NCERT Solutions for Class 6 Maths: Algebra Exercise 11.3

From Numbers to Variables: Understanding Expressions

In mathematics, we often work with combinations of numbers and operations. For example, (4 × 5) + 10 is a numerical expression because it is made up of only numbers and arithmetic operations. You can calculate its value directly.

But what if we don't know a number? Imagine your friend has some chocolates, but you don't know how many. We can use a letter, like 'x', to represent this unknown number. This 'x' is called a variable. Now, if you are given 3 more chocolates, the total number of chocolates you have can be written as x + 3. This is an algebraic expression. It's a combination of variables, numbers, and operations. Unlike a numerical expression, we can't find a single value for x + 3 unless we know the value of x. Exercise 11.3 helps you get comfortable with creating and reading these powerful algebraic expressions.

Building Expressions Step-by-Step

1. Step 1: Identify the Unknown and Assign a Variable — Read the statement carefully and find the quantity that is unknown. Choose any letter (like x, y, n, m, etc.) to represent this unknown value. For example, in 'A number increased by 10', the unknown is 'a number', so let's call it 'n'.
2. Step 2: Look for Keywords for Operations — Find words that tell you which mathematical operation to use: - Addition: 'added to', 'sum of', 'more than', 'increased by'. - Subtraction: 'subtracted from', 'less than', 'difference of', 'decreased by'. - Multiplication: 'multiplied by', 'times', 'product of'. - Division: 'divided by', 'quotient of'. In our example 'n increased by 10', the keyword is 'increased by', which means addition (+).
3. Step 3: Combine and Write the Expression — Put the variable, the number, and the operation sign together in the correct order. For 'n increased by 10', we combine 'n' and '10' with a '+' sign to get the final expression: n + 10.

Watch Out for Subtraction Order!

A very common mistake happens with subtraction. The phrases 'subtracted from' and 'less than' reverse the order of the terms.

• '7 subtracted from y' means you start with 'y' and take 7 away. So, the correct expression is y - 7.
• A common wrong answer is 7 - y. This would be the expression for 'y subtracted from 7'.

Always read subtraction statements carefully to see which number or variable is being taken away from which.

From Words to Algebra: Worked Examples

• Statement: 8 added to t. Explanation: 'added to' means addition. Expression: t + 8
• Statement: 10 subtracted from p. Explanation: We start with p and subtract 10. Expression: p - 10
• Statement: y is multiplied by 5. Explanation: 'multiplied by' means multiplication. We usually write the number first. Expression: 5y
• Statement: k is divided by 3. Explanation: 'divided by' means division, often written as a fraction. Expression: k/3
• Statement: q is multiplied by -4. Explanation: Multiplication with a negative number. Expression: -4q

Practice Questions with Solutions

• Q: Write algebraic expressions for the following cases: (a) The sum of numbers x and 4. (b) 9 less than a number p. A: Step 1 (a): The keyword 'sum' means addition. We need to add x and 4. Step 2 (a): The expression is x + 4. Step 3 (b): The keyword phrase 'less than' means subtraction, and it reverses the order. We start with p and subtract 9. Step 4 (b): The expression is p - 9. Final answer: (a) x + 4 (b) p - 9
• Q: Which of the following are expressions with numbers only? And which are expressions with variables? (a) 15 - z (b) (8 × 3) + 1 (c) 3x (d) 17 - (5 × 2) A: Step 1: An expression with a variable contains a letter (like x, y, z). An expression with numbers only does not. Step 2: Check each option. (a) 15 - z contains the letter 'z', so it's an expression with a variable. (b) (8 × 3) + 1 contains only numbers and operations, so it's an expression with numbers only. (c) 3x contains the letter 'x', so it's an expression with a variable. (d) 17 - (5 × 2) contains only numbers and operations, so it's an expression with numbers only. Final answer: Expressions with numbers only: (b) and (d). Expressions with variables: (a) and (c).
• Q: Form an expression for this situation: Ram's age is y years. His father's age is 5 years more than 3 times Ram's age. A: Step 1: First, find '3 times Ram's age'. Ram's age is y, so 3 times his age is 3 × y or 3y. Step 2: The father's age is '5 years more than' this result. 'More than' means addition. Step 3: We add 5 to the result from Step 1. So, the expression is 3y + 5. Final answer: The expression for the father's age is 3y + 5.
• Q: Give an everyday situation or statement for the expression x - 20. A: Step 1: Identify the components. We have a variable x and we are subtracting 20 from it. Step 2: Think of a real-life scenario. The variable x can be an initial amount, and 20 is an amount that is taken away, spent, or lost. Step 3: Create a statement. For example, 'I had x rupees and I spent 20 rupees. The money I have left is x - 20.' Another example: 'The temperature was x degrees, and it dropped by 20 degrees. The new temperature is x - 20.' Final answer: One possible statement is: A shopkeeper had x pens and sold 20 of them. He now has x - 20 pens left.

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