Rules for Algebraic Expressions
Algebra is like a language. Just as English has rules for making sentences, algebra has rules for writing and reading expressions.
In Class 6, you will learn how to translate word statements into algebraic expressions and understand the rules for writing them correctly.
These rules help you convert real-life situations into mathematical expressions that can be solved.
What is Rules for Algebraic Expressions - Grade 6 Maths (Algebra)?
Rules for writing algebraic expressions:
- Rule 1: The multiplication sign (×) is usually not written between a number and a variable. Write 5n, not 5 × n.
- Rule 2: The number (coefficient) is written before the variable. Write 3x, not x3.
- Rule 3: When a variable is multiplied by 1, we write just the variable. Write x, not 1x.
- Rule 4: When a variable is multiplied by −1, write −x, not −1x.
- Rule 5: Variables in a term are written in alphabetical order. Write abc, not cba.
Rules for Algebraic Expressions Formula
Translating words to expressions:
Addition words:
- "sum of x and 5" → x + 5
- "7 more than y" → y + 7
- "increased by 3" → n + 3
- "total of a and b" → a + b
Subtraction words:
- "difference of x and 4" → x − 4
- "5 less than y" → y − 5 (NOT 5 − y)
- "decreased by 2" → n − 2
- "subtract 3 from a" → a − 3
Multiplication words:
- "twice a number" → 2n
- "triple x" → 3x
- "product of a and b" → ab
Division words:
- "a number divided by 5" → n/5
- "half of x" → x/2
- "quotient of a and b" → a/b
Types and Properties
Common patterns to recognise:
- "Consecutive numbers starting from n" → n, n + 1, n + 2, ...
- "Even numbers" → 2, 4, 6, ... or 2n (for any positive integer n).
- "Odd numbers" → 1, 3, 5, ... or 2n − 1.
- "Square of a number" → n²
- "Cube of a number" → n³
Order matters for subtraction and division:
- "x less than y" means y − x (not x − y).
- "x subtracted from y" means y − x.
- "x divided by y" means x/y.
- "x divided into y" means y/x.
Solved Examples
Example 1: Simple Translation
Problem: Write an algebraic expression for "a number increased by 9".
Solution:
Let the number be x. "Increased by 9" means add 9.
Answer: x + 9
Example 2: Subtraction – Watch the Order
Problem: Write an expression for "6 less than a number p".
Solution:
"6 less than p" means we subtract 6 from p.
Answer: p − 6 (NOT 6 − p)
Example 3: Multiplication and Addition Combined
Problem: Write an expression for "5 times a number, then add 3".
Solution:
Let the number be n. Five times n = 5n. Then add 3.
Answer: 5n + 3
Example 4: Division Expression
Problem: Write an expression for "a number divided by 7, then subtract 2".
Solution:
Number divided by 7 = x/7. Then subtract 2.
Answer: x/7 − 2
Example 5: Age Problem
Problem: Priya's age is x years. Write expressions for: (a) her age 5 years later, (b) her age 3 years ago, (c) her father's age if he is twice her age plus 5.
Solution:
- (a) x + 5
- (b) x − 3
- (c) 2x + 5
Example 6: Consecutive Numbers
Problem: Write three consecutive numbers starting from n.
Solution:
The three consecutive numbers are: n, n + 1, n + 2.
Example 7: Expression from a Pattern
Problem: A pattern gives: 3, 5, 7, 9, 11, ... Write the nth term.
Solution:
The numbers increase by 2 each time, starting from 3.
1st term: 3 = 2(1) + 1
2nd term: 5 = 2(2) + 1
3rd term: 7 = 2(3) + 1
Answer: nth term = 2n + 1
Example 8: Applying the Writing Rules
Problem: Write correctly: (a) 1 × x, (b) y × 3, (c) b × a × 2.
Solution:
- (a) 1 × x = x (drop the 1)
- (b) y × 3 = 3y (number before variable)
- (c) b × a × 2 = 2ab (number first, variables in alphabetical order)
Real-World Applications
Where expression rules are used:
- Word problems: Converting "Ravi has 5 more marbles than Sita" into an expression (s + 5).
- Formulas: Writing perimeter = 2l + 2b instead of words.
- Patterns: Finding the 100th term of a sequence without listing all terms.
- Coding: Programming uses variables and expressions just like algebra.
- Science: Physics and chemistry formulas use algebraic expressions.
Key Points to Remember
- Do not write the × sign between a number and variable: write 5n, not 5 × n.
- The number (coefficient) comes before the variable: 3x, not x3.
- "Less than" and "subtracted from" reverse the order: "5 less than x" = x − 5.
- 1 × x is just x; −1 × x is just −x.
- Variables in a product are written in alphabetical order: abc, not bca.
- Consecutive numbers: n, n + 1, n + 2.
- Even numbers: 2n. Odd numbers: 2n − 1.
Practice Problems
- Write an expression for "a number multiplied by 4 and then decreased by 7".
- Write an expression for "10 subtracted from y".
- Rewrite correctly: (a) m × 6, (b) 1 × p, (c) c × b × a.
- Write the first three consecutive even numbers starting from 2n.
- The cost of one pen is Rs p. Write an expression for the cost of 12 pens.
- A pattern gives 4, 7, 10, 13, ... Write an expression for the nth term.
Frequently Asked Questions
Q1. Why do we write 3x instead of x3?
By convention, the number (coefficient) is always written before the variable. This makes expressions easier to read and compare.
Q2. What does '5 less than x' mean?
It means x − 5. You subtract 5 from x. Be careful: it does NOT mean 5 − x. The phrase 'less than' reverses the order.
Q3. Why do we drop the 1 in 1x?
Because 1 times any number is just that number. So 1x = x. Writing the 1 is unnecessary.
Q4. How do I translate words into algebraic expressions?
Identify the operation (add, subtract, multiply, divide) and the numbers/variables involved. 'More than' = add. 'Less than' = subtract. 'Times' = multiply. 'Divided by' = divide.
Q5. What is the expression for 'twice a number plus 3'?
Let the number be n. Twice n = 2n. Plus 3 gives: 2n + 3.
Q6. Can an expression have more than one variable?
Yes. Expressions like 3x + 2y − z have multiple variables. Each variable can represent a different unknown number.
