Using Expressions in Formulas
You already know formulas for perimeter and area of shapes. These formulas are actually algebraic expressions written using variables.
For example, the perimeter of a rectangle = 2(l + b). Here, l and b are variables representing length and breadth.
In Class 6, you will see how algebraic expressions are used in real formulas and how to find values by substituting numbers into them.
What is Using Expressions in Formulas - Grade 6 Maths (Algebra)?
What is a formula?
A formula is an algebraic expression that shows the relationship between different quantities. It gives a rule to calculate one quantity from others.
Common formulas you already know:
- Perimeter of a rectangle = 2(l + b)
- Perimeter of a square = 4s
- Area of a rectangle = l × b
- Area of a square = s × s = s²
- Perimeter of an equilateral triangle = 3a
Each formula uses variables (letters) so that it works for any values, not just specific numbers.
Using Expressions in Formulas Formula
How to use a formula:
- Write down the formula.
- Identify what values are given.
- Substitute (replace) the variables with the given numbers.
- Calculate the answer.
Example: Find the perimeter of a rectangle with length 8 cm and breadth 5 cm.
- Formula: P = 2(l + b)
- Substitute: P = 2(8 + 5)
- Calculate: P = 2 × 13 = 26 cm
Writing your own formulas:
- A matchstick pattern uses 3 sticks per triangle. For n triangles in a row (sharing sides): Sticks = 2n + 1.
- The total cost of n items at Rs p each = np.
- Distance = speed × time, or d = s × t.
Types and Properties
Formulas from patterns:
- Square numbers: 1, 4, 9, 16, ... → nth term = n²
- Triangular numbers: 1, 3, 6, 10, ... → nth term = n(n+1)/2
- Arithmetic patterns: If the difference between terms is constant (d), then nth term = a + (n−1)d, where a is the first term.
Formulas from geometry:
- Perimeter of regular polygon with n sides of length a = na
- Number of diagonals of a polygon with n sides = n(n−3)/2
Solved Examples
Example 1: Perimeter of a Rectangle
Problem: Find the perimeter of a rectangle with length 12 cm and breadth 7 cm.
Solution:
Formula: P = 2(l + b)
P = 2(12 + 7) = 2 × 19 = 38 cm
Example 2: Area of a Square
Problem: A square has side 9 cm. Find its area.
Solution:
Formula: A = s²
A = 9² = 9 × 9 = 81 cm²
Example 3: Perimeter of an Equilateral Triangle
Problem: An equilateral triangle has each side = 6.5 cm. Find its perimeter.
Solution:
Formula: P = 3a
P = 3 × 6.5 = 19.5 cm
Example 4: Total Cost Formula
Problem: One notebook costs Rs 35. Write a formula for the cost of n notebooks, and find the cost of 8 notebooks.
Solution:
Formula: Cost = 35n
For n = 8: Cost = 35 × 8 = Rs 280
Example 5: Pattern Formula
Problem: A pattern is: 5, 8, 11, 14, 17, ... Write an expression for the nth term and find the 10th term.
Solution:
First term = 5, common difference = 3.
nth term = 5 + (n − 1) × 3 = 5 + 3n − 3 = 3n + 2
10th term = 3(10) + 2 = 30 + 2 = 32
Example 6: Matchstick Pattern
Problem: Triangles are made in a row using matchsticks. The first triangle uses 3 sticks. Each additional triangle shares one side with the previous one (adds 2 sticks). Write a formula for n triangles.
Solution:
- 1 triangle: 3 sticks
- 2 triangles: 3 + 2 = 5 sticks
- 3 triangles: 5 + 2 = 7 sticks
Pattern: 2n + 1 sticks for n triangles.
For 10 triangles: 2(10) + 1 = 21 sticks
Example 7: Finding Side from Perimeter
Problem: The perimeter of a square is 48 cm. Find its side.
Solution:
Formula: P = 4s
48 = 4s
s = 48 ÷ 4 = 12 cm
Example 8: Regular Hexagon Perimeter
Problem: A regular hexagon has each side = 5 cm. Write a formula and find the perimeter.
Solution:
A regular hexagon has 6 equal sides.
Formula: P = 6a
P = 6 × 5 = 30 cm
Real-World Applications
Where formulas with expressions are used:
- Construction: Calculating perimeter for fencing, area for flooring, volume for filling.
- Shopping: Total cost = price per item × number of items.
- Travel: Distance = speed × time.
- Science: Formulas for force, energy, and density all use algebraic expressions.
- Technology: Spreadsheet formulas (like =A1*B1+C1) are algebraic expressions.
Key Points to Remember
- A formula is an algebraic expression that shows a relationship between quantities.
- To use a formula, substitute the given values into the variables and calculate.
- Perimeter of rectangle = 2(l + b), Area = l × b.
- Perimeter of square = 4s, Area = s².
- Patterns can be described using expressions like 2n + 1 or 3n + 2.
- Formulas work for any value of the variables — that is the power of algebra.
- You can also reverse a formula to find an unknown (e.g., find side from perimeter).
Practice Problems
- Find the area of a rectangle with length 15 cm and breadth 9 cm using the formula.
- One pencil costs Rs 8. Write a formula for n pencils and find the cost of 25 pencils.
- A pattern gives: 2, 6, 10, 14, ... Write an expression for the nth term.
- The perimeter of a regular pentagon is 45 cm. Find the length of each side.
- How many matchsticks are needed for 15 triangles in a row (sharing sides)?
- Using d = s × t, find the distance covered at 60 km/h in 3 hours.
Frequently Asked Questions
Q1. What is the difference between an expression and a formula?
An expression is a combination of variables and numbers (like 2x + 3). A formula is an expression that equals something — it shows a rule for calculating (like P = 2l + 2b). Every formula contains an expression.
Q2. How do I substitute values into a formula?
Replace each variable (letter) with the given number. Then calculate step by step. For example, in P = 2(l + b) with l = 10 and b = 4: P = 2(10 + 4) = 2(14) = 28.
Q3. Can I create my own formula?
Yes! If you see a pattern, you can write a formula for it. Find how the numbers change and express it using a variable like n.
Q4. Why are formulas useful?
Formulas let you calculate answers quickly for any values, without starting from scratch each time. One formula works for all rectangles, all squares, all patterns of that type.
Q5. What does 'nth term' mean?
The nth term is a formula that gives you any term in a sequence. If the nth term is 3n + 1, then the 1st term is 4, the 2nd is 7, the 100th is 301.
Q6. How do I find a formula from a pattern?
Look at how the numbers change. If they increase by the same amount each time, the formula is of the form an + b. Find 'a' (the common difference) and 'b' by checking with the first term.
