Important Formulae and Points to Remember: Expressions Using Letter - Numbers | Mathematics (Ganita Prakash) Class 7 - New NCERT PDF Download

1. Concept of Letter-Numbers

• Letter-numbers are letters used to represent unknown or variable values (e.g., a, n, x).

• Algebraic expressions are combinations of numbers and letter-numbers, joined by operations (+, –, ×, ÷).

2. Writing Expressions from Situations

3. Common Arithmetic Expression Properties

• Swapping (Commutative Property): Order of addition doesn't affect the result:
  a + b = b + a

• Grouping (Associative Property): Regrouping terms doesn’t change the sum:
  (a + b) + c = a + (b + c)

• Distributive Property:
  a × (b + c) = ab + ac

4. Evaluating Expressions

• Replace letter-numbers with given values.

• Follow BODMAS (Brackets, Orders, Division/Multiplication, Addition, Subtraction).

Example:
If a = 23, then a + 3 = 23 + 3 = 26.

5. Writing Your Own Expressions

6. Omitting Multiplication Sign

• 4 × n is written as 4n
• Always write the number before the letter:

7. Sequences and nth Term Formula

What is a Sequence?

A sequence is an ordered list of numbers that follow a specific pattern or rule. Each number in the sequence is called a term.

Example Sequence:

4, 8, 12, 16, ...

This sequence increases by 4 each time, so it's called an arithmetic sequence (a sequence where each term increases or decreases by the same fixed number).

For a sequence like: 4, 8, 12, 16... (multiples of 4) 

n^th term = 4n

8. Perimeter of a Rectangle

Expression:
p = l + b + l + b 

p = 2l + 2b 
(where $ll$l = length, $bb$b = breadth)

9. Like and Unlike Terms

• Like Terms: Same letter-numbers (e.g., 5c, 3c)

• Unlike Terms: Different letter-numbers (e.g.,  5c, 11d)

Rule: Only like terms can be added or subtracted.

10. Distributive Property in Use

Example: 4 (x+y) −y 
= 4x + 4y − y 
= 4x+3y

11. Non-equivalent Expressions

Sometimes, two expressions look similar but are not equal because they follow different mathematical rules or structures.

To check if two expressions are equivalent, we can:

• Simplify both expressions, or

• Substitute a value into each and compare the results

Exmaple: Check whether 5u and 5 + u are equal or not 

These two expressions look alike, but they're not the same.

Let's substitute u = 2:

• 5u = 5 × 2 = 10

• 5 + u = 5 + 2 = 7

Since 10 ≠ 7,
5u and 5 + u are not equivalent.​

12. Common Mistakes in Simplification

13. Number Machines (Input-Output Rules)

• A formula or expression represents the operation performed on two inputs.

• Example: If the machine uses inputs a and b, and performs 2a - b, then:
  Output=2a−b

• Practice finding formulas for different number machines.

14. Patterns in a Calendar (2 × 2 Squares)

In any 2 × 2 block on a calendar:

$Top-left number\mathrm{<br>}\backslash text\{Top-left\; number\}\; =\; a$Top-left number = a

Other positions:

• Top-right: $a\; +\; 1$a + 1

• Bottom-left: $a\; +\; 7$a + 7

• Bottom-right: a + 8

• Diagonal Sums:
    a + (a + 8) = (a + 1) + (a + 7) = 2a + 8

15. Matchstick Patterns

Pattern of Triangles using Matchsticks:

• Step 1:  3 matchsticks

• Each new triangle shares 1 side with the previous triangle, so you only add 2 new matchsticks each time.

• To find the number of matchsticks needed for any step y, use the formula:

  Matchsticks at Step y = 2y + 1