Area Models

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Answer Key & Explanations

Area Models · Grade 5 · Worksheet 2

1. 3/4 × 2/3 = ? Answer: 1/2 Solution: To multiply fractions, we multiply the numerators together and the denominators together. Write down the problem. 3/4 × 2/3 Multiply the numerators (the top numbers).
   Full step-by-step solution

   To multiply fractions, we multiply the numerators together and the denominators together. Step 1: Write down the problem. 3/4 × 2/3 Step 2: Multiply the numerators (the top numbers). 3 × 2 = 6 Step 3: Multiply the denominators (the bottom numbers). 4 × 3 = 12 Step 4: Write the new fraction. This gives us 6/12 Step 5: Simplify the fraction to its lowest terms. We find the greatest common factor of 6 and 12. The number 6 divides evenly into both 6 and 12. 6 ÷ 6 = 1 12 ÷ 6 = 2 Step 6: Write the simplified fraction. This gives us 1/2 Therefore, the final answer is 1/2.

2. 3/5 × 7/8 = ? Answer: 21/40 Solution: Multiply the numerators: 3 × 7 = 21 Multiply the denominators: 5 × 8 = 40 Write the resulting fraction: 21/40 Check if the fraction can be simplified.
   Full step-by-step solution

   Step 1: Multiply the numerators: 3 × 7 = 21 Step 2: Multiply the denominators: 5 × 8 = 40 Step 3: Write the resulting fraction: 21/40 Step 4: Check if the fraction can be simplified. The greatest common factor of 21 and 40 is 1, so the fraction is already in simplest form. The answer is 21/40.

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 area of the garden in square meters? Answer: 3/10 Solution: To find the area of a rectangle, we multiply the length by the width. The garden's length is 3/4 meter and its width is 2/5 meter. Write the multiplication expression for the area.
   Full step-by-step solution

   To find the area of a rectangle, we multiply the length by the width. The garden's length is 3/4 meter and its width is 2/5 meter. Step 1: Write the multiplication expression for the area. Area = (3/4) × (2/5) Step 2: Multiply the numerators together. 3 × 2 = 6 Step 3: Multiply the denominators together. 4 × 5 = 20 Step 4: Write the result as a fraction. Area = 6/20 Step 5: Simplify the fraction by dividing the numerator and denominator by their greatest common factor, which is 2. 6 ÷ 2 = 3 20 ÷ 2 = 10 So, 6/20 = 3/10 Step 6: State the final answer. The area of the garden is 3/10 square meter.

4. A rectangular solar panel is divided into a grid to show its energy collection sections. The panel is 3/5 of a meter wide and 4/7 of a meter long. Using an area model, what is the total area of the solar panel in square meters? Answer: 12/35 Solution: To find the area of a rectangle, multiply the width by the length. The width is 3/5 meter and the length is 4/7 meter.
   Full step-by-step solution

   Step 1: To find the area of a rectangle, multiply the width by the length. Step 2: The width is 3/5 meter and the length is 4/7 meter. Step 3: Multiply the fractions: (3/5) × (4/7) Step 4: Multiply the numerators: 3 × 4 = 12 Step 5: Multiply the denominators: 5 × 7 = 35 Step 6: The area is 12/35 square meter. The answer is 12/35.

5. A rectangular garden measures 3/4 of a meter in width and 2/3 of a meter in length. Draw an area model to represent the garden. What is the area of the garden in square meters? Answer: 1/2 Solution: We have a rectangular garden. Its width is 3/4 meter and its length is 2/3 meter. We need to find the area.
   Full step-by-step solution

   Step 1: Understand the problem. We have a rectangular garden. Its width is 3/4 meter and its length is 2/3 meter. We need to find the area. Step 2: Recall the area formula for a rectangle. The area of a rectangle is found by multiplying its length by its width. So, Area = length × width. Step 3: Substitute the given values into the formula. Length = 2/3 meter Width = 3/4 meter Area = (2/3) × (3/4) Step 4: Multiply the fractions. To multiply fractions, multiply the numerators together and the denominators together. Numerators: 2 × 3 = 6 Denominators: 3 × 4 = 12 So, Area = 6/12 Step 5: Simplify the fraction. Find the greatest common factor (GCF) of 6 and 12. The GCF is 6. Divide both the numerator and the denominator by 6: 6 ÷ 6 = 1 12 ÷ 6 = 2 So, 6/12 simplifies to 1/2. Step 6: State the final answer. The area of the garden is 1/2 square meter. Explanation of the area model: Imagine a rectangle. Its width is divided into 4 equal parts, and we shade 3 of them to represent 3/4. Its length is divided into 3 equal parts, and we shade 2 of them to represent 2/3. The overlapping shaded region forms a grid of smaller rectangles. There are 4 × 3 = 12 small rectangles in the whole unit square. The overlapping area covers 2 × 3 = 6 of these small rectangles. So the area is 6/12 of the unit square, which simplifies to 1/2.

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

   To find the area of a rectangle, we multiply its width by its length. The width is 3/4 meter. The length is 2/5 meter. So, the area is: (3/4) * (2/5) When multiplying fractions, we multiply the numerators (top numbers) together and the denominators (bottom numbers) together. Step 1: Multiply the numerators. 3 * 2 = 6 Step 2: Multiply the denominators. 4 * 5 = 20 This gives us the fraction 6/20. Step 3: Simplify the fraction. Both 6 and 20 can be divided by 2. 6 divided by 2 is 3. 20 divided by 2 is 10. So, 6/20 simplifies to 3/10. Therefore, the total area of the garden is 3/10 square meters.

7. Liam is baking cookies for a school fundraiser. He needs to make 3/4 of a batch of chocolate chip cookies and 2/3 of a batch of oatmeal raisin cookies. Using an area model, what fraction of a full single batch of cookies will Liam bake in total? Answer: 17/12 Solution: - 3/4 of a batch of chocolate chip cookies - 2/3 of a batch of oatmeal raisin cookies We want the total fraction of a full single batch that he bakes. 3/4 + 2/3 The denominators are 4 and 3.
   Full step-by-step solution

   Let's solve step-by-step. --- **Step 1: Understand the problem** Liam is making: - 3/4 of a batch of chocolate chip cookies - 2/3 of a batch of oatmeal raisin cookies We want the **total** fraction of a **full single batch** that he bakes. So we need to add: 3/4 + 2/3 --- **Step 2: Find a common denominator** The denominators are 4 and 3. The least common denominator is 12. --- **Step 3: Convert each fraction to have denominator 12** For 3/4: Multiply numerator and denominator by 3: 3/4 = (3 × 3)/(4 × 3) = 9/12 For 2/3: Multiply numerator and denominator by 4: 2/3 = (2 × 4)/(3 × 4) = 8/12 --- **Step 4: Add the fractions** 9/12 + 8/12 = (9 + 8)/12 = 17/12 --- **Step 5: Interpret the result** 17/12 means 1 full batch and 5/12 of another batch, but the problem just asks for the total fraction of a full single batch, so 17/12 is the answer. --- **Final Answer:** 17/12

8. A farmer plants a rectangular field that is 7/8 of a kilometer long and 5/6 of a kilometer wide. What fraction of a square kilometer is the area of the field? Answer: 35/48 Solution: The area of a rectangle is found by multiplying length × width Multiply the fractions: (7/8) × (5/6) Multiply the numerators: 7 × 5 = 35 Multiply the denominators: 8 × 6 = 48 The area is 35/48 of a square kilometer Check if the fraction can be simplified: 35 and 48 have no common factors other…
   Full step-by-step solution

   Step 1: The area of a rectangle is found by multiplying length × width Step 2: Multiply the fractions: (7/8) × (5/6) Step 3: Multiply the numerators: 7 × 5 = 35 Step 4: Multiply the denominators: 8 × 6 = 48 Step 5: The area is 35/48 of a square kilometer Step 6: Check if the fraction can be simplified: 35 and 48 have no common factors other than 1, so 35/48 is already in simplest form The answer is 35/48.

9. 7/8 × 4/5 = ? Answer: 7/10 Solution: Multiply the numerators: 7 × 4 = 28 Multiply the denominators: 8 × 5 = 40 Write the resulting fraction: 28/40 Simplify by dividing numerator and denominator by their greatest common factor (4): 28 ÷ 4 = 7, 40 ÷ 4 = 10 The simplified fraction is 7/10 The answer is 7/10.
   Full step-by-step solution

   Step 1: Multiply the numerators: 7 × 4 = 28 Step 2: Multiply the denominators: 8 × 5 = 40 Step 3: Write the resulting fraction: 28/40 Step 4: Simplify by dividing numerator and denominator by their greatest common factor (4): 28 ÷ 4 = 7, 40 ÷ 4 = 10 Step 5: The simplified fraction is 7/10 The answer is 7/10.