Expressions Using Letter-Numbers

4.1 The Notion of Letter-Numbers

In mathematics we often need to describe relationships between different quantities. Writing every possible value separately is difficult, so mathematicians use letters to represent numbers.

These letters help us create general rules that work for many situations instead of only one.

Why Do We Use Letters?

Letters make mathematical expressions shorter, easier to understand and useful for solving many different problems.

Example – Age Relationship

Suppose Shabnam is 3 years older than Aftab.

If Aftab's age is represented by the letter a, then Shabnam's age can be written as:

The same expression works whether Aftab is 10 years old, 13 years old, or any other age.

What is a Variable?

A variable is a letter that represents an unknown or changing number.

Matchstick Pattern

Patterns help us understand how numbers change. Imagine making the same shape again and again using matchsticks.

Instead of counting every matchstick, we use an expression.

Shopping Example

A shopkeeper sells coconuts for ₹35 each and jaggery for ₹60 per kg.

Common Mistakes

• Confusing letters with fixed numbers.
• Forgetting to explain what the variable represents.
• Writing expressions without units where required.
• Using different letters for the same quantity.

4.1 Writing Algebraic Expressions

An algebraic expression is a mathematical statement made using numbers, letters (variables) and mathematical operations such as addition, subtraction, multiplication and division.

Instead of writing long sentences repeatedly, we can express relationships in a short mathematical form.

Example 1 – Cost of Coconuts

Suppose one coconut costs ₹35. If a person buys c coconuts, then the total cost is

Example 2 – Cost of Jaggery

Suppose jaggery costs ₹60 per kilogram. If a customer buys j kilograms, then

Relationship Between Quantity and Expression

Understanding the Expression

In the expression 35c + 60j

• 35 represents the price of one coconut.
• c represents the number of coconuts.
• 60 represents the price of one kilogram of jaggery.
• j represents the quantity of jaggery.
• The symbol (+) adds both costs.

Formulae Around Us

Many mathematical rules are written as formulas. A formula is simply an algebraic expression that represents a relationship.

Worked Example

Find the perimeter of a square whose side is 7 cm.

Real-Life Applications

Common Errors

• Writing 35 + c instead of 35c.
• Using different variables for the same quantity.
• Forgetting to explain the meaning of variables.
• Adding unlike quantities.

4.2 Revisiting Arithmetic Expressions

Before learning more about algebraic expressions, let us revise arithmetic expressions. An arithmetic expression contains only numbers and mathematical operations such as addition (+), subtraction (−), multiplication (×) and division (÷).

To get the correct answer, mathematical operations must be performed in the proper order.

What is an Arithmetic Expression?

An arithmetic expression is formed using numbers, brackets and mathematical operations.

Order of Operations

Whenever an expression contains different mathematical operations, follow this sequence.

Worked Example 1

Evaluate 23 − 10 ÷ 2

Worked Example 2

Evaluate 68 − (18 + 13)

Worked Example 3

Evaluate 20 + 8 × (16 − 6)

Understanding Brackets

Brackets tell us which operation should be completed first. They help avoid confusion and make expressions easier to read.

Common Mistakes

• Ignoring brackets.
• Adding before multiplication.
• Performing operations from left to right without following the rules.
• Forgetting to simplify the expression step by step.

Think About It

Observe the expressions below.

Why are the answers different? Because the brackets change the order in which operations are performed.

4.3 Omission of the Multiplication Symbol in Algebraic Expressions

In arithmetic, multiplication is usually written using the symbol ×. However, in algebra, mathematicians generally omit this symbol when multiplying a number by a variable. This makes algebraic expressions shorter and easier to read.

Why is the Multiplication Symbol Removed?

Removing the multiplication symbol makes mathematical expressions simpler and avoids unnecessary symbols. It also provides a standard way of writing algebraic expressions that is used all over the world.

Finding the Pattern

Observe the following sequence.

Each number is obtained by multiplying 4 with the position number.

Worked Example

Find the 7th term of the sequence 4, 8, 12, 16, ...

Writing Expressions

The multiplication symbol is omitted only when multiplying numbers with variables.

Application in Geometry

Variables are commonly used in geometry formulas.

Common Mistakes

• Writing x5 instead of 5x.
• Using × unnecessarily in algebra.
• Confusing addition with multiplication.
• Using different variables for the same quantity.

4.4 Simplification of Algebraic Expressions

Earlier we learnt how to write algebraic expressions using variables. Sometimes an expression contains many terms that can be combined to make it shorter and easier to understand. This process is called simplification.

Using the Distributive Property

Suppose a large rectangle is divided into two smaller rectangles. Instead of finding the area separately, we can write one algebraic expression for the total area.

This rule is known as the Distributive Property.

Example 1

Expand the expression

Shopping Example

A furniture shop sells chairs for ₹40 and tables for ₹75. Suppose a customer purchases x chairs and y tables.

Like Terms and Unlike Terms

Only terms having exactly the same variables with the same powers can be combined.

Combining Like Terms

Add or subtract only the numerical coefficients while keeping the variable unchanged.

Worked Example

Simplify

Example

Simplify

Common Mistakes

• ❌ 5x + 4 = 9x
• ❌ 8a + 3b = 11ab
• ❌ 2x × 3 = 5x

Equal Algebraic Expressions

Two algebraic expressions are called equivalent expressions if they produce the same value for every value of the variable.

Finding the Value of an Expression

Sometimes the value of a variable is given. Replace the variable with its value and calculate the answer.

Worked Example

Find the value of 7a − 9 when a = 8

Comparing Expressions

Observe the expressions below.

Mind the Mistake

Students often make these mistakes while simplifying expressions.

Real-Life Applications

4.5 Pick Patterns and Reveal Relationships

Mathematics is full of interesting patterns. We often observe patterns in numbers, shapes and daily life. Instead of writing every term separately, we can describe the entire pattern using an algebraic expression.

An algebraic expression helps us predict any term of a pattern without writing all previous terms.

Formula Detective

Suppose a machine takes a number as input and performs the following steps.

Input → ×2 → −2 → Output

Let the input number be represented by x.

This single formula works for every input value.

Understanding Number Machines

A number machine performs the same mathematical operations every time. Observe the table below.

Notice that every output is obtained by multiplying the input by 2 and then subtracting 2.

Worked Example

Find the output when the input is 12.

Finding the Rule

Study each input-output table carefully and identify the formula.

Algebraic Expressions Describe Patterns

Suppose flowers are arranged in a repeating design. Instead of counting every flower separately, we can observe the pattern and write a rule.

The variable n represents the position number.

Pattern Examples

Real-Life Uses of Patterns

Chapter Summary

In this chapter, we learnt how letters can be used to represent numbers and how algebra helps us describe mathematical relationships in a simple and powerful way. Instead of writing long calculations, we use variables and algebraic expressions to write general rules.

Quick Revision Notes

Practice Questions

Short Answer

1. Write an expression for five more than x.
2. Write an expression for three times a number decreased by 7.
3. Simplify 8x + 5x.
4. Simplify 4(a+6).
5. Find the value of 6x−4 when x=9.
6. Find the perimeter of a square of side y.
7. Identify the coefficient in 9m.
8. Write the formula for the nth multiple of 7.

Everyday Applications

Frequently Asked Questions

Design Patterns Using Algebra

Patterns appear everywhere around us. Some designs repeat after a fixed number of positions. Instead of checking every position one by one, we can use algebra to predict which design will appear.

Every third position repeats the same pattern.

Understanding the Pattern

The designs repeat after every 3 positions.

Finding Any Position

To know which design appears at a particular position, divide the position number by 3.

Worked Examples

Patterns in a Calendar

Calendars also contain interesting mathematical patterns. Consider any 2 × 2 block of dates.

 12   13

 19   20

Now add the numbers along each diagonal.

Both diagonal sums are equal.

Why Does This Happen?

Suppose the top-left number is represented by a.

a      a+1

a+7    a+8

Now compare the diagonal sums.

Since both expressions are equal, every 2 × 2 square in a calendar has equal diagonal sums.

Where Are Patterns Used?

Matchstick Patterns

Patterns can also be formed using matchsticks. Instead of counting every matchstick one by one, we can discover a mathematical rule that works for every step.

Observe the Pattern

Step 1      △

Step 2     △△

Step 3    △△△

Step 4   △△△△

The number of triangles increases by 1 in every step. The number of matchsticks also increases in a regular way.

Matchstick Table

Finding the Rule

Observe the matchsticks carefully.

• Step 1 → 3 matchsticks
• Step 2 → 5 matchsticks
• Step 3 → 7 matchsticks
• Step 4 → 9 matchsticks

The number of matchsticks increases by 2 every time.

General Formula

Let the step number be represented by n.

Since every step adds two matchsticks, the algebraic expression becomes

This formula gives the number of matchsticks for any step.

Worked Example

Find the number of matchsticks required for Step 12.

Another Way to Think

Instead of remembering the entire pattern, notice that the first triangle always needs 3 matchsticks. Every additional triangle shares one side and therefore adds only 2 more matchsticks.

Simplifying,

Both methods give exactly the same answer.

Compare the Two Expressions

Therefore, both expressions are equivalent.

Real-Life Applications

Figure It Out (Questions 1–5)

Now apply what you have learnt about algebraic expressions, patterns, and simplification to solve real-life situations.

Question 1

One plate of food costs ₹30 and one plate of dosa costs ₹20. If x plates of puri and y plates of dosa are ordered, write an algebraic expression for the total amount.

Question 2

Pushpita sells two kinds of flowers. One bunch contains p roses. Another bunch contains q roses. If she sells x bunches of the first type and y bunches of the second type, write an expression for the total number of flowers sold.

Question 3

A snail climbs a wall. On the first day it climbs d metres. Each following day it climbs 1 metre more than the previous day. Write expressions for the distance travelled.

Question 4

Radha practises cycling. First week she rides x km daily. Every next week she increases her distance by 2 km. Write expressions for four weeks.

Question 5

Complete the missing operations in the number machines. Observe the rule carefully before filling the blanks.

Input → Operation → Output

Everyday Applications

Figure It Out (Questions 6–10)

In this section we solve algebraic expression problems involving travelling, simplifying expressions, addition, subtraction and writing expressions from everyday situations.

Question 6 : Train Journey

A local train travels through three stations. The distance between every two stations is t km. The train stops for 2 minutes at every station. Write an expression for the total travelling time.

Station A ---- t km ---- Station B ---- t km ---- Station C

Question 7 : Simplify the Expressions

Simplify each algebraic expression by combining like terms.

Question 8 : Add the Expressions

Add the following algebraic expressions carefully.

Question 9 : Subtract the Expressions

Subtract one algebraic expression from another.

Question 10 : Write Algebraic Expressions

Convert each verbal statement into an algebraic expression.

Common Mistakes

• Ignoring negative signs while simplifying.
• Adding unlike terms.
• Removing brackets incorrectly.
• Writing words instead of algebraic expressions.
• Forgetting to combine constant terms.

Everyday Applications