Area Models

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Answer Key & Explanations

Area Models · Grade 5 · Worksheet 3

1. 2/3 × 3/4 = ? Answer: 1/2 Solution: Write down the multiplication problem. We are multiplying two fractions: 2/3 × 3/4. Multiply the numerators (the top numbers).
   Full step-by-step solution

   Step 1: Write down the multiplication problem. We are multiplying two fractions: 2/3 × 3/4. Step 2: Multiply the numerators (the top numbers). Numerators: 2 × 3 = 6. So, the new numerator is 6. Step 3: Multiply the denominators (the bottom numbers). Denominators: 3 × 4 = 12. So, the new denominator is 12. This gives us the new fraction 6/12. Step 4: Simplify the fraction. Find the greatest common factor (GCF) of 6 and 12. The factors of 6 are: 1, 2, 3, 6. The factors of 12 are: 1, 2, 3, 4, 6, 12. The greatest common factor is 6. Step 5: Divide both the numerator and denominator by the GCF. 6 ÷ 6 = 1 12 ÷ 6 = 2 This gives us the simplified fraction 1/2. Final Answer: 1/2

2. Noah is painting a mural on a rectangular wall that is 3/4 meters tall and 2/5 meters wide. He wants to calculate the area of the wall he needs to paint. What is the area of the wall in square meters? Answer: 3/10 Solution: To find the area of a rectangle, you multiply its length by its width. For instance, the area of a 1/2 meter by 2/3 meter rectangle would be (1/2) * (2/3) = 2/6, which simplifies to 1/3 square meter.
   Full step-by-step solution

   To find the area of a rectangle, you multiply its length by its width. When multiplying fractions, you multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator. For instance, the area of a 1/2 meter by 2/3 meter rectangle would be (1/2) * (2/3) = 2/6, which simplifies to 1/3 square meter.

3. A rectangular garden is divided into a grid to show planting sections. The garden is 3/4 meter long and 2/5 meter wide. Using an area model, what is the total area of the garden in square meters? Answer: 3/10 Solution: We have a rectangular garden with length = 3/4 meter and width = 2/5 meter. We need to find the area using an area model.
   Full step-by-step solution

   Step 1: Understand the problem We have a rectangular garden with length = 3/4 meter and width = 2/5 meter. We need to find the area using an area model. Step 2: Recall the area formula for a rectangle Area = length × width Step 3: Write the multiplication Area = (3/4) × (2/5) Step 4: Multiply the numerators Numerator: 3 × 2 = 6 Step 5: Multiply the denominators Denominator: 4 × 5 = 20 Step 6: Write the resulting fraction Area = 6/20 Step 7: Simplify the fraction Find the greatest common factor (GCF) of 6 and 20. The GCF is 2. Divide numerator and denominator by 2: 6 ÷ 2 = 3 20 ÷ 2 = 10 So, 6/20 = 3/10 Step 8: State the final answer The total area of the garden is 3/10 square meter.

4. Emma is making a quilt for her grandmother. She needs 2/3 yard of blue fabric and 3/5 yard of yellow fabric for each square. Using an area model, what fraction of a square yard of fabric does Emma need for one quilt square? Answer: 2/5 Solution: To find the total fabric needed, multiply 2/3 by 3/5. Draw a rectangle and divide it vertically into 3 equal parts. Shade 2 of these parts to represent 2/3.
   Full step-by-step solution

   Step 1: To find the total fabric needed, multiply 2/3 by 3/5. Step 2: Draw a rectangle and divide it vertically into 3 equal parts. Shade 2 of these parts to represent 2/3. Step 3: Divide the same rectangle horizontally into 5 equal parts. Shade 3 of these parts to represent 3/5. Step 4: The overlapping shaded area represents the product. Count the total number of small rectangles: 3 × 5 = 15. Step 5: Count the overlapping shaded rectangles: 2 × 3 = 6. Step 6: The fraction is 6/15, which simplifies to 2/5. The answer is 2/5.

5. 2/3 × 5/8 = ? Answer: 5/12 Solution: Multiply the numerators: 2 × 5 = 10 Multiply the denominators: 3 × 8 = 24 Write the resulting fraction: 10/24 Simplify the fraction by dividing numerator and denominator by their greatest common factor (2): 10÷2 = 5, 24÷2 = 12 The simplified fraction is 5/12 The answer is 5/12.
   Full step-by-step solution

   Step 1: Multiply the numerators: 2 × 5 = 10 Step 2: Multiply the denominators: 3 × 8 = 24 Step 3: Write the resulting fraction: 10/24 Step 4: Simplify the fraction by dividing numerator and denominator by their greatest common factor (2): 10÷2 = 5, 24÷2 = 12 Step 5: The simplified fraction is 5/12 The answer is 5/12.

6. A rectangular garden measures 3/4 of a kilometer in length and 5/6 of a kilometer in width. What fraction of a square kilometer is the area of the garden? Answer: 5/8 Solution: We have a rectangular garden. Length = 3/4 km Width = 5/6 km We need to find the area in square kilometers and express it as a fraction.
   Full step-by-step solution

   Step 1: Understand the problem We have a rectangular garden. Length = 3/4 km Width = 5/6 km We need to find the area in square kilometers and express it as a fraction. Step 2: Recall the formula for area of a rectangle Area = Length × Width Step 3: Substitute the given values Area = (3/4) × (5/6) Step 4: Multiply the fractions To multiply fractions, multiply the numerators together and the denominators together. Numerator: 3 × 5 = 15 Denominator: 4 × 6 = 24 So Area = 15/24 Step 5: Simplify the fraction Find the greatest common divisor (GCD) of 15 and 24. Factors of 15: 1, 3, 5, 15 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 GCD is 3. Divide numerator and denominator by 3: 15 ÷ 3 = 5 24 ÷ 3 = 8 So Area = 5/8 Step 6: Conclusion The area of the garden is 5/8 of a square kilometer. Final answer: 5/8

7. A rectangular garden is divided into 4 equal sections. Each section is planted with different vegetables. If the garden is 12 meters long and 8 meters wide, what is the area of one vegetable section? Answer: 24 Solution: Find the total area of the garden. The garden is a rectangle with length 12 meters and width 8 meters. Area of rectangle = length × width Total area = 12 × 8 = 96 square meters.
   Full step-by-step solution

   Step 1: Find the total area of the garden. The garden is a rectangle with length 12 meters and width 8 meters. Area of rectangle = length × width Total area = 12 × 8 = 96 square meters. Step 2: Understand the division of the garden. The garden is divided into 4 equal sections. So, each section has the same area. Step 3: Find the area of one section. Divide the total area by the number of sections. Area of one section = Total area ÷ 4 Area of one section = 96 ÷ 4 = 24 square meters. Final answer: The area of one vegetable section is 24 square meters.