In Class 7, the initial chapter in Mathematics introduces students to Integers. Integer are a collection of numbers that comprise positive and negative numbers, along with zero. Integer are a crucial aspect in the real world because they enable us to represent situations involving height, temperatures, money transactions, among others.
This blog post is dedicated to Exercise 1.1 of Chapter 1 on Integers, and it gives step-by-step solutions to the problems along with explanations to make you understand the concepts in depth. The chapter will discuss integer operations like addition, subtraction, multiplication, and division, which are basic to understand how integers function.
Let’s proceed with the exercise, go through each question, and discuss the solutions step by step.
Exercise 1.1: Integer
This exercise focuses on understanding and applying the basic operations of integers: writing integers, solving equations, and understanding the rules governing the addition, subtraction, multiplication, and division of integers.
Q1. Write the following numbers in their integer form:
a) 5 meters below the surface of the Earth
b) 15 degrees Celsius above zero
c) 50 kilometers to the south of the city
d) 20 kilometers to the north of the city
Solution and Explanation:
- 5 meters below the surface of the Earth:
When you talk about a location below the surface, the number is negative because we are going down. Hence, the integer representation is -5. This is an example of negative integers being used to represent depth or position below a reference point. - 15 degrees Celsius above zero:
A temperature above zero is considered positive, and the integer representation of 15 degrees above zero is +15 or simply 15. This highlights that temperatures above the freezing point are represented as positive integers. - 50 kilometers to the south of the city:
The word “south” indicates a direction, and by convention, moving south is considered a negative direction in this case. So, the integer representation is -50. - 20 kilometers to the north of the city:
Similarly, “north” is considered a positive direction. Therefore, the integer representation of 20 kilometers north is +20 or simply 20.
Q2. Write the integers represented by the following situations:
a) A loss of ₹200
b) A temperature drop of 10°C
c) A deposit of ₹500
d) A rise of 15 meters in height
Solution and Explanation:
- A loss of ₹200:
A loss is represented by a negative integer, as it reflects a reduction in value. The integer representation is -200. - A temperature drop of 10°C:
A temperature drop represents a decrease in temperature, so it is denoted by a negative integer. The integer form is -10. - A deposit of ₹500:
A deposit means an increase in the amount, representing a positive value. The integer form is +500 or simply 500. - A rise of 15 meters in height:
A rise is an increase in height, and increases are represented as positive integers. Thus, the integer representation is +15 or simply 15.
Q3. Identify whether the following statements are true or false:
a) The sum of two positive integers is always negative.
b) The product of two negative integers is positive.
c) The difference between two positive integers is always positive.
d) The sum of a positive integer and a negative integer is always negative.
Solution and Explanation:
- The sum of two positive integers is always negative → This statement is False.
The sum of two positive integers is always positive, as adding positive numbers will only result in a larger positive number. - The product of two negative integers is positive → This statement is True.
According to the rules of multiplication, multiplying two negative integers results in a positive integer. For example, (-2) × (-3) = 6. - The difference between two positive integers is always positive → This statement is False.
If the larger number is subtracted from the smaller number, the result will be negative. For example, 5 – 8 = -3, which is negative. - The sum of a positive integer and a negative integer is always negative → This statement is False.
The sum of a positive and a negative integer can be negative, positive, or even zero, depending on the values of the integers. For instance, 5 + (-3) = 2, which is positive.
Q4. Find the sum of the following integers:
a) -7 + 3
b) 12 + (-5)
c) -8 + (-4)
d) 6 + (-9)
Solution and Explanation:
- -7 + 3:
To add these integers, we start by observing that -7 is negative and 3 is positive. We subtract the smaller absolute value from the larger one, and keep the sign of the number with the larger absolute value.
So, -7 + 3 = -4. - 12 + (-5):
Here, we are adding a positive number (12) to a negative number (-5). Since the positive number is greater in absolute value, the result will be positive.
Therefore, 12 + (-5) = 7. - -8 + (-4):
Both numbers are negative, so we add the absolute values and keep the negative sign.
Hence, -8 + (-4) = -12. - 6 + (-9):
Here, we add a positive number (6) to a negative number (-9). The absolute value of -9 is greater, so the result is negative.
Therefore, 6 + (-9) = -3.
Q5. Subtract the following integers:
a) 5 – 8
b) -6 – 3
c) 7 – (-4)
d) -12 – (-6)
Solution and Explanation:
- 5 – 8:
This subtraction involves a smaller positive number (5) being subtracted from a larger positive number (8). When subtracting a larger number from a smaller one, the result is negative.
So, 5 – 8 = -3. - -6 – 3:
Subtracting a positive integer from a negative integer results in a more negative value.
Therefore, -6 – 3 = -9. - 7 – (-4):
Subtracting a negative integer is equivalent to adding the positive version of that integer.
So, 7 – (-4) = 7 + 4 = 11. - -12 – (-6):
Subtracting a negative number is the same as adding the positive counterpart.
Thus, -12 – (-6) = -12 + 6 = -6.
Q6. Multiply the following integers:
a) 4 × (-3)
b) -5 × (-2)
c) -7 × 6
d) 9 × (-1)
Solution and Explanation:
- 4 × (-3):
When you multiply a positive integer by a negative integer, the result is negative.
So, 4 × (-3) = -12. - -5 × (-2):
When you multiply two negative integers, the result is positive.
Therefore, -5 × (-2) = +10. - -7 × 6:
The product of a negative and a positive integer is negative.
Hence, -7 × 6 = -42. - 9 × (-1):
The product of any number and -1 is the negative of the original number.
So, 9 × (-1) = -9.
Q7. Divide the following integers:
a) 12 ÷ (-4)
b) -20 ÷ 5
c) -15 ÷ (-3)
d) 8 ÷ (-2)
Solution and Explanation:
- 12 ÷ (-4):
Dividing a positive integer by a negative integer gives a negative result.
Therefore, 12 ÷ (-4) = -3. - -20 ÷ 5:
Dividing a negative integer by a positive integer gives a negative result.
Hence, -20 ÷ 5 = -4. - -15 ÷ (-3):
Dividing two negative integers results in a positive integer.
Therefore, -15 ÷ (-3) = +5. - 8 ÷ (-2):
Dividing a positive integer by a negative integer results in a negative integer.
Hence, 8 ÷ (-2) = -4.
Conclusion
The concept of Integers is critical to understanding various real-life scenarios, such as changes in temperature, financial transactions, and even movement across different altitudes or depths. Mastering the operations of addition, subtraction, multiplication, and division of integers is a fundamental skill in mathematics that will serve students well in future studies.
By working through Exercise 1.1, students gain a strong foundation for tackling more complex problems involving integers. With consistent practice and understanding, you will find that operations on integers become much easier to handle.
This concludes the detailed solution of Class 7 Maths Chapter 1: Exercise 1.1. I hope this helps you better understand integers and the associated operations. If you have any more questions or need further clarification, feel free to leave a comment or reach out!
Also Read: Class 6 Maths Chapter 4: Decimals – MCQs with Answers
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Sidhant Singh is a highly accomplished professional educator with a diverse academic background. He holds a Master’s degree in History, an MSc in Electrical Engineering and a PhD, demonstrating his extensive knowledge and expertise in both the humanities and the sciences. His unique combination of skills enables him to offer a well-rounded perspective in his teaching and research, making him a valuable resource for students across various fields.