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Algebra

We know that letter of the English alphabet like a, b, c, x, y, z, l, m, n, etc. can be used as variables to form expressions.

In our daily life, we see various patterns of objects or numbers which help us to predict or determine some information.

For example, observe the letter . It consists of five match sticks. 

Now, observe the following design made by a number of As.

We can find the total number of match sticks involved in this design by counting them, which is a time consuming process.

There is one more way to solve such problems.   

To understand the concept, look at the following video.

Now, we can easily find the number of match sticks involved in the above design as follows:

Number of match sticks in one A = 5

Number of A's in the design = 8

Number of match sticks involved in design = 5 × Number of A's in design = 5 × 8 = 40

Now, let us learn how to use variables to generalize patterns found in real life.

Suppose there are 15 flowering plants in one row in a garden. How many plants are there in the garden, if there are 30 such rows?

The total number of flowering plants in the garden depends on the number of rows in the garden.

If the number of rows = 3 then total number of plants = 15 × 3 = 45

If number of rows = 4 then total number of plants = 15 × 4 = 60

Total number of plants = 15 × number of rows in the garden

If we substitute the variable n in place of the number of rows, then the above rule becomes:

Total number of plants = 15 × n = 15n, where n = 1, 2, 3, 4 …

According to the considered situation, n = 30

Thus, total number of plants = 15 × 30 = 450

In the same way, we can apply the concept of variables in many real life situations.

Let us now understand this concept better by solving some more examples.

Example 1:

In a class, students have been seated in few rows such that 5 students are there in each row. Generalize this pattern for n rows. Also, find the total number of students in the class if there are 8 rows in total. 

Solution:

We can find the total number of  students as follows:

Total number of students in class = Number of students in one row × Number of rows

⇒ Total number of students in class = 5 × Number of rows

When the number of rows is n.

Total number of students in class = 5 × n = 5n

When the number of rows is 8.

Total number of students in class = 5n = 5 × 8 = 40

Example 2:

Observe the following pattern made by match sticks.

Generalize the given pattern and find the number of match sticks in 20th column. 

Solution: 

From the figure, we have

Number of match stic…

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