CBSE Class 7 Maths: Integers Exercise 1.3

Welcome, Class 7 students, to the exciting world of Integers! In Class 6, you learned what integers are and how to add and subtract them. Now, in Chapter 1: Integers, we are going to dive deeper into two more fundamental operations: multiplication and division. Exercise 1.3 specifically focuses on mastering these skills, which are crucial not just for your current math lessons but for all future topics in mathematics and science. You'll learn the simple yet powerful rules for multiplying and dividing positive and negative numbers. By the end of this page, you will confidently solve problems involving products and quotients of integers, understand their properties, and avoid common sign errors. Let's conquer Integers Ex 1.3 together!

Understanding Multiplication and Division of Integers

Integers include positive numbers, negative numbers, and zero. When we multiply or divide integers, the most important thing to remember are the rules for signs. These rules are very consistent and easy to apply once you practice them.

Multiplication of Integers:

  1. Positive × Positive: The product is always positive. For example, 3 × 5 = 15.
  2. Negative × Negative: The product is always positive. For example, (-3) × (-5) = 15.
  3. Positive × Negative: The product is always negative. For example, 3 × (-5) = -15.
  4. Negative × Positive: The product is always negative. For example, (-3) × 5 = -15.

In short, if the two integers have the same sign (both positive or both negative), their product is positive. If they have different signs, their product is negative.

Division of Integers:

  1. Positive ÷ Positive: The quotient is always positive. For example, 15 ÷ 5 = 3.
  2. Negative ÷ Negative: The quotient is always positive. For example, (-15) ÷ (-5) = 3.
  3. Positive ÷ Negative: The quotient is always negative. For example, 15 ÷ (-5) = -3.
  4. Negative ÷ Positive: The quotient is always negative. For example, (-15) ÷ 5 = -3.

The rules for signs in division are identical to those in multiplication: if the two integers have the same sign, their quotient is positive; if they have different signs, their quotient is negative. Remember that division by zero is undefined.

Step-by-Step: Multiplying Integers

  1. Step 1: Multiply the absolute values. — Ignore the signs for a moment and multiply the numbers as if they were positive whole numbers. For example, if you have (-4) × 6, first multiply 4 × 6 = 24.
  2. Step 2: Determine the sign of the product. — Now, look at the signs of the original integers: If both integers have the same sign (both positive or both negative), the product is positive. If the integers have different signs (one positive and one negative), the product is negative. Using our example (-4) × 6: one is negative, one is positive (different signs), so the product is negative. Thus, (-4) × 6 = -24.
  3. Example with multiple integers — When multiplying more than two integers, count the number of negative signs: If there is an even number of negative signs, the final product is positive. If there is an odd number of negative signs, the final product is negative. For example, (-2) × (-3) × (-4). First, multiply absolute values: 2 × 3 × 4 = 24. Next, count negative signs: There are three negative signs (an odd number). So, the final product is negative. (-2) × (-3) × (-4) = -24.

Step-by-Step: Dividing Integers

  1. Step 1: Divide the absolute values. — Ignore the signs and perform the division of the numbers as if they were positive whole numbers. For example, if you have (-45) ÷ 9, first divide 45 ÷ 9 = 5.
  2. Step 2: Determine the sign of the quotient. — Similar to multiplication, check the signs of the original integers: If both integers have the same sign (both positive or both negative), the quotient is positive. If the integers have different signs (one positive and one negative), the quotient is negative. Using our example (-45) ÷ 9: one is negative, one is positive (different signs), so the quotient is negative. Thus, (-45) ÷ 9 = -5.
  3. Special Case: Division by Zero — Remember, division by zero is undefined. You can never divide any number by zero. For example, 10 ÷ 0 or (-5) ÷ 0 are both undefined.

Applying the Rules: Worked Examples

  • Example 1: Find the product of (-18) × (-10) × 9. Step 1: Multiply the absolute values: 18 × 10 × 9 = 180 × 9 = 1620. Step 2: Count the number of negative signs: There are two negative signs ((-18) and (-10)). Since two is an even number, the final product will be positive. * Final Answer: (-18) × (-10) × 9 = 1620.
  • Example 2: Evaluate (-30) ÷ 10. Step 1: Divide the absolute values: 30 ÷ 10 = 3. Step 2: Determine the sign: One integer is negative (-30) and the other is positive (10). They have different signs, so the quotient will be negative. * Final Answer: (-30) ÷ 10 = -3.
  • Example 3: Simplify [(-6) + 5] ÷ [(-2) + 1]. Step 1: Solve the expression inside the first bracket: (-6) + 5 = -1. Step 2: Solve the expression inside the second bracket: (-2) + 1 = -1. Step 3: Perform the division: (-1) ÷ (-1). Step 4: Divide absolute values (1 ÷ 1 = 1) and determine sign (both are negative, same signs, so result is positive). * Final Answer: 1.

Exam Tip: Avoiding Common Sign Errors

The most frequent mistake students make in integer operations is incorrect sign assignment. Always double-check your signs! When multiplying or dividing:

  1. Focus on the operation: Is it multiplication or division?
  2. Multiply/Divide the numbers first: Ignore signs initially to get the numerical part of the answer.
  3. Apply the sign rule:
  • Same signs (+, + or -, -) always give a positive result.
  • Different signs (+, - or -, +) always give a negative result.

Also, remember that 0 × any integer = 0 and any integer ÷ 0 is undefined, not 0. Don't confuse division by zero with multiplying by zero. Use BODMAS/PEMDAS correctly for expressions involving multiple operations.

Practice Questions with Solutions

  • Q: Find the product: (-25) × 7. A: Step 1: Multiply the absolute values: 25 × 7 = 175. Step 2: Determine the sign. One integer is negative (-25) and one is positive (7). Since they have different signs, the product is negative. Final answer: -175.
  • Q: Evaluate: 132 ÷ (-11). A: Step 1: Divide the absolute values: 132 ÷ 11 = 12. Step 2: Determine the sign. One integer is positive (132) and one is negative (-11). Since they have different signs, the quotient is negative. Final answer: -12.
  • Q: Calculate: (-1) × (-1) × (-1) × (-1) × (-1). A: Step 1: Multiply the absolute values: 1 × 1 × 1 × 1 × 1 = 1. Step 2: Count the number of negative signs. There are five negative signs. Since five is an odd number, the final product will be negative. Final answer: -1.
  • Q: Simplify: [(-48) ÷ 12] ÷ (-4). A: Step 1: Solve the expression inside the bracket first: (-48) ÷ 12. Sub-step 1.1: Divide absolute values: 48 ÷ 12 = 4. Sub-step 1.2: Determine sign: (-48) is negative, 12 is positive (different signs), so the result is -4. Step 2: Now, we have (-4) ÷ (-4). Sub-step 2.1: Divide absolute values: 4 ÷ 4 = 1. Sub-step 2.2: Determine sign: Both are negative (same signs), so the result is positive. Final answer: 1.

Frequently Asked Questions

What are the rules for multiplying integers with different signs?

When multiplying integers with different signs (one positive and one negative), the product is always negative. For example, 5 × (-3) = -15 and (-7) × 2 = -14. Remember to first multiply the numbers and then apply the negative sign.

How do I divide two negative integers?

When you divide two negative integers, the quotient is always positive. For instance, (-20) ÷ (-5) = 4. Think of it as 'a negative divided by a negative gives a positive'.

What happens if I multiply three or more integers with negative signs?

If you multiply three or more integers, count the total number of negative signs. If there's an even number of negative signs, the final product will be positive. If there's an odd number of negative signs, the final product will be negative.

Is division by zero allowed for integers?

No, division by zero is undefined for any number, including integers. You cannot divide any integer by zero. For example, 10 ÷ 0 is not a number and has no value.