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Sagot :
Let's analyze Lori's multiplication of [tex]\( 29 \times 31 \)[/tex] step-by-step to identify her mistake and find the correct solution.
### Correct Multiplication Breakdown:
1. We begin by breaking down the numbers 29 and 31:
[tex]\[ 29 = 20 + 9 \][/tex]
[tex]\[ 31 = 30 + 1 \][/tex]
2. Rewrite the multiplication using the distributive property:
[tex]\[ (20 + 9) \times (30 + 1) \][/tex]
3. Apply the distributive property to expand the expression:
[tex]\[ 29 \times 31 = (20 + 9) \times (30 + 1) = 20 \times 30 + 20 \times 1 + 9 \times 30 + 9 \times 1 \][/tex]
4. Calculate each term separately:
[tex]\[ 20 \times 30 = 600 \][/tex]
[tex]\[ 20 \times 1 = 20 \][/tex]
[tex]\[ 9 \times 30 = 270 \][/tex]
[tex]\[ 9 \times 1 = 9 \][/tex]
5. Sum these results to obtain the final product:
[tex]\[ 600 + 20 + 270 + 9 = 899 \][/tex]
### Lori's Mistake:
Lori arrived at the incorrect result of 699 when multiplying [tex]\( 29 \times 31 \)[/tex]. Let's identify what she could have done wrong.
1. If we compare Lori's result of 699 to our correct result of 899, there's a difference of 200.
2. This difference indicates a possible error in the multiplication or the addition of partial products.
If Lori correctly did [tex]\( 20 \times 30 = 600 \)[/tex] and [tex]\( 9 \times 9 = 81 \)[/tex], but somewhere in the process she might have misplaced a number or miscalculated the sum:
- Lori might have skipped adding [tex]\( 270 \)[/tex] from [tex]\( 9 \times 30 \)[/tex], possibly confusing it with a smaller number or placing the result incorrectly.
### Summary:
- The correct value of [tex]\( 29 \times 31 \)[/tex] is [tex]\( 899 \)[/tex].
- Lori's mistake resulted in an incorrect product of [tex]\( 699 \)[/tex], which has a difference of 200 from the correct result.
Identifying exactly that she might have placed a zero in the ones column before multiplying [tex]\( 3 \times 9 \)[/tex], leading her astray in subsequent steps, is another possible point of error. This incorrect step could cause an accrued difference of 200 given [tex]\( 3 \times 9 = 270 \)[/tex].
### Final Answer:
- The correct value for [tex]\( 29 \times 31 \)[/tex] is [tex]\( 899 \)[/tex].
- Lori got [tex]\( 699 \)[/tex] likely due to misplacing digits or miscalculating during the expansion and summation stages.
- The precise difference due to Lori's mistake is [tex]\( 200 \)[/tex].
### Correct Multiplication Breakdown:
1. We begin by breaking down the numbers 29 and 31:
[tex]\[ 29 = 20 + 9 \][/tex]
[tex]\[ 31 = 30 + 1 \][/tex]
2. Rewrite the multiplication using the distributive property:
[tex]\[ (20 + 9) \times (30 + 1) \][/tex]
3. Apply the distributive property to expand the expression:
[tex]\[ 29 \times 31 = (20 + 9) \times (30 + 1) = 20 \times 30 + 20 \times 1 + 9 \times 30 + 9 \times 1 \][/tex]
4. Calculate each term separately:
[tex]\[ 20 \times 30 = 600 \][/tex]
[tex]\[ 20 \times 1 = 20 \][/tex]
[tex]\[ 9 \times 30 = 270 \][/tex]
[tex]\[ 9 \times 1 = 9 \][/tex]
5. Sum these results to obtain the final product:
[tex]\[ 600 + 20 + 270 + 9 = 899 \][/tex]
### Lori's Mistake:
Lori arrived at the incorrect result of 699 when multiplying [tex]\( 29 \times 31 \)[/tex]. Let's identify what she could have done wrong.
1. If we compare Lori's result of 699 to our correct result of 899, there's a difference of 200.
2. This difference indicates a possible error in the multiplication or the addition of partial products.
If Lori correctly did [tex]\( 20 \times 30 = 600 \)[/tex] and [tex]\( 9 \times 9 = 81 \)[/tex], but somewhere in the process she might have misplaced a number or miscalculated the sum:
- Lori might have skipped adding [tex]\( 270 \)[/tex] from [tex]\( 9 \times 30 \)[/tex], possibly confusing it with a smaller number or placing the result incorrectly.
### Summary:
- The correct value of [tex]\( 29 \times 31 \)[/tex] is [tex]\( 899 \)[/tex].
- Lori's mistake resulted in an incorrect product of [tex]\( 699 \)[/tex], which has a difference of 200 from the correct result.
Identifying exactly that she might have placed a zero in the ones column before multiplying [tex]\( 3 \times 9 \)[/tex], leading her astray in subsequent steps, is another possible point of error. This incorrect step could cause an accrued difference of 200 given [tex]\( 3 \times 9 = 270 \)[/tex].
### Final Answer:
- The correct value for [tex]\( 29 \times 31 \)[/tex] is [tex]\( 899 \)[/tex].
- Lori got [tex]\( 699 \)[/tex] likely due to misplacing digits or miscalculating during the expansion and summation stages.
- The precise difference due to Lori's mistake is [tex]\( 200 \)[/tex].
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