Number Patterns

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Number Patterns · Grade 4 · Worksheet 3

1. Emma is arranging numbers in a pattern. The pattern follows a rule where each number is multiplied by the same odd number to get the next term. The first four terms of her pattern are: 3, 9, 27, 81. If she continues the pattern, what will be the 7th term in the sequence? Answer: 2187 Solution: Identify the rule. 3 times 3 = 9, 9 times 3 = 27, 27 times 3 = 81. The rule is multiply by 3 each time.
   Full step-by-step solution

   Step 1: Identify the rule. 3 times 3 = 9, 9 times 3 = 27, 27 times 3 = 81. The rule is multiply by 3 each time. Step 2: The 4th term is 81. To find the 5th term: 81 times 3 = 243. Step 3: The 5th term is 243. To find the 6th term: 243 times 3 = 729. Step 4: The 6th term is 729. To find the 7th term: 729 times 3 = 2187. The 7th term is 2187.

2. Aroha is drawing a number pattern on a grid. She starts at 108 and follows the rule: subtract 12 each time. What is the 5th number in her pattern? Answer: 60 Solution: Start at 108. Subtract 12 to get the 2nd number: 108 - 12 = 96. Subtract 12 to get the 3rd number: 96 - 12 = 84.
   Full step-by-step solution

   Step 1: Start at 108. Step 2: Subtract 12 to get the 2nd number: 108 - 12 = 96. Step 3: Subtract 12 to get the 3rd number: 96 - 12 = 84. Step 4: Subtract 12 to get the 4th number: 84 - 12 = 72. Step 5: Subtract 12 to get the 5th number: 72 - 12 = 60. The 5th number in the pattern is 60.

3. Liam is building a rectangular garden in his backyard. The length of the garden is 12 feet, and the width is 8 feet. If he wants to put a fence around the entire garden, how many feet of fencing does he need? Answer: 40 Solution: To find the total length of fencing needed, we need to calculate the perimeter of the rectangular garden. Recall the formula for the perimeter of a rectangle. The perimeter P is given by: P = 2 * (length + width) Identify the given measurements from the problem.
   Full step-by-step solution

   To find the total length of fencing needed, we need to calculate the perimeter of the rectangular garden. Step 1: Recall the formula for the perimeter of a rectangle. The perimeter P is given by: P = 2 * (length + width) Step 2: Identify the given measurements from the problem. Length = 12 feet Width = 8 feet Step 3: Substitute the given values into the formula. P = 2 * (12 + 8) Step 4: Perform the calculation inside the parentheses first. 12 + 8 = 20 Step 5: Multiply the result by 2. P = 2 * 20 = 40 Step 6: State the final answer. Liam needs 40 feet of fencing.

4. (48 + 32) ÷ 8 = ? Answer: 10 Solution: Calculate inside the parentheses: 48 + 32 = 80 Divide the result by 8: 80 ÷ 8 = 10 The answer is 10.
   Full step-by-step solution

   Step 1: Calculate inside the parentheses: 48 + 32 = 80 Step 2: Divide the result by 8: 80 ÷ 8 = 10 The answer is 10.

5. A rectangular swimming pool is drawn on a grid where each square unit represents 1 square meter. The pool has a length of 15 meters and a width of 10 meters. A concrete walkway is built around the entire outside of the pool. The walkway has a uniform width of 2 meters. What is the total area of the pool and the walkway combined? Answer: 266 Solution: The pool has length = 15 m and width = 10 m. The walkway adds 2 meters to both sides of the length, so total length = 15 + 2 + 2 = 19 m.
   Full step-by-step solution

   Step 1: The pool has length = 15 m and width = 10 m. Step 2: The walkway adds 2 meters to both sides of the length, so total length = 15 + 2 + 2 = 19 m. Step 3: The walkway adds 2 meters to both sides of the width, so total width = 10 + 2 + 2 = 14 m. Step 4: Total area = length × width = 19 × 14. Step 5: Calculate 19 × 14 = 19 × (10 + 4) = (19 × 10) + (19 × 4) = 190 + 76 = 266. The total area is 266 square meters.

6. Hana is drawing a number pattern on a grid. She starts at 104 and follows the rule: subtract 12 each time. What is the 6th number in her pattern? Answer: 44 Solution: The first number is 104. Subtract 12 each time, so the second number is 104 - 12 = 92. The third number is 92 - 12 = 80.
   Full step-by-step solution

   Step 1: The first number is 104. Step 2: Subtract 12 each time, so the second number is 104 - 12 = 92. Step 3: The third number is 92 - 12 = 80. Step 4: The fourth number is 80 - 12 = 68. Step 5: The fifth number is 68 - 12 = 56. Step 6: The sixth number is 56 - 12 = 44. The answer is 44.

7. Pattern: 450, 375, 300, 225, __ Rule? Answer: 150 Solution: Look at the pattern: 450, 375, 300, 225 Find the relationship between consecutive terms: 450 - 375 = 75, 375 - 300 = 75, 300 - 225 = 75 The rule is 'subtract 75' Apply the rule to the last term: 225 - 75 = 150 The next term in the pattern is 150
   Full step-by-step solution

   Step 1: Look at the pattern: 450, 375, 300, 225 Step 2: Find the relationship between consecutive terms: 450 - 375 = 75, 375 - 300 = 75, 300 - 225 = 75 Step 3: The rule is 'subtract 75' Step 4: Apply the rule to the last term: 225 - 75 = 150 Step 5: The next term in the pattern is 150

8. A rectangular garden is drawn with a length of 12 meters and a width of 8 meters. A path is built around the outside of the garden that is 2 meters wide. What is the total area of the garden and the path combined? Answer: 192 Solution: - Length = 12 m - Width = 8 m A path 2 m wide is built around the outside of the garden. We need the total area of garden + path. The path goes around all four sides, so it adds 2 m to the top, bottom, left, and right.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a rectangular garden: - Length = 12 m - Width = 8 m A path 2 m wide is built around the outside of the garden. We need the total area of garden + path. --- **Step 2: Visualize the path** The path goes around all four sides, so it adds 2 m to the top, bottom, left, and right. That means: - New length = original length + 2 m (left) + 2 m (right) = 12 + 2 + 2 = 16 m - New width = original width + 2 m (top) + 2 m (bottom) = 8 + 2 + 2 = 12 m --- **Step 3: Calculate the total area (garden + path)** Total area = new length × new width = 16 × 12 --- **Step 4: Perform the multiplication** 16 × 12 = 16 × (10 + 2) = 160 + 32 = 192 --- **Step 5: Final answer** The total area of the garden and the path combined is 192 square meters. --- **Answer:** 192

9. Pattern: 128, 192, 288, 432, __ Rule? Answer: 648 Solution: Look at the pattern: 128, 192, 288, 432 Find the relationship between consecutive terms: 192 ÷ 128 = 1.5, 288 ÷ 192 = 1.5, 432 ÷ 288 = 1.5 The rule is 'multiply by 1.5' Apply the rule to the last term: 432 × 1.5 = 648 The next term in the pattern is 648
   Full step-by-step solution

   Step 1: Look at the pattern: 128, 192, 288, 432 Step 2: Find the relationship between consecutive terms: 192 ÷ 128 = 1.5, 288 ÷ 192 = 1.5, 432 ÷ 288 = 1.5 Step 3: The rule is 'multiply by 1.5' Step 4: Apply the rule to the last term: 432 × 1.5 = 648 Step 5: The next term in the pattern is 648