NCERT Solutions for Class 7 Maths Chapter 1: Integers Exercise 1.4

Welcome to your YoLearn math sketchpad! In this guide, we are focusing on integers ex 1 4 class 7 ncert, which centers on the operation of dividing integers. You have already mastered addition, subtraction, and multiplication of negative numbers. Now, we will master division—the inverse operation of multiplication. Division allows us to calculate changes in elevation, temperature drops, or divide quantities into negative and positive intervals. By understanding the sign rules, BODMAS bracket sequence, and division properties, you will confidently solve every question in Exercise 1.4. Let's work through these rules with simple, clear steps!

Understanding Division of Integers

Division is the opposite of multiplication. For example, since 3 × 4 = 12, we can say 12 ÷ 4 = 3. Similarly, this inverse relationship applies to negative integers! If (-3) × 4 = -12, then it must mean that (-12) ÷ 4 = -3 and (-12) ÷ (-3) = 4. When dividing integers, we calculate the quotient of their absolute values first, and then apply simple sign rules to determine whether our answer is positive or negative.

The Step-by-Step Division Process

  1. Divide Absolute Values — Ignore the positive or negative signs for a moment. Divide the absolute numerical values just like ordinary whole numbers.
  2. Determine the Signs — Check the signs of the dividend and divisor. If both signs are identical (both positive or both negative), the result is positive. If the signs are different, the result is negative.
  3. Write the Final Quotient — Combine the numerical result with the determined sign. For example, (-45) ÷ 9 gives -5 because the signs are different, while (-45) ÷ (-9) gives 5 because the signs are the same.

Properties of Integer Division

Closure Property
Integers are not closed under division. If we divide -8 by 3, the result is a fraction, which is not an integer.
Commutative Property
Division is not commutative. This means a ÷ b is not equal to b ÷ a. For example, (-10) ÷ 2 = -5, but 2 ÷ (-10) is not equal to -5.
Division by Zero
For any integer 'a', dividing by zero is undefined. (a ÷ 0) is not defined, while dividing zero by any non-zero integer is always zero (0 ÷ a = 0).
Division by One
Dividing any integer by 1 gives the same integer itself (a ÷ 1 = a). However, dividing an integer by -1 changes its sign (a ÷ -1 = -a).

BODMAS and Double Signs Tip

Always solve calculations inside brackets first! In questions like [(-30) ÷ 5] ÷ 2, prioritize the inner square brackets first: (-30) ÷ 5 = -6. Then, divide by the outer integer: -6 ÷ 2 = -3. Don't skip bracket boundaries or you will get incorrect quotients!

Practice Questions with Solutions

  • Q: Evaluate: (-30) ÷ 10 A: Step 1: Divide the numerical values without signs: 30 ÷ 10 = 3. Step 2: Compare the signs of the original numbers. One is negative (-30) and the other is positive (10). Different signs yield a negative result. Step 3: Write the quotient with the correct sign. Final answer: -3
  • Q: Evaluate: [(-36) ÷ 12] ÷ 3 A: Step 1: Solve the division inside the square brackets first: (-36) ÷ 12. Step 2: 36 ÷ 12 = 3. Since the signs are opposite, the result is -3. Step 3: Substitute the value back into the expression: (-3) ÷ 3. Step 4: 3 ÷ 3 = 1. Since the signs are opposite, the quotient is negative. Final answer: -1
  • Q: Write five pairs of integers (a, b) such that a ÷ b = -3. A: Step 1: We need to find pairs of numbers where the absolute value division is 3, and one number in each pair must be negative. Step 2: Create normal pairs that divide to 3: (6, 2), (9, 3), (12, 4), (15, 5), (18, 6). Step 3: Apply a negative sign to one integer in each pair to make the result -3. Final answer: (-6, 2), (9, -3), (-12, 4), (15, -5), and (-18, 6)
  • Q: An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above ground level, how long will it take to reach -350 m? A: Step 1: Find the total vertical distance. Starting height = +10 m. Final depth = -350 m. Step 2: Total distance = Final Position - Initial Position = -350 m - 10 m = -360 m. Step 3: The descent rate is 6 m/min downwards, represented as -6 m/min. Step 4: Time = Total Distance ÷ Rate of descent = (-360) ÷ (-6) = 60 minutes. Final answer: 60 minutes (or 1 hour)

Frequently Asked Questions

What is the rule of signs for dividing integers?

When dividing integers, if both numbers have the same sign, the quotient is positive. If the numbers have different signs, the quotient is negative.

Why is dividing an integer by zero undefined?

Division is the opposite of multiplication. Since multiplying any number by 0 can never yield a non-zero integer, division by zero is mathematically impossible and thus undefined.

Are the associative properties applicable to the division of integers?

No, division is not associative for integers. Changing the grouping of the terms changes the final quotient.