Number Patterns

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Answer Key & Explanations

Number Patterns · Grade 4 · Worksheet 1

1. Liam is organizing his trading cards into albums. Each album page holds 9 cards. If Liam has 135 cards total, how many full pages will he be able to fill? Answer: 15 Solution: - Each album page holds 9 cards. - Liam has 135 cards total. - We want the number of full pages he can fill.
   Full step-by-step solution

   We are told: - Each album page holds 9 cards. - Liam has 135 cards total. - We want the number of **full** pages he can fill. --- **Step 1: Understand the operation needed** To find how many full pages of 9 cards each can be made from 135 cards, we divide the total number of cards by the number of cards per page: 135 ÷ 9 --- **Step 2: Perform the division** We can think of this as: How many times does 9 go into 135? 9 × 10 = 90 135 − 90 = 45 9 × 5 = 45 45 − 45 = 0 So, 10 + 5 = 15 times. Alternatively, directly: 9 × 15 = 135, so 135 ÷ 9 = 15. --- **Step 3: Interpret the result** Since 135 ÷ 9 = 15 exactly, with no remainder, Liam will fill exactly 15 full pages. --- **Final answer:** 15

2. Noah is building a fence around his rectangular vegetable garden. The garden is 15 feet long and 12 feet wide. If fencing costs $8 per foot, how much will Noah spend on fencing for the entire garden? Answer: 432 Solution: Perimeter = 2 × (length + width) Perimeter = 2 × (15 + 12) Perimeter = 2 × 27 Perimeter = 54 feet Cost = perimeter × cost per foot Cost = 54 × 8 Cost = 432 Noah will spend $432 on fencing for his garden.
   Full step-by-step solution

   Step 1: Find the perimeter of the rectangular garden Perimeter = 2 × (length + width) Perimeter = 2 × (15 + 12) Perimeter = 2 × 27 Perimeter = 54 feet Step 2: Calculate the total cost of fencing Cost = perimeter × cost per foot Cost = 54 × 8 Cost = 432 Noah will spend $432 on fencing for his garden.

3. Emma is drawing a number pattern on a grid. She starts at 120 and uses the rule 'subtract 15' to create a sequence. What are the next three numbers in her pattern after 120? Answer: 105, 90, 75 Solution: Start with the first number: 120. Subtract 15 from 120: 120 - 15 = 105. This is the second number.
   Full step-by-step solution

   Step 1: Start with the first number: 120. Step 2: Subtract 15 from 120: 120 - 15 = 105. This is the second number. Step 3: Subtract 15 from 105: 105 - 15 = 90. This is the third number. Step 4: Subtract 15 from 90: 90 - 15 = 75. This is the fourth number. The next three numbers after 120 are 105, 90, and 75.

4. Kaia is arranging tiles in a pattern on a grid. The first row has 1 tile, the second row has 3 tiles, the third row has 5 tiles, and the fourth row has 7 tiles. If the pattern continues in the same way, how many tiles will be in the ninth row? Answer: 17 Solution: List the number of tiles in each row: Row 1: 1, Row 2: 3, Row 3: 5, Row 4: 7. Find the rule. Subtract consecutive terms: 3 - 1 = 2, 5 - 3 = 2, 7 - 5 = 2.
   Full step-by-step solution

   Step 1: List the number of tiles in each row: Row 1: 1, Row 2: 3, Row 3: 5, Row 4: 7. Step 2: Find the rule. Subtract consecutive terms: 3 - 1 = 2, 5 - 3 = 2, 7 - 5 = 2. The rule is add 2 each time. Step 3: Continue the pattern from row 4 to row 9: Row 5: 7 + 2 = 9 Row 6: 9 + 2 = 11 Row 7: 11 + 2 = 13 Row 8: 13 + 2 = 15 Row 9: 15 + 2 = 17 The ninth row has 17 tiles.

5. 256 ÷ 8 = ? Answer: 32 Solution: We are dividing 256 into 8 equal groups. We want to find how many are in each group. Break down the number 256.
   Full step-by-step solution

   Let's solve 256 ÷ 8 step by step. Step 1: Understand the problem. We are dividing 256 into 8 equal groups. We want to find how many are in each group. Step 2: Break down the number 256. We can think of 256 as 240 + 16. This is helpful because both 240 and 16 are easy to divide by 8. Step 3: Divide 240 by 8. We know that 24 ÷ 8 = 3. Therefore, 240 ÷ 8 = 30. So, 240 split into 8 groups gives 30 in each group. Step 4: Divide 16 by 8. We know that 16 ÷ 8 = 2. So, 16 split into 8 groups gives 2 in each group. Step 5: Add the two results together. From the 240 part we got 30, and from the 16 part we got 2. 30 + 2 = 32. Step 6: Conclusion. Therefore, 256 ÷ 8 = 32. We can check our work: 8 × 32 = 256, which is correct.

6. Pattern: 625, 500, 400, 320, __ Rule? Answer: 256 Solution: Look at the pattern: 625, 500, 400, 320 Find the relationship between consecutive terms: 500 ÷ 625 = 0.8, 400 ÷ 500 = 0.8, 320 ÷ 400 = 0.8 The rule is 'multiply by 0.8' Apply the rule to the last term: 320 × 0.8 = 256 The next term in the pattern is 256
   Full step-by-step solution

   Step 1: Look at the pattern: 625, 500, 400, 320 Step 2: Find the relationship between consecutive terms: 500 ÷ 625 = 0.8, 400 ÷ 500 = 0.8, 320 ÷ 400 = 0.8 Step 3: The rule is 'multiply by 0.8' Step 4: Apply the rule to the last term: 320 × 0.8 = 256 Step 5: The next term in the pattern is 256

7. 245 × 6 = ? Answer: 1470 Solution: Write the multiplication vertically for clarity. 245 × 6 Multiply the ones place. 6 × 5 = 30 Write down 0 in the ones place, and carry over 3 to the tens place.
   Full step-by-step solution

   Let's multiply 245 by 6 step by step. Step 1: Write the multiplication vertically for clarity. 245 × 6 Step 2: Multiply the ones place. 6 × 5 = 30 Write down 0 in the ones place, and carry over 3 to the tens place. 245 × 6 ----- 0 (carry 3) Step 3: Multiply the tens place. 6 × 4 = 24 Add the carried-over 3: 24 + 3 = 27 Write down 7 in the tens place, and carry over 2 to the hundreds place. 245 × 6 ----- 70 (carry 2) Step 4: Multiply the hundreds place. 6 × 2 = 12 Add the carried-over 2: 12 + 2 = 14 Write down 4 in the hundreds place and 1 in the thousands place. 245 × 6 ----- 1470 Step 5: Combine the result. The final answer is 1470. Therefore, 245 × 6 = 1470.