## Thinking Mathematically (6th Edition)

The difference in the indicated base is ${{345}_{\text{seven}}}$.
Start by subtracting the numbers in the right-hand column:${{4}_{\text{seven}}}-{{6}_{\text{seven}}}$. ${{6}_{\text{seven}}}$is greater than ${{4}_{\text{seven}}}$. So, we need to borrow from the preceding column. Now, borrow one group of 7 because in the provided question we are working in base seven. This gives a sum of $4+7$ or 11 in base ten. Now, subtract 6 from 11: \begin{align} & \text{ 1} \\ & \begin{matrix} \text{ 62}{{\text{4}}_{\text{seven}}} \\ -{{246}_{\text{seven}}} \\ 5 \\ \end{matrix} \\ \end{align} Now, subtract the numbers in the second column from the right: ${{1}_{\text{seven}}}-{{4}_{\text{seven}}}$. ${{4}_{\text{seven}}}$is greater than ${{1}_{\text{seven}}}$. So, we need to borrow from the preceding column. Now, borrow one group of 7 because in the provided question we are working in base seven. This gives a sum of $1+7$ or 8 in base ten. Now, subtract 4 from 8: \begin{align} & \text{ 51} \\ & \begin{matrix} \text{ 62}{{\text{4}}_{\text{seven}}} \\ -{{246}_{\text{seven}}} \\ 45 \\ \end{matrix} \\ \end{align} Now, perform the subtraction in the third column from the right. \begin{align} & \text{ 51} \\ & \begin{matrix} \text{ 62}{{\text{4}}_{\text{seven}}} \\ -{{246}_{\text{seven}}} \\ {{345}_{\text{seven}}} \\ \end{matrix} \\ \end{align} The difference in the indicated base is ${{345}_{\text{seven}}}$.