Algebra Ex 11.1 NCERT Solutions for Class 6 Maths

What is Algebra? The Power of Variables

Imagine you have a secret number in your mind, and you ask a friend to add 5 to it. How would you write this idea down if you don't know the number? This is where algebra comes in! Algebra is a part of mathematics that uses letters (like x, y, a, b, or any other letter) to represent numbers that we don't know yet, or numbers that can change. These letters are called variables.

Why are variables so powerful? They help us write general rules that work for any number. For example, if you want to find the perimeter of any square, you know it's side + side + side + side. If we let the length of one side be 's', then the perimeter is simply s + s + s + s, which we can write as 4s. This rule 4s works for a square of any size! Exercise 11.1 primarily focuses on helping you identify such patterns and express them using these flexible variable letters. You'll observe patterns in matchsticks, numbers, and geometric shapes, and then write a general rule for them using a variable.

How to Find Rules and Form Algebraic Expressions

1. Step 1: Observe the Pattern Carefully — Look at the given figures or numbers in the pattern. Pay attention to how each figure or term changes from the previous one. For example, if you have matchstick figures, count the number of matchsticks in Figure 1, Figure 2, Figure 3, and so on.
2. Step 2: List the Relationship — Create a small table or list showing the 'figure number' (or term number) and the 'number of items' (like matchsticks, dots, etc.). Figure 1: 3 matchsticks Figure 2: 5 matchsticks Figure 3: 7 matchsticks
3. Step 3: Look for a Connection/Rule — Try to find a mathematical operation (addition, subtraction, multiplication, division) that connects the figure number to the number of items. In our example (3, 5, 7), notice that each number is 2 more than twice the figure number: Figure 1: (2 × 1) + 1 = 3 Figure 2: (2 × 2) + 1 = 5 Figure 3: (2 × 3) + 1 = 7
4. Step 4: Use a Variable to Generalize the Rule — Once you've found the connection, replace the 'figure number' with a variable (like 'n' or 'x'). This gives you the general rule. So, if 'n' represents the figure number, the rule for the number of matchsticks is 2n + 1.

Worked Examples from Algebra Ex 11.1

• Example 1: Matchstick Pattern (Letter 'U') Observe the pattern of the letter 'U' made from matchsticks: Figure 1: U (3 matchsticks) Figure 2: UU (6 matchsticks) Figure 3: UUU (9 matchsticks) Find a general rule for the number of matchsticks required to make 'n' such figures. Solution: 1. Count matchsticks: Figure 1 has 3 matchsticks. Figure 2 has 6 matchsticks. Figure 3 has 9 matchsticks. 2. Look for a pattern: We can see that the number of matchsticks is always 3 times the figure number. For Figure 1: 3 × 1 = 3 For Figure 2: 3 × 2 = 6 For Figure 3: 3 × 3 = 9 3. Generalize with a variable: If 'n' represents the number of figures, the rule for the number of matchsticks needed is 3 × n or simply 3n.
• Example 2: Cost of Items Reshma buys 'p' pens. If each pen costs ₹10, what is the total cost of the pens she bought? Solution: 1. Identify knowns: Cost of one pen = ₹10. Number of pens = 'p'. 2. Think about calculation: If Reshma bought 1 pen, cost = ₹10. If 2 pens, cost = ₹10 × 2 = ₹20. If 3 pens, cost = ₹10 × 3 = ₹30. 3. Form the expression: Since she bought 'p' pens, the total cost will be ₹10 × p. 4. Final Expression: The total cost is 10p rupees.
• Example 3: Age Relationship Raman is 5 years older than his sister, Simran. If Simran's current age is 'y' years, what is Raman's age? Solution: 1. Identify Simran's age: Simran's age = 'y' years. 2. Understand 'older than': 'Older than' means we need to add. Raman is 5 years more than Simran. 3. Form the expression: Raman's age = Simran's age + 5. 4. Substitute variable: Raman's age = y + 5 years.

Exam Tip: Avoiding Common Mistakes in Algebra Ex 11.1

When working with algebraic expressions in Exercise 11.1, students often make a few common errors. Here’s how to avoid them:

1. Confusing 'times' and 'more than':

• "5 times a number x" means 5 * x or 5x.
• "5 more than a number x" means x + 5.
• Always read the problem carefully to understand the operation required.

1. Not defining your variable: Before you write an expression like 4s, always state what 's' represents (e.g., "Let 's' be the side length of the square"). This makes your answer clear and complete.

1. Not checking your rule: For pattern problems (like matchsticks), always test your derived rule with the first two or three figures. If your rule 2n + 1 works for Figure 1 (21+1=3) and Figure 2 (22+1=5), you are likely correct!

Practice Questions with Solutions

• Q: Observe the pattern of the letter 'T' made from matchsticks: Figure 1: T (2 matchsticks) Figure 2: TT (4 matchsticks) Figure 3: TTT (6 matchsticks) Find a general rule for the number of matchsticks required to make 'n' such figures. A: Step 1: Count matchsticks for each figure. Figure 1: 2 matchsticks Figure 2: 4 matchsticks Figure 3: 6 matchsticks Step 2: Identify the relationship. The number of matchsticks is always double the figure number. For Figure 1: 2 × 1 = 2 For Figure 2: 2 × 2 = 4 For Figure 3: 2 × 3 = 6 Step 3: Express this rule using a variable 'n' for the figure number. Final answer: The general rule is 2n.
• Q: The side of a regular pentagon is 'a' units. Write the expression for its perimeter. A: Step 1: Recall what a regular pentagon is. A pentagon has 5 sides, and 'regular' means all its sides are equal in length. Step 2: Understand perimeter. Perimeter is the total length of all sides. Step 3: Since there are 5 equal sides, each of length 'a', the perimeter will be a + a + a + a + a. Step 4: Simplify the expression. Final answer: The perimeter of the regular pentagon is 5a units.
• Q: What is the expression for "12 less than a number x"? A: Step 1: Understand the phrase "less than". This means we need to subtract. Step 2: Identify the number from which we are subtracting. We are taking 12 away from 'x'. Step 3: Write the subtraction expression. Final answer: The expression is x - 12.
• Q: A bird flies 1 kilometer in 1 minute. Can you express the distance covered by the bird in 't' minutes? A: Step 1: Identify the given rate: distance covered in 1 minute = 1 km. Step 2: Think about how distance changes with time. If it flies for 2 minutes, it covers 1 km/min * 2 min = 2 km. If for 5 minutes, 5 km. Step 3: Generalize this using the variable 't' for time in minutes. Final answer: The distance covered by the bird in 't' minutes is 1 × t km, or simply t km.

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