Welcome to Sid Classes. In this comprehensive guide, we provide accurate, step-by-step solutions for NCERT Class 9 Maths Ganita Manjari Chapter 2 Exercise 2.3. This exercise is crucial for understanding how real-world situations (like savings, dropping counts, and geometric dimensions) translate into linear patterns and algebraic expressions.
Our solutions use clean standard mathematical notation (\(\text{LaTeX}\)) and breakdown logic to ensure you score 100% marks in your school tests and board examinations.
🔑 Key Concepts to Remember:
- Linear Pattern: A sequence where the difference between any two consecutive terms remains constant.
- General Formula: \(\text{Total Value} = \text{Initial Value} \pm (\text{Rate of Change} \times n)\)
- Use a plus sign (\(+\)) if the quantity is increasing (e.g., bank savings).
- Use a minus sign (\(-\)) if the quantity is decreasing (e.g., dropping members, reading pages).
NCERT Class 9 Maths Ganita Manjari Chapter 2 Exercise 2.3 Solutions
Question 1
A student has ₹500 in her savings bank account. She gets ₹150 every month as pocket money. How much money will she have at the end of every month from the second month onwards? Find a linear expression to represent the amount she will have in the \(n^{\text{th}}\) month.
Solution:
- Initial Balance (Month 0): \(\text{₹}500\)
- End of 1st Month: \(500 + 150 = \text{₹}650\)
- End of 2nd Month: \(650 + 150 = 500 + (2 \times 150) = \text{₹}800\)
- End of 3rd Month: \(800 + 150 = 500 + (3 \times 150) = \text{₹}950\)
The total money she will have at the end of every month starting from the second month onwards forms the following sequence: ₹800, ₹950, ₹1100, ₹1250, ...
Finding the Linear Expression:
Let \(n\) represent the number of months.
$$\text{Amount after } n \text{ months} = \text{Initial Savings} + (n \times \text{Pocket Money Per Month})$$
$$\text{Amount} = 500 + 150n$$
Final Answer: The linear expression is \(\mathbf{500 + 150n}\).
Question 2
A rally starts with 120 members. Each hour, 9 members drop out of the group. How many members will remain after 1, 2, 3, … hours? Find a linear expression to represent the number of members at the end of the \(n^{\text{th}}\) hour.
Solution:
Here, the total count is decreasing at a constant rate each hour. Let's list the values:
- Initial Strength: \(120\text{ members}\)
- Remaining after 1 hour: \(120 - 9 = \mathbf{111\text{ members}}\)
- Remaining after 2 hours: \(120 - (2 \times 9) = 120 - 18 = \mathbf{102\text{ members}}\)
- Remaining after 3 hours: \(120 - (3 \times 9) = 120 - 27 = \mathbf{93\text{ members}}\)
The number of remaining members after 1, 2, 3, ... hours forms the sequence: 111, 102, 93, 84, ...
Finding the Linear Expression:
Let \(n\) represent the total hours passed.
$$\text{Remaining Members} = \text{Starting Count} - (\text{Drop Rate per hour} \times n)$$
$$\text{Remaining Members} = 120 - 9n$$
Final Answer: The linear expression is \(\mathbf{120 - 9n}\).
Question 3
Suppose the length of a rectangle is 13 cm. Find the area if the breadth is (i) 12 cm, (ii) 10 cm, (iii) 8 cm. Find the linear pattern representing the area of the rectangle.
Solution:
We know that the formula for calculating the area of a rectangle is given by:
$$\text{Area} = \text{Length} \times \text{Breadth}$$
Given that the Length is constant at \(13\text{ cm}\):
(i) When Breadth = 12 cm:
$$\text{Area} = 13 \times 12 = \mathbf{156\text{ cm}^2}$$
(ii) When Breadth = 10 cm:
$$\text{Area} = 13 \times 10 = \mathbf{130\text{ cm}^2}$$
(iii) When Breadth = 8 cm:
$$\text{Area} = 13 \times 8 = \mathbf{104\text{ cm}^2}$$
Finding the Linear Pattern:
If we treat the breadth as a variable denoted by \(b\), the area scales directly with changes in breadth.
$$\text{Area} = 13 \times b = 13b$$
Final Answer: The linear pattern representing the area is \(\mathbf{13b}\).
Question 4
Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (i) 5 cm, (ii) 9 cm, (iii) 13 cm. Find the linear pattern representing the volume of the rectangular box.
Solution:
The standard formula for the volume of a rectangular prism/box is:
$$\text{Volume} = \text{Length} \times \text{Breadth} \times \text{Height}$$
First, let's find the constant base area: \(\text{Length} \times \text{Breadth} = 7 \times 11 = 77\text{ cm}^2\).
(i) When Height = 5 cm:
$$\text{Volume} = 77 \times 5 = \mathbf;{385\text{ cm}^3}$$
(ii) When Height = 9 cm:
$$\text{Volume} = 77 \times 9 = \mathbf{693\text{ cm}^3}$$
(iii) When Height = 13 cm:
$$\text{Volume} = 77 \times 13 = \mathbf{1001\text{ cm}^3}$$
Finding the Linear Pattern:
Letting the variable height be denoted by \(h\):
$$\text{Volume} = 77 \times h = 77h$$
Final Answer: The linear pattern representing the volume is \(\mathbf{77h}\).
Question 5
Sarita is reading a book of 500 pages. She reads 20 pages every day. How many pages will be left after 15 days? Express this as a linear pattern.
Solution:
Let's compile the information provided to calculate the remaining pages:
- Total Pages in Book: \(500\)
- Pages read per single day: \(20\)
- Total pages read across 15 days: \(15 \times 20 = 300\text{ pages}\)
Now, subtract the read pages from the total length of the book:
$$\text{Pages left after 15 days} = 500 - 300 = \mathbf{200\text{ pages}}$$
Formulating the Linear Pattern:
To find a general rule for pages remaining on any arbitrary day \(d\):
$$\text{Pages Remaining} = \text{Total Pages} - (\text{Pages Read Per Day} \times d)$$
$$\text{Pages Remaining} = 500 - 20d$$
Final Answer: The remaining pages after 15 days is 200, and its linear pattern is \(\mathbf{500 - 20d}\).
Frequently Asked Questions (FAQs)
Q1: What is a linear expression in Class 9 algebraic patterns?
A: A linear expression is a mathematical statement where the highest exponent of the variable involved is exactly 1. For example, in the expression \(120 - 9n\), the power of variable \(n\) is 1.
Q2: How do you identify whether a sequence follows a linear pattern?
A: A sequence is linear if the differences between successive numbers are identical throughout. If a value grows or drops by the exact same steady number each step, it is linear.
Q3: What textbook does Exercise 2.3 belong to?
A: These answers strictly follow the new Ganita Manjari Part I textbook issued for NCERT Grade 9 Mathematics.
Q4: What do variables like 'n', 'b', or 'd' stand for in these solutions?
A: These variables represent independent, changing quantities. For instance, \(n\) refers to the number of months or hours, \(b\) stands for breadth, and \(d\) indicates the number of days.
Q5: Why is the pattern in Question 2 written with a minus sign (\(120 - 9n\))?
A: The minus sign indicates a decreasing pattern. Because participants are constantly dropping out of the rally group over time, the total number subtracts from the original balance.
Q6: Can the variable in a linear pattern have a square power like \(n^2\)?
A: No. If an expression includes a squared term like \(n^2\), it becomes a quadratic pattern, not a linear pattern. Linear patterns must only have a maximum degree power of 1.
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