# Multi-Digit Multiplication: Place Value and Area Models
Section A: Multiple Choice Questions
Q1: What is the value of the digit 5 in the number 4,568? (a) 5 (b) 50 (c) 500 (d) 5,000
Solution:
Ans: (c) Explanation: The digit 5 is in the hundreds place, so its value is \(5 \times 100 = 500\). The place value determines the actual value of the digit in a number.
Ans: (a) Explanation: Using the distributive property, we break 34 into \(30 + 4\), then multiply both parts by 12: \((30 + 4) \times 12 = (30 \times 12) + (4 \times 12)\). This helps make multiplication easier.
Q3: An area model for 23 × 15 is divided into four rectangles. Which of the following is NOT one of the partial products? (a) 20 × 10 (b) 20 × 5 (c) 3 × 10 (d) 23 × 15
Solution:
Ans: (d) Explanation: In an area model, we break the numbers into tens and ones: \(23 = 20 + 3\) and \(15 = 10 + 5\). The four partial products are \(20 \times 10\), \(20 \times 5\), \(3 \times 10\), and \(3 \times 5\). The expression \(23 \times 15\) is the final product, not a partial product.
Q4: Using an area model, what is 40 × 30? (a) 120 (b) 700 (c) 1,200 (d) 12,000
Solution:
Ans: (c) Explanation: To multiply \(40 \times 30\), we can think of it as \(4 \times 3 = 12\), then add the two zeros to get 1,200. This shows how place value affects multiplication.
Q5: What is the sum of the partial products in 26 × 14 when using the area model? (a) 364 (b) 340 (c) 200 + 80 + 60 + 24 (d) 40
Solution:
Ans: (c) Explanation: Breaking down \(26 \times 14\): \(20 \times 10 = 200\), \(20 \times 4 = 80\), \(6 \times 10 = 60\), and \(6 \times 4 = 24\). The sum of these partial products equals the total: \(200 + 80 + 60 + 24 = 364\).
Q6: When multiplying 52 × 36, what is the partial product of 50 × 30? (a) 150 (b) 1,500 (c) 180 (d) 1,800
Solution:
Ans: (b) Explanation: To find \(50 \times 30\), multiply \(5 \times 3 = 15\), then count the zeros (two total), giving 1,500. This is one of the four partial products in the area model for \(52 \times 36\).
Q7: Which shows the correct way to break apart 47 × 23 using place value? (a) (40 + 7) × (20 + 3) (b) (47 + 23) (c) (4 + 7) × (2 + 3) (d) 40 × 20 only
Solution:
Ans: (a) Explanation: Using place value, we break 47 into \(40 + 7\) and 23 into \(20 + 3\). Then we multiply: \((40 + 7) \times (20 + 3)\), which creates four partial products in the area model.
Q8: If one rectangle in an area model for 35 × 42 shows 30 × 40, what is that partial product? (a) 120 (b) 1,200 (c) 70 (d) 12
Solution:
Ans: (b) Explanation: Multiplying \(30 \times 40\): Think \(3 \times 4 = 12\), then add two zeros (one from each number) to get 1,200. This is the largest partial product in the area model for \(35 \times 42\).
Section B: Fill in the Blanks
Q9:In the number 3,456, the digit 4 is in the __________ place.
Solution:
Ans: hundreds Explanation: The place value system shows that moving from right to left: ones, tens, hundreds, thousands. The digit 4 is in the hundreds place, representing 400.
Q10:An area model breaks a multiplication problem into smaller __________ that are easier to solve.
Solution:
Ans: partial products Explanation: The area model strategy divides a rectangle into sections, each representing a partial product. These are then added together to find the total product.
Q11:When you multiply 60 × 50, you can multiply 6 × 5 and then add __________ zeros to the product.
Solution:
Ans: two Explanation: Since \(60 = 6 \times 10\) and \(50 = 5 \times 10\), there are two zeros total. Multiply \(6 \times 5 = 30\), then add two zeros to get 3,000.
Q12:The distributive property states that \(a \times (b + c) = (a \times b) + (a \times __________)\).
Solution:
Ans: c Explanation: The distributive property allows us to multiply a number by a sum by multiplying the number by each addend separately: \(a \times (b + c) = (a \times b) + (a \times c)\).
Q13:To find 28 × 15 using an area model, you need to find __________ partial products.
Solution:
Ans: four Explanation: Breaking \(28 = 20 + 8\) and \(15 = 10 + 5\) creates a 2 × 2 grid in the area model, resulting in four partial products: \(20 \times 10\), \(20 \times 5\), \(8 \times 10\), and \(8 \times 5\).
Q14:In 34 × 12, one partial product is 30 × 10 = __________.
Solution:
Ans: 300 Explanation: Multiplying \(30 \times 10\) gives 300. This is one of the four partial products when using the area model to solve \(34 \times 12\).
Section C: Word Problems
Q15:A school library has 24 shelves. Each shelf holds 18 books. How many books can the library hold in total? Use an area model to solve.
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