Introduction to Variables
In mathematics, a variable is a letter or symbol that stands for an unknown number. Instead of writing a blank or a question mark, mathematicians use letters like x, y, n, or a to represent numbers we do not know yet.
In Class 5, you are introduced to the concept of variables for the first time. This is the beginning of algebra — one of the most powerful tools in mathematics.
Variables help us write general rules, describe patterns, and solve problems where a number is missing.
What is Introduction to Variables - Class 5 Maths (Patterns and Algebra)?
A variable is a letter or symbol used to represent an unknown or changing number.
- In x + 5 = 12, the letter x is a variable. It stands for the number 7 (since 7 + 5 = 12).
- In n × 3 = 24, the letter n is a variable. It stands for 8.
Constants vs Variables:
- A constant is a fixed number (like 5, 12, 100). It does not change.
- A variable can take different values depending on the situation.
Types and Properties
How Variables Are Used:
- As unknowns: x + 7 = 15 (find x)
- In formulas: Area = l × b (l and b are variables for length and breadth)
- In patterns: The nth term of 3, 6, 9, 12 is 3 × n
- In general statements: "A number plus 10" can be written as n + 10
Solved Examples
Example 1: Example 1: Identifying a Variable
Problem: In the expression y + 8 = 20, identify the variable.
Solution:
The letter y is the variable. It represents an unknown number.
To find y: y = 20 − 8 = 12
Answer: Variable = y, and y = 12
Example 2: Example 2: Writing Statements as Expressions
Problem: Write "a number increased by 6" using a variable.
Solution:
Let the unknown number be n.
"A number increased by 6" = n + 6
Answer: n + 6
Example 3: Example 3: Variable in Multiplication
Problem: Write "5 times a number" using a variable.
Solution:
Let the number be x.
"5 times a number" = 5 × x or simply 5x
Answer: 5x
Example 4: Example 4: Evaluating an Expression
Problem: If x = 4, find the value of 3x + 7.
Solution:
Step 1: Replace x with 4: 3 × 4 + 7
Step 2: = 12 + 7 = 19
Answer: When x = 4, 3x + 7 = 19
Example 5: Example 5: Mangoes Problem
Problem: Ria has some mangoes. Her mother gives her 8 more. Now she has 15 mangoes. Write this using a variable and find how many mangoes she had.
Solution:
Let the original number of mangoes = m
Equation: m + 8 = 15
m = 15 − 8 = 7
Answer: Ria had 7 mangoes originally.
Example 6: Example 6: Age Problem
Problem: Aman is 'a' years old. His father is 30 years older. Write the father's age using a variable. If Aman is 10, find his father's age.
Solution:
Father's age = a + 30
If a = 10, father's age = 10 + 30 = 40
Answer: Father's age = a + 30. When a = 10, father is 40 years old.
Example 7: Example 7: Perimeter Formula
Problem: The perimeter of a square is 4 × s, where s is the side length. Find the perimeter when s = 9 cm.
Solution:
Step 1: Perimeter = 4 × s = 4 × 9 = 36
Answer: Perimeter = 36 cm
Example 8: Example 8: Pattern Rule with Variable
Problem: The pattern 5, 10, 15, 20, 25 follows the rule: term = 5 × n, where n is the position. Find the 12th term.
Solution:
Step 1: 12th term = 5 × 12 = 60
Answer: The 12th term = 60
Example 9: Example 9: Money Problem
Problem: Priya has ₹p. She spends ₹45 on a book. Write how much money she has left. If p = 100, find the amount left.
Solution:
Money left = p − 45
If p = 100: money left = 100 − 45 = 55
Answer: Money left = p − 45. When p = 100, she has ₹55 left.
Example 10: Example 10: Two Variables
Problem: The area of a rectangle is l × b. Find the area when l = 8 cm and b = 5 cm.
Solution:
Step 1: Area = l × b = 8 × 5 = 40
Answer: Area = 40 cm²
Key Points to Remember
- A variable is a letter that represents an unknown or changing number.
- Common variables: x, y, n, a, b, l, b, h.
- A constant is a fixed number that does not change.
- Variables are used in equations, expressions, formulas, and patterns.
- To evaluate an expression, replace the variable with the given value and calculate.
- 5 × x is written as 5x (the multiplication sign is omitted).
- Variables form the foundation of algebra.
Practice Problems
- Identify the variable in the expression: 2n + 9 = 25.
- Write "a number decreased by 12" using a variable.
- If x = 6, find the value of 4x − 5.
- Kavi has 'b' books. He buys 15 more. Write the total number of books using a variable.
- The cost of one pen is ₹p. Write the cost of 8 pens using a variable. If p = 12, find the total cost.
- A rectangle has length 'l' and breadth 5 cm. Write the perimeter using a variable. Find the perimeter when l = 12 cm.
- If y = 10, find the value of y² + 3.
- The pattern follows the rule: term = 2n + 1. Find the 7th term.
Frequently Asked Questions
Q1. What is a variable in maths?
A variable is a letter (like x, y, or n) that stands for an unknown number. It can take different values depending on the problem.
Q2. Why do we use letters instead of numbers?
Letters allow us to write general rules that work for any number. For example, the perimeter of a square = 4 × s works for any side length s.
Q3. What is the difference between a variable and a constant?
A variable can change or take different values (like x or n). A constant is a fixed number (like 5, 100, or 3.14) that never changes.
Q4. What does it mean to evaluate an expression?
Evaluating means replacing the variable with a given number and calculating the result. For example, if x = 3, then 2x + 4 = 2(3) + 4 = 10.
Q5. Can a variable represent more than one number?
In an equation like x + 5 = 12, the variable x has exactly one value (7). But in a formula like Area = l × b, the variables l and b can take many different values.
Q6. Why is 5x written without the multiplication sign?
In algebra, when a number is multiplied by a variable, the multiplication sign is dropped for simplicity. So 5 × x is written as 5x.
Q7. What is an expression?
An expression is a mathematical phrase using numbers, variables, and operations. Examples: 3x + 5, n − 7, 2a + 3b. An expression does not have an equals sign.
Q8. Is this the same as algebra?
Yes, this is the beginning of algebra. Variables, expressions, and equations are the core concepts of algebra, which you will study in greater detail in higher classes.
