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Answer :
Sure! Let's go through the solution step-by-step.
### Part a: Determine the numbers for the letters
We have a multiplication problem involving the number 143 and 852, and we want to identify the missing digits represented by letters (A, C, D) in a column multiplication setup.
Here's how the breakdown works:
- 143 × 2 gives us 286, which is the first partial product.
- 143 × 5 gives us 715, and when you shift this one position to the left (because it's actually 143 × 50), it becomes 7150.
- 143 × 8 gives us 1144, and when you shift this two positions to the left (because it's 143 × 800), it becomes 114400.
By following this breakdown:
- D should be replaced with 2.
- A should be replaced with 5.
- C should be replaced with 8.
### Part b: Explain how the column method uses the distributive law
The column method of multiplication uses the distributive property of multiplication to break the numbers into parts that are easier to work with.
Here’s a brief explanation of how this works:
- The operation 143 × 852 is decomposed using the distributive law as follows:
- 143 × (800 + 50 + 2)
Applying the distributive law:
- 143 × 852 = (143 × 800) + (143 × 50) + (143 × 2)
In the column method:
1. 143 × 2 = 286, which corresponds to the term (143 × 2).
2. 143 × 50 = 7150 (calculated as 143 × 5, shifted one position for the tens place), corresponding to (143 × 50).
3. 143 × 800 = 114400 (calculated as 143 × 8, shifted two positions for the hundreds place), corresponding to (143 × 800).
Finally, you add up all these partial products:
286 + 7150 + 114400 = 121836 which is the final product.
This approach makes use of the distributive property (a × (b + c) = (a × b) + (a × c)) to simplify the multiplication into more manageable parts.
### Part a: Determine the numbers for the letters
We have a multiplication problem involving the number 143 and 852, and we want to identify the missing digits represented by letters (A, C, D) in a column multiplication setup.
Here's how the breakdown works:
- 143 × 2 gives us 286, which is the first partial product.
- 143 × 5 gives us 715, and when you shift this one position to the left (because it's actually 143 × 50), it becomes 7150.
- 143 × 8 gives us 1144, and when you shift this two positions to the left (because it's 143 × 800), it becomes 114400.
By following this breakdown:
- D should be replaced with 2.
- A should be replaced with 5.
- C should be replaced with 8.
### Part b: Explain how the column method uses the distributive law
The column method of multiplication uses the distributive property of multiplication to break the numbers into parts that are easier to work with.
Here’s a brief explanation of how this works:
- The operation 143 × 852 is decomposed using the distributive law as follows:
- 143 × (800 + 50 + 2)
Applying the distributive law:
- 143 × 852 = (143 × 800) + (143 × 50) + (143 × 2)
In the column method:
1. 143 × 2 = 286, which corresponds to the term (143 × 2).
2. 143 × 50 = 7150 (calculated as 143 × 5, shifted one position for the tens place), corresponding to (143 × 50).
3. 143 × 800 = 114400 (calculated as 143 × 8, shifted two positions for the hundreds place), corresponding to (143 × 800).
Finally, you add up all these partial products:
286 + 7150 + 114400 = 121836 which is the final product.
This approach makes use of the distributive property (a × (b + c) = (a × b) + (a × c)) to simplify the multiplication into more manageable parts.
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