e.g. "Simplifying ÷, ×, ( )", "Removing ( )") to make it clear you understand the order of operations.

Question 6. Arrange the numbers −5 + 8, 3 × (−4), −15 ÷ 5, |−5 - 7|, −7 - 3 × (−2) and −7 - 4 ÷ (−2) in descending order.
Answer: First, work out each expression: −5 + 8 = 3; 3 × (−4) = −12; −15 ÷ 5 = −3; |−5 − 7| = |−12| = 12; −7 − 3 × (−2) = −7 − (−6) = −7 + 6 = −1; −7 − 4 ÷ (−2) = −7 − (−2) = −7 + 2 = −5. The numbers are now: 3, −12, −3, 12, −1, −5. Arranging them in descending order (largest to smallest): 12 > 3 > −1 > −3 > −5 > −12. In terms of the original expressions, this is: |−5 − 7| > −5 + 8 > −7 − 3 × (−2) > −15 ÷ 5 > −7 − 4 ÷ (−2) > 3 × (−4).
In simple words: Evaluate each expression to get a single number, then arrange all the numbers from biggest to smallest. Remember that positive numbers come before negative numbers, and numbers closer to zero are bigger than numbers further from zero.

Exam Tip: Be careful with absolute value - |−12| becomes 12 (positive). Always evaluate inside absolute value bars before removing them, and double-check your signs when simplifying expressions with negatives.

Question 7. Write a pair of integers whose product is −12 and there lies seven integers between them.
Answer: We need two integers whose product is −12 with exactly seven integers between them. If seven integers lie between two integers a and b, then the absolute difference between a and b must be 8. The factor pairs of −12 are: (1, −12), (−1, 12), (2, −6), (−2, 6), (3, −4), (−3, 4). Now check the absolute difference for each pair: |1 − (−12)| = 13; |(−1) − 12| = 13; |2 − (−6)| = 8; |(−2) − 6| = 8; |3 − (−4)| = 7; |(−3) − 4| = 7. Only two pairs have an absolute difference of 8, so the required pairs are (2, −6) or (−2, 6). We can verify: 2 × (−6) = −12, and the seven integers between 2 and −6 are −5, −4, −3, −2, −1, 0, 1.
In simple words: Find two numbers that multiply to give −12. Count how many whole numbers fit between them - if there are exactly 7, that is your answer. If not, try the next factor pair.

Exam Tip: Check all factor pairs of the given product. For each pair, use absolute value to find the difference, and remember that if the difference is n, then there are n - 1 integers between them.

Question 8. A shopkeeper earns a profit of Rs.2 by selling a pen and incurs a loss of 50 paise per pencil and loss of 15 paise per eraser while selling pencils and erasers of old stock. On a particular day, he earns a profit of Rs.10. If he sold 10 pens and the number of pencils and erasers he sold are in the ratio 7 : 10, then find the number of pencils and erasers he sold on that day.
Answer: Convert all amounts to paise for consistency: Profit per pen = Rs.2 = 200 paise; Loss per pencil = 50 paise (written as −50 paise); Loss per eraser = 15 paise (written as −15 paise). The shopkeeper sold 10 pens. Profit from pens = 10 × 200 = 2000 paise. Total profit for the day = Rs.10 = 1000 paise. The number of pencils and erasers are in the ratio 7 : 10. Let pencils sold = 7x and erasers sold = 10x. Loss from pencils = 7x × 50 = 350x paise. Loss from erasers = 10x × 15 = 150x paise. Total loss = 350x + 150x = 500x paise. Using the formula: Net profit = Profit from pens - Total loss, we have 1000 = 2000 − 500x. Solving: 500x = 2000 − 1000 = 1000, so x = 2. Therefore, pencils sold = 7 × 2 = 14 and erasers sold = 10 × 2 = 20. The shopkeeper sold 14 pencils and 20 erasers on that day.
In simple words: Work out how much profit came from pens. Subtract this from the total profit to find the total loss. Use the ratio to set up an equation and solve for the unknown.

Exam Tip: Always convert all currency to the same unit (paise or rupees) before combining. Use the ratio correctly by introducing a variable (like x) for each part, and set up the profit/loss equation carefully.