Solved Exercises on Perimeter and Area with Multiplicative Comparisons in Grade 4

Master perimeter and area with multiplicative comparisons: understanding relationships between dimensions and measurements through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rectangle Area Comparison
Exercise 1
Rectangle A has a length of 8 units and a width of 3 units. Rectangle B has a length of 12 units and a width of 2 units. Which rectangle has a greater area? How many times greater is the larger area compared to the smaller area?
Definition:

Area: The amount of space inside a shape, measured in square units

Rectangle area: Area = length × width

Multiplicative comparison: Comparing quantities by multiplication

Solution method:
  1. Calculate the area of each rectangle
  2. Compare the areas to find which is larger
  3. Divide the larger area by the smaller area to find the multiplicative relationship
  4. State the comparison in terms of "times as much"
Calculate Areas
A: 8×3=24, B: 12×2=24
Compare
24 vs 24
Result
Equal areas
Step 1: Calculate area of Rectangle A

Area of A = length × width = 8 units × 3 units = 24 square units

Step 2: Calculate area of Rectangle B

Area of B = length × width = 12 units × 2 units = 24 square units

Step 3: Compare the areas

24 square units vs 24 square units

Step 4: Find the multiplicative relationship

24 ÷ 24 = 1

Step 5: State the comparison

Both rectangles have equal areas. Neither is greater.

Both rectangles have area = 24 square units
Final answer:

Both rectangles have equal areas of 24 square units. Neither rectangle has a greater area.

Applied rules:

• Area Formula: Area of rectangle = length × width

• Multiplicative Comparison: Divide larger by smaller to find "times as much"

• Equality Rule: When areas are equal, the ratio is 1

Tip: Rectangles with different dimensions can have the same area
2 Perimeter Multiplicative Comparison
Exercise 2
Square X has sides of length 6 units. Square Y has sides of length 3 units. How many times greater is the perimeter of Square X compared to Square Y?
Definition:

Perimeter: The distance around the outside of a shape

Square perimeter: Perimeter = 4 × side length

Multiplicative relationship: The factor by which one quantity is larger than another

Calculate Perimeters
X: 4×6=24, Y: 4×3=12
Compare
24 vs 12
Result
24÷12=2
Step 1: Calculate perimeter of Square X

Perimeter of X = 4 × side length = 4 × 6 units = 24 units

Step 2: Calculate perimeter of Square Y

Perimeter of Y = 4 × side length = 4 × 3 units = 12 units

Step 3: Find the multiplicative relationship

Ratio = Perimeter of X ÷ Perimeter of Y = 24 ÷ 12 = 2

Step 4: State the comparison

The perimeter of Square X is 2 times greater than the perimeter of Square Y

Square X perimeter is 2 times greater
Final answer:

The perimeter of Square X is 2 times greater than the perimeter of Square Y.

Applied rules:

• Perimeter Formula: Perimeter of square = 4 × side length

• Multiplicative Comparison: Divide larger by smaller to find "times as much"

• Linear Relationship: Perimeter changes linearly with side length

Tip: When side lengths double, perimeters also double
3 Area and Side Length Relationship
Exercise 3
Square P has a side length of 4 units. Square Q has a side length that is 3 times as long as Square P. How many times greater is the area of Square Q compared to Square P?
Definition:

Square area: Area = side length × side length = side²

Side length relationship: Square Q has side length = 3 × Square P's side length

Area scaling: Area changes by the square of the linear dimension change

Find Dimensions
P: 4, Q: 3×4=12
Calculate Areas
P: 4²=16, Q: 12²=144
Compare
144÷16=9
Step 1: Find side length of Square Q

Side length of Q = 3 × side length of P = 3 × 4 units = 12 units

Step 2: Calculate area of Square P

Area of P = side² = 4² = 16 square units

Step 3: Calculate area of Square Q

Area of Q = side² = 12² = 144 square units

Step 4: Find the multiplicative relationship

Ratio = Area of Q ÷ Area of P = 144 ÷ 16 = 9

Step 5: State the comparison

The area of Square Q is 9 times greater than the area of Square P

Square Q area is 9 times greater
Final answer:

The area of Square Q is 9 times greater than the area of Square P.

Applied rules:

• Area Scaling Rule: When linear dimensions scale by a factor, area scales by the square of that factor

• Square Area Formula: Area = side length²

• Multiplicative Comparison: Divide larger by smaller to find "times as much"

Tip: If side length increases by factor of n, area increases by factor of n²
Perimeter and Area with Multiplicative Comparisons
Area = length × width
Rectangle Area Formula
Perimeter
P = 2(l+w)
Rectangle Perimeter
Area
A = l × w
Rectangle Area
Scaling
Area scales by n²
When linear dimensions scale by n
Key definitions:

Perimeter: The total distance around the boundary of a shape

Area: The amount of space inside a two-dimensional shape

Multiplicative comparison: Comparing quantities by finding how many times one is larger than another

Linear dimension: A measurement along a straight line (length, width, height)

Scale factor: The number by which a dimension is multiplied

Comparison methodology:
  1. Identify dimensions: Note the length and width of each shape
  2. Calculate measurements: Find perimeter and/or area for each shape
  3. Compare values: Determine which measurement is larger
  4. Find relationship: Divide the larger by the smaller to find the multiplicative factor
  5. Express comparison: State how many times greater one measurement is
Tip 1: When side lengths increase by a factor of n, perimeters increase by factor of n.
Tip 2: When side lengths increase by a factor of n, areas increase by factor of n².
Tip 3: Draw diagrams to visualize the shapes and their relationships.
Tip 4: Always check that your answer makes sense in the context of the problem.
Common errors: Forgetting to square the scale factor for area, confusing perimeter with area, miscalculating formulas.
Memory aid: "Linear changes affect perimeter directly, area quadratically."
Geometric rules:

• Perimeter of rectangle = 2(length + width)

• Area of rectangle = length × width

• Perimeter of square = 4 × side length

• Area of square = side length²

• When linear dimensions scale by factor n, perimeter scales by n, area scales by n²

Solution: Exercises 4 to 5
4 Rectangle Dimension Changes
Exercise 4
Rectangle C has a length of 10 units and a width of 4 units. Rectangle D has the same length as Rectangle C but its width is 2 times as wide. How many times greater is the area of Rectangle D compared to Rectangle C?
Definition:

Dimension change: Only the width is doubled while length remains the same

Area change: When only one dimension changes, area changes by the same factor

Find Dimensions
C: 10×4, D: 10×8
Calculate Areas
C: 40, D: 80
Compare
80÷40=2
Step 1: Identify dimensions of Rectangle C

Length = 10 units, Width = 4 units

Step 2: Find dimensions of Rectangle D

Length = 10 units (same as C), Width = 2 × 4 = 8 units

Step 3: Calculate area of Rectangle C

Area of C = 10 × 4 = 40 square units

Step 4: Calculate area of Rectangle D

Area of D = 10 × 8 = 80 square units

Step 5: Find the multiplicative relationship

Ratio = Area of D ÷ Area of C = 80 ÷ 40 = 2

Step 6: State the comparison

The area of Rectangle D is 2 times greater than the area of Rectangle C

Rectangle D area is 2 times greater
Final answer:

The area of Rectangle D is 2 times greater than the area of Rectangle C.

Applied rules:

• Partial Scaling Rule: When only one dimension changes by factor n, area changes by factor n

• Area Formula: Area of rectangle = length × width

• Multiplicative Comparison: Divide larger by smaller to find "times as much"

Tip: When only one dimension changes, area changes by the same factor
5 Combined Perimeter and Area Comparison
Exercise 5
A rectangular garden has a length of 8 feet and a width of 5 feet. Another garden has dimensions that are each 3 times as large as the first garden. How many times greater are the perimeter and area of the second garden compared to the first garden?
Definition:

Combined scaling: Both length and width are scaled by the same factor

Perimeter scaling: Linear dimensions scale by the same factor as the scale factor

Area scaling: Area scales by the square of the scale factor

Find Dimensions
Garden 1: 8×5, Garden 2: 24×15
Calculate Measurements
P₁:26, P₂:78; A₁:40, A₂:360
Compare
P: 78÷26=3, A: 360÷40=9
Step 1: Identify dimensions of Garden 1

Length = 8 feet, Width = 5 feet

Step 2: Find dimensions of Garden 2

Length = 3 × 8 = 24 feet, Width = 3 × 5 = 15 feet

Step 3: Calculate perimeter of Garden 1

Perimeter of Garden 1 = 2(8 + 5) = 2(13) = 26 feet

Step 4: Calculate perimeter of Garden 2

Perimeter of Garden 2 = 2(24 + 15) = 2(39) = 78 feet

Step 5: Calculate area of Garden 1

Area of Garden 1 = 8 × 5 = 40 square feet

Step 6: Calculate area of Garden 2

Area of Garden 2 = 24 × 15 = 360 square feet

Step 7: Find perimeter comparison

Perimeter ratio = 78 ÷ 26 = 3

Step 8: Find area comparison

Area ratio = 360 ÷ 40 = 9

Step 9: State the comparisons

The perimeter of Garden 2 is 3 times greater than Garden 1

The area of Garden 2 is 9 times greater than Garden 1

Perimeter: 3 times greater, Area: 9 times greater
Final answer:

The perimeter of the second garden is 3 times greater than the first garden. The area of the second garden is 9 times greater than the first garden.

Applied rules:

• Linear Scaling Rule: Perimeter scales by the same factor as linear dimensions

• Area Scaling Rule: Area scales by the square of the linear scale factor

• Perimeter Formula: Perimeter of rectangle = 2(length + width)

• Area Formula: Area of rectangle = length × width

Tip: When all dimensions scale by factor n, perimeter scales by n and area scales by n²
Perimeter and Area with Multiplicative Comparisons Fundamentals
A = l × w, P = 2(l+w)
Rectangle Formulas
Key definitions:

Perimeter: The total length of the boundary of a shape

Area: The amount of space enclosed by a shape

Multiplicative comparison: A comparison that shows how many times larger one quantity is than another

Scale factor: The number by which a dimension is multiplied to create a similar figure

Complete methodology:
  1. Identify the shapes: Determine the type of shapes and their dimensions
  2. Calculate measurements: Compute perimeter and area for each shape
  3. Find relationships: Determine how the dimensions relate to each other
  4. Apply scaling rules: Use knowledge of how perimeter and area scale
  5. Make comparisons: Divide larger measurements by smaller ones
  6. State results: Express comparisons as "times as much" or "times greater"
Tip 1: Draw diagrams to visualize the shapes and their relationships.
Tip 2: Remember: perimeter scales linearly, area scales quadratically with dimension changes.
Tip 3: Always write down the formulas before substituting values.
Tip 4: Check that your answer makes sense by estimating first.
Common errors: Forgetting to square the scale factor for area, confusing perimeter with area, miscalculating formulas.
Memory aid: "Linear dimensions → perimeter by same factor, area by square of factor."
Scaling rules:

• When all linear dimensions scale by factor n: Perimeter scales by n, Area scales by n²

• When only one dimension changes by factor n: Area changes by factor n

• Perimeter of rectangle = 2(length + width)

• Area of rectangle = length × width

• Perimeter of square = 4 × side length

• Area of square = side length²

Perimeter and Area Relationships Visualization
Understanding Scaling Effects
Visual representation of how perimeter and area change with scaling:
Scale factor 1: P=12, A=9
Scale factor 2: P=24, A=36
Scale factor 3: P=36, A=81

Analysis: The chart shows how perimeter and area change when dimensions are scaled.

  • Perimeter increases linearly with scale factor
  • Area increases quadratically with scale factor
  • When dimensions double, area increases by 4 (2²)

Questions & Answers

Question: Why does the area increase by the square of the scale factor but perimeter only increases by the scale factor?

Answer: This happens because perimeter and area measure different things:

  • Perimeter: Measures distance around the shape (linear measurement), so it changes the same way as the sides
  • Area: Measures space inside the shape (square measurement), so it changes by length × width

If you double the sides of a square from 2 to 4: Perimeter: (2×4) = 8 vs (4×4) = 16 (doubled) Area: (2×2) = 4 vs (4×4) = 16 (quadrupled = 2²)

Think of it this way: if you make a room twice as long and twice as wide, you need twice as much paint for the walls (perimeter) but four times as much carpet for the floor (area).

Question: How can I remember the formulas for perimeter and area?

Answer: Here are some memory aids:

  • Perimeter: "Peri" sounds like "around" - it measures around the outside
  • Area: "Are" sounds like "square" - it measures the square units inside
  • Rectangle perimeter: Add all 4 sides: 2×length + 2×width = 2(length + width)
  • Rectangle area: Just multiply length × width

Draw rectangles and count the units around the edge for perimeter and count the squares inside for area. This visual approach helps you remember what each measurement represents.

For squares: Perimeter = 4 × side, Area = side × side = side²

Question: What if only one dimension changes? How does that affect area?

Answer: When only one dimension changes, the area changes by the same factor as that dimension:

  • If length doubles and width stays the same, area doubles
  • If width triples and length stays the same, area triples
  • The area changes by the same factor as the changed dimension

For example, if you have a rectangle 4×3 (area=12) and you change it to 8×3 (keeping width the same but doubling length), the new area is 24, which is double the original area.

This is different from when both dimensions change, where area changes by the product of the scale factors.