Name: [Your Name Here]
Date: [Today's Date Here]
---
To find the product of 523 and 416, we will use the area model to find the partial products and then use the standard algorithm for multiplication.
Step 1: Using the Area Model
First, we decompose each number into parts that are easier to multiply and then compute the partial products.
[tex]\[ 523 = 500 + 20 + 3 \][/tex]
[tex]\[ 416 = 400 + 10 + 6 \][/tex]
We set up an area model that multiplies each part of 523 by each part of 416:
[tex]\[
\begin{array}{|c|c|c|c|}
\hline
& 400 & 10 & 6 \\
\hline
500 & 500 \times 400 = 200000 & 500 \times 10 = 5000 & 500 \times 6 = 3000 \\
20 & 20 \times 400 = 8000 & 20 \times 10 = 200 & 20 \times 6 = 120 \\
3 & 3 \times 400 = 1200 & 3 \times 10 = 30 & 3 \times 6 = 18 \\
\hline
\end{array}
\][/tex]
Now, let's list the partial products calculated:
1. [tex]\( 500 \times 400 = 200000 \)[/tex]
2. [tex]\( 500 \times 10 = 5000 \)[/tex]
3. [tex]\( 500 \times 6 = 3000 \)[/tex]
4. [tex]\( 20 \times 400 = 8000 \)[/tex]
5. [tex]\( 20 \times 10 = 200 \)[/tex]
6. [tex]\( 20 \times 6 = 120 \)[/tex]
7. [tex]\( 3 \times 400 = 1200 \)[/tex]
8. [tex]\( 3 \times 10 = 30 \)[/tex]
9. [tex]\( 3 \times 6 = 18 \)[/tex]
We now sum all the partial products to find the total product using the area model:
[tex]\[
200000 + 5000 + 3000 + 8000 + 200 + 120 + 1200 + 30 + 18 = 217568
\][/tex]
Step 2: Using the Standard Algorithm
Next, we will use the standard multiplication algorithm to verify our result:
1. Multiply 523 by 6:
[tex]\[ 523 \times 6 = 3138 \][/tex]
2. Multiply 523 by 10 (shift one place to the left):
[tex]\[ 523 \times 10 = 5230 \][/tex]
3. Multiply 523 by 400 (shift two places to the left):
[tex]\[ 523 \times 400 = 209200 \][/tex]
4. Sum the results of each individual multiplication:
[tex]\[
3138 + 5230 + 209200 = 217568
\][/tex]
So, the result of [tex]\( 523 \times 416 \)[/tex] using the standard algorithm is also 217568.
Conclusion:
The product obtained from both the area model and the standard algorithm is the same, verifying that our calculations are correct.
[tex]\[ \boxed{217568} \][/tex]
