Let's carefully go through the process of multiplying 29 by 31 step-by-step to identify any mistakes Lori might have made and to arrive at the correct answer.
### Step 1: Multiply 29 by 1 (the units digit of 31)
[tex]\[
\begin{array}{c}
\ \ 2\ 9 \\
\times \ \ \ \ 1 \\
\hline
\ \ 2\ 9 \ \ \ \ \ \ \ \text{(29 multiplied by 1)}
\end{array}
\][/tex]
### Step 2: Multiply 29 by 3 (the tens digit of 31)
Remember to place a zero in the units place because we are multiplying by the tens place:
[tex]\[
\begin{array}{c}
\ \ \ \ \ 2\ 9 \ \ \ \ \ \ \ \ \ \ \ \ \text{(29 multiplied by 1, previously calculated)} \\
+\ \ \ \ 8\ 7\ 0 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{(29 multiplied by 3 )}
\end{array}
\][/tex]
Break this down:
- [tex]\(29 \times 3 = 87\)[/tex]
- Place "87" one place to the left: "870".
### Step 3: Add the two partial products
[tex]\[
\begin{array}{c}
\ \ 2\ 9 \ \ \ \ \ \ \ \text{(partial product from step 1)} \\
+\ \ 8\ 7\ 0 \ \ \ \ \text{(partial product from step 2)} \\
\hline
9\ 0\ 9
\end{array}
\][/tex]
Thus, the correct product of 29 and 31 is [tex]\(899\)[/tex].
### Step 4: Verification
It is helpful to verify our final multiplication:
[tex]\(
\begin{aligned}
29 \times 31 & = 29 \times (30 + 1) \\
& = 29 \times 30 + 29 \times 1 \\
& = 870 + 29 \\
& = 899.
\end{aligned}
\)[/tex]
Lori's mistake was indeed not placing a zero in the ones column before multiplying 29 by the 3 in the tens place properly and, potentially, errors in subsequent regrouping and addition.
By performing each step carefully and checking our work, we confirm that [tex]\(29 \times 31 = 899\)[/tex].
