NCERT Solutions for Class 7 Maths Chapter 1: Integers — Complete Guide with Step-by-Step Solutions

The number line is your best friend when working with integers. Here is how operations look on it:

Addition: To add a positive integer, move right. To add a negative integer, move left.
- $3 + 5$: Start at $3$, move $5$ steps right $\to 8$.
- $3 + (-5)$: Start at $3$, move $5$ steps left $\to -2$.

Subtraction: Subtracting is the same as adding the opposite.
- $3 - 5 = 3 + (-5) = -2$.
- $3 - (-5) = 3 + 5 = 8$.

This simple idea — subtracting a negative is the same as adding — is one of the most important concepts in this chapter.

These rules apply to both multiplication and division:

Memory aid: Same signs $\to$ positive. Different signs $\to$ negative.

Problem: Using the number line, find $(-5) + 7$.

Solution:
Start at $-5$ on the number line. Since we are adding $+7$ (a positive integer), move $7$ steps to the right.

Answer: $2$.

Problem: Find $(-8) - (-3)$.

Solution:
Subtracting $(-3)$ is the same as adding $(+3)$:

On the number line: start at $-8$, move $3$ steps right to reach $-5$.

Answer: $-5$.

Problem: Verify that integers are closed under addition using $a = -15$ and $b = 9$.

Solution:

Is $-6$ an integer? Yes.

This means the sum of two integers is always an integer. Therefore, integers are closed under addition.

Note: Closure under addition means: if $a$ and $b$ are integers, then $a + b$ is also an integer. This is ALWAYS true.

Problem: Verify that $a + b = b + a$ for $a = -11$ and $b = 7$.

Solution:

Since $a + b = b + a = -4$, the commutative property of addition is verified.

Important: This property says the order of addition does not matter. $(-11) + 7$ and $7 + (-11)$ give the same result.

Problem: Verify that $(a + b) + c = a + (b + c)$ for $a = -3$, $b = 5$, $c = -8$.

Solution:

Since both sides equal $-6$, the associative property of addition is verified.

This property says you can group additions in any way without changing the result.

Problem: What is the additive identity? Verify for $a = -13$.

Solution:
The additive identity is $0$, because adding $0$ to any integer leaves it unchanged.

In both cases, the result is $-13$ itself. Hence $0$ is the additive identity.

Problem: Find the additive inverse of $-17$. Verify your answer.

Solution:
The additive inverse of an integer $a$ is the integer that, when added to $a$, gives $0$.

Additive inverse of $-17$ is $17$, because:

General rule: The additive inverse of $a$ is $-a$. The additive inverse of $-a$ is $a$.

Problem: Check whether $a - b = b - a$ for $a = 5$ and $b = 9$.

Solution:

Since $-4 \neq 4$, we conclude that $a - b \neq b - a$ in general.

Therefore, subtraction is NOT commutative for integers.

Problem: Check whether $(a - b) - c = a - (b - c)$ for $a = 10$, $b = 3$, $c = 2$.

Solution:

Since $5 \neq 9$, subtraction is NOT associative for integers.

Problem: The temperature at noon was $5°C$. It dropped by $8°C$ by midnight. What was the temperature at midnight?

Solution:
Temperature at midnight $= 5 + (-8) = 5 - 8 = -3°C$.

Answer: The temperature at midnight was $-3°C$.

Interpretation: A drop of $8°C$ from $5°C$ takes the temperature below zero.

Problem: Find $3 \times (-1)$.

Solution:
Positive $\times$ Negative $=$ Negative.

Pattern: Multiplying any integer by $(-1)$ simply changes its sign. So $(-1) \times a = -a$ for every integer $a$.

Problem: Find $(-1) \times 225$.

Solution:

Again, multiplying by $(-1)$ flips the sign.

Problem: Find $(-21) \times (-30)$.

Solution:
Negative $\times$ Negative $=$ Positive.

Multiply the absolute values: $21 \times 30 = 630$. Since both signs are negative (same sign), the result is positive.

Problem: Evaluate $(-8) \times (-2) \times (-3)$.

Solution:
Step 1: $(-8) \times (-2) = 16$ (negative $\times$ negative $=$ positive)

Step 2: $16 \times (-3) = -48$ (positive $\times$ negative $=$ negative)

Rule: When you multiply an odd number of negative integers, the result is negative. When you multiply an even number of negative integers, the result is positive.

Problem: Find $(-125) \times 0 \times (-14) \times 7$.

Solution:
Any product involving $0$ is $0$, regardless of the other factors.

Key fact: $a \times 0 = 0$ for every integer $a$. This is called the zero product property.

Problem: Find the product $(-4) \times (-5) \times (-2) \times (-1)$.

Solution:
We have $4$ negative factors (even count), so the final product is positive.

Step 1: $(-4) \times (-5) = 20$
Step 2: $20 \times (-2) = -40$
Step 3: $(-40) \times (-1) = 40$

Answer: $40$.

Problem: The product of two integers is $-24$. If one integer is $4$, find the other.

Solution:
Let the other integer be $x$.

Verification: $4 \times (-6) = -24$. Correct.

Answer: The other integer is $-6$.

Problem: Verify that $a \times (b + c) = a \times b + a \times c$ for $a = 12$, $b = -4$, $c = 2$.

Solution:
LHS:

RHS:

Since LHS $=$ RHS $= -24$, the distributive property is verified.

Problem: Observe the pattern and fill in the blanks:
$3 \times (-1) = -3$
$3 \times (-2) = -6$
$3 \times (-3) = -9$
$3 \times (-4) = \_\_$
$3 \times (-5) = \_\_$

Solution:
The pattern shows that each time we multiply $3$ by the next negative integer, the product decreases by $3$:

This pattern confirms the rule: positive $\times$ negative $=$ negative, and the magnitude increases.

Problem: A submarine descends $3$ metres every minute. If it starts at sea level, what is its position after $7$ minutes?

Solution:
Each minute, the submarine moves $-3$ m (downward is negative).

After $7$ minutes:

The submarine is $21$ metres below sea level.

Answer: The position is $-21$ m (or $21$ m below sea level).

Problem: Verify that integers are closed under multiplication using $a = -7$ and $b = 8$.

Solution:

Is $-56$ an integer? Yes.

Therefore, the product of two integers is always an integer, confirming closure under multiplication.

Problem: Verify $a \times b = b \times a$ for $a = -9$ and $b = -6$.

Solution:

Since both equal $54$, multiplication is commutative for integers.

This means: The order in which you multiply two integers does not affect the product.

Problem: Verify $(a \times b) \times c = a \times (b \times c)$ for $a = -2$, $b = 3$, $c = -5$.

Solution:

Both sides equal $30$. Multiplication is associative for integers.

This means: You can group factors in any way you like.

Problem: Evaluate $(-7) \times 48$ using the distributive property.

Solution:
Break $48 = 50 - 2$:

Answer: $-336$.

Why this works: The distributive property says $a \times (b + c) = a \times b + a \times c$. This makes mental arithmetic much easier.

Problem: Evaluate $(-25) \times 102$ using the distributive property.

Solution:
Break $102 = 100 + 2$:

Answer: $-2550$.

Problem: Evaluate $(-1) \times (-1) \times (-1) \times \ldots$ (100 times).

Solution:
Since $100$ is even, the product of an even number of $(-1)$s is $+1$.

General rule:

Problem: What is the multiplicative identity? Verify for $a = -13$.

Solution:
The multiplicative identity is $1$, because multiplying any integer by $1$ leaves it unchanged.

In both cases, the result is $-13$ itself. Hence $1$ is the multiplicative identity.

Note: $0$ is the additive identity, while $1$ is the multiplicative identity. Do not confuse them.

Problem: Evaluate $(-3) \times (-2) \times (-1) \times 0 \times 1 \times 2 \times 3$.

Solution:
Any product involving $0$ is $0$, no matter how many other factors there are.

Tip: In an exam, if you spot a $0$ anywhere in a multiplication chain, immediately write the answer as $0$. Do not waste time computing the other factors.

Problem: Which of the following is NOT true for integers?
(a) Closure under multiplication
(b) Commutativity of multiplication
(c) Associativity of multiplication
(d) Division is commutative

Solution:
Options (a), (b), and (c) are all true for integers.

Option (d) is NOT true. Division is not commutative because $6 \div 3 = 2$ but $3 \div 6 = 0.5$, and $2 \neq 0.5$.

Problem: Simplify: $(-15) \times 8 + (-15) \times (-2)$.

Solution:
Using the distributive property in reverse (factoring out $-15$):

Answer: $-90$.

Key insight: When the same factor appears in both terms, factor it out using the distributive property to simplify the calculation.

Problem: Evaluate the following:
(i) $(-36) \div (-4)$
(ii) $(-201) \div 3$
(iii) $0 \div (-5)$

Solution:

(i) Negative $\div$ Negative $=$ Positive:

(ii) Negative $\div$ Positive $=$ Negative:

(iii) $0$ divided by any non-zero integer is always $0$:

Problem: Verify that $a \div b \neq b \div a$ for $a = 8$ and $b = -4$.

Solution:

Since $-2 \neq -0.5$, division is NOT commutative.

Problem: Verify that $a \div (b \div c) \neq (a \div b) \div c$ for $a = 8$, $b = -4$, $c = 2$.

Solution:
LHS:

RHS:

Since $-4 \neq -1$, division is NOT associative for integers.

Problem: Show that integers are not closed under division.

Solution:
Take $a = 7$ and $b = 2$ (both integers).

$3.5$ is not an integer. Therefore, the quotient of two integers is not always an integer.

This means integers are NOT closed under division.

Key difference: Integers ARE closed under addition, subtraction, and multiplication, but NOT under division.

Problem: Why is division by zero undefined?

Solution:
If $a \div 0 = x$, then $x \times 0 = a$. But $x \times 0 = 0$ for every $x$.
- If $a \neq 0$, no value of $x$ works. Contradiction.
- If $a = 0$, every value of $x$ works. No unique answer.

In both cases, division by $0$ gives no meaningful result. Therefore, division by zero is undefined.

Remember: $0 \div a = 0$ (for $a \neq 0$), but $a \div 0$ is undefined.

Problem: Evaluate $[(-16) \div 4] + [(-16) \div (-8)]$.

Solution:

Answer: $-2$.

Problem: $\_\_ \div (-9) = -5$. Find the missing integer.

Solution:
Let the missing integer be $x$.

Verification: $45 \div (-9) = -5$. Correct.

Answer: $45$.

Problem: What happens when you divide an integer by $1$? By $(-1)$? By itself?

Solution:
- $a \div 1 = a$ (dividing by $1$ leaves the integer unchanged)
- $a \div (-1) = -a$ (dividing by $-1$ changes the sign)
- $a \div a = 1$ for any $a \neq 0$ (any non-zero integer divided by itself is $1$)

Examples:

Problem: Simplify: $(-30) \div 5 + (-18) \div (-3) - (-2) \times 4$.

Solution:
Follow BODMAS (division and multiplication before addition and subtraction):

Step 1: $(-30) \div 5 = -6$
Step 2: $(-18) \div (-3) = 6$
Step 3: $(-2) \times 4 = -8$

Now combine:

Answer: $8$.

Problem: The temperature dropped by $24°C$ in $6$ hours. If the drop was uniform, what was the change in temperature per hour?

Solution:
Total drop $= -24°C$ (negative because it dropped).
Time $= 6$ hours.

The temperature dropped by $4°C$ each hour.

Answer: $-4°C$ per hour.

You might wonder why we spend so much time on abstract properties like commutativity and associativity. The answer is that these properties are the foundation of algebra.

When you learn to solve equations in Chapter 4 (Simple Equations), you will use these properties constantly. For example:
- Commutativity lets you rearrange terms: $x + 3 = 3 + x$.
- Associativity lets you group terms: $(2 + x) + 5 = 2 + (x + 5)$.
- Distributive property lets you expand or factor: $3(x + 2) = 3x + 6$.
- Additive inverse lets you "cancel" terms: $x + 5 - 5 = x$.

Every property you verify in this chapter is a tool you will use throughout mathematics.

Answer 1:

Group positives: $15 + 6 = 21$. Group negatives: $-12 - 8 - 3 = -23$. Total: $21 + (-23) = -2$.

Answer 2:
LHS: $(-8) \times (3 + (-5)) = (-8) \times (-2) = 16$.
RHS: $(-8) \times 3 + (-8) \times (-5) = -24 + 40 = 16$.
LHS $=$ RHS. Verified.

Answer 3:
$75$ is odd, so $(-1)^{75} = -1$.

Answer 4:

Answer 5:

Answer 6:

Answer 7:
LHS: $b \div c = (-6) \div (-2) = 3$. $a \div (b \div c) = 24 \div 3 = 8$.
RHS: $(a \div b) \div c = [24 \div (-6)] \div (-2) = (-4) \div (-2) = 2$.
$8 \neq 2$, so division is NOT associative.

Answer 8:

Problem: Observe the pattern and find the next two terms:
$(-3) \times (-4) = 12$
$(-3) \times (-3) = 9$
$(-3) \times (-2) = 6$
$(-3) \times (-1) = 3$
$(-3) \times 0 = ?$
$(-3) \times 1 = ?$
$(-3) \times 2 = ?$

Solution:
The pattern decreases by $3$ each time:

This pattern beautifully illustrates why a negative times a positive is negative.

Problem: Find two integers whose sum is $-5$ and whose product is $6$.

Solution:
We need two numbers that add to $-5$ and multiply to $6$.

Since the product is positive and the sum is negative, both numbers must be negative.

Try $-2$ and $-3$:
- Sum $= (-2) + (-3) = -5$ (correct)
- Product $= (-2) \times (-3) = 6$ (correct)

Answer: The two integers are $-2$ and $-3$.

Problem: Replace $*$ with $<$, $>$, or $=$ in: $(-8) \times (-9) \times (-10) \;*\; 8 \times (-9) \times 10$.

Solution:
LHS: $(-8) \times (-9) \times (-10)$
$3$ negative factors (odd) $\to$ negative result.
Magnitude: $8 \times 9 \times 10 = 720$.
So LHS $= -720$.

RHS: $8 \times (-9) \times 10$
$1$ negative factor (odd) $\to$ negative result.
Magnitude: $8 \times 9 \times 10 = 720$.
So RHS $= -720$.

Since $-720 = -720$, the answer is $=$.

Key insight: Both products have the same absolute value ($720$) and the same sign (negative, due to an odd count of negative factors in each case).