**Middle School Number Sense Lesson 94: Subtracting in a Base other than 10**

This concept is similar to, but a little more difficult than,

**BaseAdd**. It has appeared

**6 times**this year, with a median placement at

**question # 56**.

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**No matter what base we are in, we cannot write a single digit that is (at least) as large as the base itself. So in base 10, the largest digit is 9. In base 8, the largest digit we can write is 7. In base 4, the largest digit we can write is 3, etc. Other than that, this works the same way as subtracting whole numbers in base 10.**

__Remember__:**How to Solve:**

- Within each number, work from right to left. Subtract the units digits first. If their difference is 0 or greater, write it down.
- If the difference is <0, regroup (borrow) the base (number of units) from the digit to the left.
- Move to the next digits. If their difference is 0 or greater, write it down. Otherwise, go back to step 2. (Repeat this process until you run out of digits).

**Example 1: 82 (base 9) – 34 (base 9) = ___ (base 9)**

- Start with the units digit. We cannot subtract
**2 – 4**(the answer is negative). - Regroup (borrow) 1 unit of
**9**from the 8. Now subtract**(9 + 2) – 4**, which is**7**. Write this down. - Move to the left. We are left with
**(8 – 1) – 3**, which is**4**. Write this in front. The answer is**47**.

**Example 2: 63 (base 7) – 15 (base 7) = ___ (base 7)**

- Start with the units digit. We cannot subtract
**3 – 5**(the answer is negative). - Regroup (borrow) 1 unit of
**7**from the 6. Now subtract**(7 + 3) – 5**, which is**5**. Write this down. - Move to the left. We are left with
**(6 – 1) – 1**, which is**4**. Write this in front. The answer is**45**.

**Example 3: 31 (base 5) – 13 (base 5) = ___ (base 5)**

- Start with the units digit. We cannot subtract
**1 – 3**. - Regroup (borrow) 1 unit of
**5**from the 3. Now subtract**(5 + 1) – 3**, which is**3**. Write this down. - Move to the left. We are left with (
**3 – 1) – 1**, which is**1**. Write this in front. The answer is**13**.

**Example 4: 312 (base 8) – 25 (base 8) = ___ (base 8)**

- Start with the units digit. We cannot subtract
**2 – 5**. - Regroup 1 unit of
**8**from the 1. Now subtract**(8 + 2) – 5**, which is**5**. Write this down. - Move to the digits to the left. We are left with
**(1 – 1) – 2**, which is negative. - Regroup 1 unit of
**8**from the 3. Now subtract**(8 + 0) – 2**, which is**6**. Write this in front. - Move to the left. We are left with
**(3 – 1) –**an implied**0**, which is**2**. Write this in front. - The answer is
**265**.

**Example 5: 525 (base 6) – 251 (base 6) = ___ (base 6)**

- Start with the units digit.
**5 – 1**is**4**; write this down. - Move to the middle digits. We cannot subtract
**2 – 5**. - Regroup 1 unit of
**6**from the 5. Now subtract**(6 + 2) – 5**, which is**3**. Write this in front. - Move to the digits on the left. We are left with
**(5 – 1) – 2**, which is**2**. Write this in front. - The answer is
**234**.

**Up Next for Middle School: MultNear100+-**