Solved Exercises on Finding Perimeter of a Rectangle with Variables in Grade 3

Master finding perimeter of rectangles with variables: algebraic expressions, geometric formulas through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic variable rectangle
Exercise 1
Find the perimeter of a rectangle with length x units and width y units.
Definition:

Variable: A symbol (like x, y, z) that represents a number

Rectangle: A four-sided shape with opposite sides equal and all angles 90°

Perimeter: The total distance around the rectangle

Perimeter method with variables:
  1. Use the same formula: Perimeter = 2 × (length + width)
  2. Substitute the variables: Perimeter = 2 × (x + y)
  3. Simplify: Perimeter = 2(x + y)
Given
Length = x, Width = y
Formula
P = 2(l+w)
Result
P = 2(x+y)
Step 1: Identify the dimensions

Length = x units, Width = y units

These are variables representing unknown numbers

Step 2: Apply the perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × (x + y)

Step 3: Write the algebraic expression

Perimeter = 2(x + y) units

Perimeter = 2(x + y) units
Final answer:

The perimeter of the rectangle is 2(x + y) units

Applied rules:

• Variable Property: Variables represent numbers

• Rectangle Formula: P = 2(l+w) works with variables

• Algebraic Expression: Keep variables in the answer

2 Rectangle with expressions
Exercise 2
Find the perimeter of a rectangle with length (a + 2) units and width (b - 1) units.
Definition:

Algebraic Expression: A combination of variables, numbers, and operations

Algebraic Perimeter: Perimeter expressed with variables and expressions

Given
Length = a+2, Width = b-1
Formula
P = 2(l+w)
Result
P = 2(a+b+1)
Step 1: Identify the dimensions

Length = (a + 2) units, Width = (b - 1) units

These are algebraic expressions involving variables

Step 2: Apply the perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × ((a + 2) + (b - 1))

Step 3: Simplify the expression

Perimeter = 2 × (a + 2 + b - 1)

Perimeter = 2 × (a + b + 1)

Perimeter = 2(a + b + 1)

Perimeter = 2(a + b + 1) units
Final answer:

The perimeter of the rectangle is 2(a + b + 1) units

Applied rules:

• Expression Addition: Combine like terms when adding expressions

• Distributive Property: 2(a + b + 1) = 2a + 2b + 2

• Algebraic Simplification: Combine constants and like terms

3 Number times variable
Exercise 3
Find the perimeter of a rectangle with length 3x units and width 2y units.
Definition:

Coefficient: A number multiplied by a variable (3 in 3x)

Term: A single number, variable, or number and variable multiplied together

Given
Length = 3x, Width = 2y
Formula
P = 2(l+w)
Result
P = 2(3x+2y)
Step 1: Identify the dimensions

Length = 3x units, Width = 2y units

These are terms with coefficients 3 and 2 respectively

Step 2: Apply the perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × (3x + 2y)

Step 3: Distribute the 2

Perimeter = 2(3x + 2y)

Perimeter = 6x + 4y units

Step 4: Alternative form

You can also write it as: Perimeter = 2(3x + 2y) units

Perimeter = 6x + 4y units
Final answer:

The perimeter of the rectangle is 6x + 4y units

Applied rules:

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

• Coefficient Multiplication: 2 × 3x = 6x, 2 × 2y = 4y

• Like Terms: Cannot combine unlike terms (x and y)

Rules and methods, laws,...
\(P = 2(l + w)\)
Perimeter of Rectangle with Variables
Rectangle
\(P = 2(l + w)\)
Basic formula
With Variables
\(P = 2(x + y)\)
Substituted
With Coefficients
\(P = 2(ax + by)\)
Expanded
Key definitions:

Variable: A symbol (usually a letter) that represents an unknown number

Algebraic Expression: A mathematical phrase containing numbers, variables, and operations

Coefficient: A number multiplied by a variable (the 3 in 3x)

Term: A single number, variable, or number and variable multiplied together

Like Terms: Terms that contain the same variable raised to the same power

Perimeter: The total distance around the outside of a shape

Complete methodology:
  1. Identify the shape: Confirm it's a rectangle
  2. Identify dimensions: Find the expressions for length and width
  3. Substitute into formula: Replace l and w in P = 2(l + w) with the expressions
  4. Simplify: Combine like terms if possible
  5. Apply distributive property: If needed to expand
  6. Write final answer: With variables included
Tip 1: The perimeter formula P = 2(l + w) works the same with variables.
Tip 2: Remember the distributive property: a(b + c) = ab + ac.
Tip 3: You cannot combine unlike terms (x and y cannot be added).
Tip 4: Always keep variables in your final answer when asked for an expression.
Common errors: Forgetting to multiply by 2, combining unlike terms, not distributing properly.
Memory aids: "Perimeter is the fence around the yard" and "distribute to each term".
Formulas to know by heart:

• Perimeter of rectangle: P = 2(l + w)

• With variables: P = 2(x + y)

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

• Area of rectangle: A = l × w

Solution: Exercises 4 to 5
4 Rectangle with complex expressions
Exercise 4
Find the perimeter of a rectangle with length (2m + 3) units and width (m - 1) units.
Definition:

Complex Expression: An algebraic expression with multiple terms

Combining Like Terms: Adding or subtracting terms with the same variable

Given
Length = 2m+3, Width = m-1
Formula
P = 2(l+w)
Result
P = 6m+4
Step 1: Identify the dimensions

Length = (2m + 3) units, Width = (m - 1) units

Both expressions involve the variable m

Step 2: Apply the perimeter formula

Perimeter = 2 × (length + width)

Perimeter = 2 × ((2m + 3) + (m - 1))

Step 3: Simplify inside parentheses

Perimeter = 2 × (2m + 3 + m - 1)

Combine like terms: 2m + m = 3m, and 3 - 1 = 2

Perimeter = 2 × (3m + 2)

Step 4: Distribute the 2

Perimeter = 2(3m + 2)

Perimeter = 6m + 4 units

Step 5: Verify by substitution

If m = 1: Length = 5, Width = 0, Perimeter = 10 (but width can't be 0)

If m = 2: Length = 7, Width = 1, Perimeter = 16, Formula: 6(2)+4 = 16 ✓

Perimeter = 6m + 4 units
Final answer:

The perimeter of the rectangle is 6m + 4 units

Applied rules:

• Combining Like Terms: Terms with the same variable can be combined

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

• Algebraic Simplification: Combine like terms to simplify expressions

5 Comparing perimeters with variables
Exercise 5
Rectangle A has length (x + 4) units and width (x - 2) units. Rectangle B has length 2x units and width 3 units. Which rectangle has a larger perimeter?
Definition:

Comparison Problem: Calculating and comparing expressions with variables

Algebraic Comparison: Determining which expression is larger

Rectangle A
P = 2((x+4)+(x-2)) = 4x+4
Rectangle B
P = 2(2x+3) = 4x+6
Result
B larger by 2
Step 1: Calculate Rectangle A's perimeter

Rectangle A: length = (x + 4), width = (x - 2)

Perimeter = 2 × (length + width) = 2 × ((x + 4) + (x - 2))

Perimeter = 2 × (x + 4 + x - 2) = 2 × (2x + 2) = 4x + 4 units

Step 2: Calculate Rectangle B's perimeter

Rectangle B: length = 2x, width = 3

Perimeter = 2 × (length + width) = 2 × (2x + 3) = 4x + 6 units

Step 3: Compare the perimeters

Rectangle A: 4x + 4

Rectangle B: 4x + 6

Since 4x + 6 > 4x + 4 (the constant 6 is greater than 4), Rectangle B has the larger perimeter

Step 4: State the conclusion

Rectangle B has a larger perimeter than Rectangle A

Rectangle B's perimeter is 2 units larger than Rectangle A's perimeter

Rectangle B has larger perimeter (4x+6 vs 4x+4)
Final answer:

Rectangle B has a larger perimeter of 4x + 6 units compared to Rectangle A's perimeter of 4x + 4 units

Applied rules:

• Comparison strategy: Calculate both expressions before comparing

• Like terms: When comparing, focus on the differences

• Algebraic reasoning: Since 6 > 4, then 4x+6 > 4x+4 for all values of x

Complete Guide: Laws, Methods, Rules, Definitions, Expressions
\(P = 2(l + w)\)
Perimeter of Rectangle with Variables
Key definitions:

Variable: A symbol (typically a letter) that represents an unknown number. In geometry problems, variables often represent measurements like length or width.

Algebraic Expression: A mathematical phrase containing numbers, variables, and operations (+, -, ×, ÷). For example: 2x + 3, (a + b), 4y - 1.

Coefficient: The number multiplied by a variable in a term. In 5x, the coefficient is 5.

Term: A single number, variable, or number and variable multiplied together. Examples: 3, x, 4y, 2ab.

Like Terms: Terms that contain the same variable raised to the same power. For example, 3x and 5x are like terms, but 3x and 3y are not.

Perimeter: The total distance around the outside edge of a two-dimensional shape.

Complete methodology:
  1. Identify the shape: Confirm that the figure is a rectangle
  2. Identify dimensions: Determine the algebraic expressions for length and width
  3. Substitute into formula: Replace l and w in P = 2(l + w) with the variable expressions
  4. Simplify inside parentheses: Combine like terms if possible
  5. Apply distributive property: Multiply the 2 by each term inside the parentheses
  6. Combine like terms: Simplify the final expression
  7. State the answer: Include the appropriate units (expressed in terms of variables)
Tip 1: The perimeter formula P = 2(l + w) works exactly the same with variables as with numbers.
Tip 2: Remember the distributive property: a(b + c) = ab + ac. This is essential when expanding expressions.
Tip 3: You can only combine like terms. Terms with different variables (like x and y) cannot be added together.
Tip 4: Always keep variables in your final answer when working with algebraic expressions.
Rectangle Properties: 4 right angles, opposite sides equal, diagonals equal and bisect each other, sum of interior angles is 360°.
Variable Characteristics: Represent unknown values, follow same arithmetic rules as numbers.
Common Errors: Forgetting to multiply by 2 in perimeter formula, combining unlike terms, not distributing properly.
Essential formulas:

• Perimeter of rectangle: P = 2(l + w)

• With variables: P = 2(x + y)

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

• Area of rectangle: A = l × w

• Combining like terms: ax + bx = (a + b)x

Rectangle Perimeter with Variables
:variables
Rectangle Structure

• 4 sides: 2 lengths (l) + 2 widths (w)

• 4 right angles (90°)

• Opposite sides equal

• Perimeter = 2(l + w)

Variable Perimeter

• P = 2(x + y)

• P = 2(ax + by)

• P = 2((x+a) + (y-b))

• Always distribute 2

Algebra Rules

• Like terms: x + x = 2x

• Unlike terms: x + y ≠ combine

• Distributive: a(b+c) = ab+ac

• Keep variables in answer

Variable Perimeter Steps
1
Identify Variables
2
Substitute Formula
3
Simplify
4
Distribute
5
Final Answer
Visual Representation

Rectangle with length (x) and width (y)

Perimeter = distance around the rectangle

P = 2(x + y) = x + y + x + y

Remember: Variables follow the same rules as numbers!

Questions & Answers

Question: What is a variable? Why do we use letters in math problems?

Answer: A variable is a symbol (usually a letter like x, y, or z) that represents an unknown number. We use variables for several reasons:

  • To represent unknown quantities: When we don't know a specific number, we can use a letter to stand for it
  • To show relationships: Variables help us express how different quantities relate to each other
  • To make general statements: We can write formulas that work for any numbers

For example, in perimeter problems, we might say the length is x and the width is y. This allows us to write a general formula that works no matter what specific numbers we later discover.

Think of a variable like a placeholder - it's waiting for us to fill in the actual number when we have more information.

Question: Why do we multiply by 2 in the perimeter formula even when using variables?

Answer: The reason we multiply by 2 is because it's a fundamental property of rectangles that doesn't change whether we're using numbers or variables. Here's why:

  • A rectangle always has 4 sides: top, bottom, left, right
  • The top and bottom sides are always equal (both equal to the length)
  • The left and right sides are always equal (both equal to the width)
  • So the total perimeter is: length + width + length + width = 2×length + 2×width = 2(length + width)

This is true regardless of whether the length is 5, or x, or (2m + 3). The geometric properties of rectangles remain the same, so the formula P = 2(l + w) applies universally.

Variables are just placeholders for numbers, so all the same mathematical relationships apply.

Question: Can I add x and y together in the perimeter formula? Like 2(x + y) = 2x + 2y?

Answer: Yes, absolutely! The expression 2(x + y) is equal to 2x + 2y. This is due to the distributive property of multiplication over addition.

  • Distributive property: a(b + c) = ab + ac
  • Applying to our case: 2(x + y) = 2×x + 2×y = 2x + 2y

Both 2(x + y) and 2x + 2y are correct ways to write the perimeter. Some people prefer 2(x + y) because it clearly shows the original formula structure, while others prefer 2x + 2y because it's fully expanded.

However, you CANNOT combine x and y into a single term because they are different variables (unlike terms). So x + y stays as x + y - you can't simplify it further.