NCERT Class 7 Ganita Prakash 1 Chapter 04 Expressions using Letter-Numbers PDF Download

Chapter 4: Expressions Using Letter-Numbers

4.1 The Notion of Letter-Numbers

In this chapter we shall look at a concise way of expressing mathematical relations and patterns. We shall see how this helps us in thinking about these relationships and patterns, and in explaining why they may hold true.

Example 1: Shabnam is 3 years older than Aftab. When Aftab's age 10 years, Shabnam's age will be 13 years. Now Aftab's age is 18 years, what will Shabnam's age be? _______

? Given Aftab's age, how will you find out Shabnam's age?

Easy: We add 3 to Aftab's age to get Shabnam's age.

? Can we write this as an expression?

Shabnam's age is 3 years more than Aftab's. In short, this can be written as:Shabnam's age = Aftab's age + 3.

Such mathematical relations are generally represented in a shorthand form. In the relation above, instead of writing the phrase 'Aftab's Age', the convention is to use a convenient symbol. Usually, letters or short phrases are used for this purpose.

Let us say we use the letter \( a \) to denote Aftab's age (we could have used any other letter), and \( s \) to denote Shabnam's age. Then the expression to find Shabnam's age will be \( a + 3 \), which can be written as

\[ s = a + 3. \]

? If \( a \) is 23 (Aftab's age in years), then what is Shabnam's age?

[Figure 4.1: Table showing Aftab's age (4, 10, 23, ?, a) and Expression for Shabnam's age (4 + 3, 10 + 3, 23 + 3, ? + 3, a + 3), See in your textbook]

Replacing \( a \) by 23 in the expression \( a + 3 \), we get, \( s = 23 + 3 = 26 \) years. Letters such as \( a \) and \( s \) that are used to represent numbers are called letter-numbers. Mathematical expressions containing letter-numbers, such as the expression \( a + 3 \), are called algebraic expressions.

Teacher's Note

A letter-number is just a placeholder for an unknown value. When you replace the letter with a specific number, you get a specific answer. In Example 1, \( a \) can be any age, but once you know Aftab is 23, you substitute 23 for \( a \) and calculate. This substitution process is the key to using algebraic expressions.

? Given the age of Shabnam, write an expression to find Aftab's age.

We know that Aftab is 3 years younger than Shabnam. So, Aftab's age will be 3 less than Shabnam's. This can be described as Aftab's age = Shabnam's age - 3.

If we again use the letter \( a \) to denote Aftab's age and the letter \( s \) to denote Shabnam's age, then the algebraic expression would be: \( a = s - 3 \), meaning 3 less than \( s \).

? Use this expression to find Aftab's age if Shabnam's age is 20.

Example 2: Parthiv is making matchstick patterns. He repeatedly places Ls next to each other. Each L has two matchsticks as shown in Figure 4.2.

[Figure 4.2: Matchsticks arranged in L shapes, See in your textbook]

How many matchsticks are needed to make 5 Ls? It will be \( 5 \times 2 \). How many matchsticks are needed to make 7 Ls? It will be \( 7 \times 2 \). How many matchsticks are needed to make 45 Ls? It will be \( 45 \times 2 \).

Now, what is the relation between the number of Ls and the number of sticks?

First, let us describe the relationship or the pattern here. Every L needs 2 matchsticks. So the number of matchsticks needed will be 2 times the number of L's. This can be written as: Number of matchsticks = \( 2 \times \) Number of L's

Now, we can use any letter to denote the number of L's. Let's use \( n \). The algebraic expression for the number of matchsticks will be: \[ 2 \times n. \]

This expression tells us how many matchsticks are needed to make \( n \) L's. To find the number of matchsticks, we just replace \( n \) by the number of L.

Example 3: Ketaki prepares and supplies coconut-jaggery laddus. The price of a coconut is Rs. 35 and the price of 1 kg jaggery is Rs. 60.

? How much should she pay if she buys 10 coconuts and 5 kg jaggery?

Cost of 10 coconuts = \( 10 \times \) Rs. 35 Cost of 5 kg jaggery = \( 5 \times \) Rs. 60 Total cost = \( 10 \times \) Rs. 35 + \( 5 \times \) Rs. 60 = Rs. 350 + Rs. 300 = Rs. 650.

? How much should she pay if she buys 8 coconuts and 9 kg jaggery?

? Write an algebraic expression to find the total amount to be paid for a given number of coconuts and quantity of jaggery.

Let us identify the relationships and then write the expressions.

Here, 'c' represents the number of coconuts and 'j' represents the number of kgs of jaggery. The total amount to be paid will be: Cost of coconuts + Cost of jaggery.

The corresponding algebraic expression can be written as: \[ c \times 35 + j \times 60 \]

? Use this expression (or formula) to find the total amount to be paid for 7 coconuts and 4 kg jaggery.

Notice that for different values of 'c' and 'j', the value of the expression also changes. Writing this expression as a sum of terms we get: \[ c \times 35 + j \times 60 \]

Teacher's Note

When an algebraic expression has multiple letter-numbers (like \( c \) and \( j \) here), you must substitute carefully. For 7 coconuts and 4 kg jaggery: replace \( c \) with 7 and \( j \) with 4, then calculate \( 7 \times 35 + 4 \times 60 = 245 + 240 = 485 \). Check that you have replaced every occurrence of the letter.

Example 4: We are familiar with calculating the perimeters of simple shapes. Write expressions for perimeters.

The perimeter of a square is 4 times the length of its side. This can be written as the expression: \( 4 \times q \), where \( q \) stands for the sidelength.

? What is the perimeter of a square with sidelength 7 cm? Use the expression to find out.

You must have realised how the use of letter-numbers and algebraic expressions allows us to express general mathematical relations in

Key Points

• A letter-number is a letter that represents an unknown number. We use it to write general relationships that work for any value of that number.
• An algebraic expression is a mathematical statement using letter-numbers, like \( a + 3 \) or \( 2 \times n \). To find its value, replace the letter with a number and calculate.
• You can write algebraic expressions for real-life situations by first identifying the pattern or relationship, then using letters to stand for the varying quantities.
• The same algebraic expression can be used for many different values, making it a shorthand way to describe a whole family of calculations.

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