Now answer the following questions. Evaluate: $$ 141_{ ext {six }}+233_{ ext {six }}-102_{ ext {six }} $$ A. $$ 412_{ ext {six }} $$ B. $$ 405_{ ext {six }} $$ C. $$ \quad 312_{ ext {six }} $$ D. $$ 305_{ ext {six }} $$

Question

Now answer the following questions. Evaluate: $$ 141_{	ext {six }}+233_{	ext {six }}-102_{	ext {six }} $$ A. $$ 412_{	ext {six }} $$ B. $$ 405_{	ext {six }} $$ C. $$ \quad 312_{	ext {six }} $$ D. $$ 305_{	ext {six }} $$

UpStudy AI · Accepted Answer

To evaluate the expression $$ 141_{\text{six}} + 233_{\text{six}} - 102_{\text{six}} $$, we'll perform the calculations in base six.

### Step 1: Convert Each Number to Base Ten

1. **Convert $$ 141_{\text{six}} $$ to base ten:**
   $$
   1 \times 6^2 + 4 \times 6^1 + 1 \times 6^0 = 1 \times 36 + 4 \times 6 + 1 \times 1 = 36 + 24 + 1 = 61_{\text{ten}}
   $$

2. **Convert $$ 233_{\text{six}} $$ to base ten:**
   $$
   2 \times 6^2 + 3 \times 6^1 + 3 \times 6^0 = 2 \times 36 + 3 \times 6 + 3 \times 1 = 72 + 18 + 3 = 93_{\text{ten}}
   $$

3. **Convert $$ 102_{\text{six}} $$ to base ten:**
   $$
   1 \times 6^2 + 0 \times 6^1 + 2 \times 6^0 = 1 \times 36 + 0 \times 6 + 2 \times 1 = 36 + 0 + 2 = 38_{\text{ten}}
   $$

### Step 2: Perform the Calculation in Base Ten

$$
61_{\text{ten}} + 93_{\text{ten}} - 38_{\text{ten}} = 61 + 93 = 154 \
154 - 38 = 116_{\text{ten}}
$$

### Step 3: Convert the Result Back to Base Six

To convert $$ 116_{\text{ten}} $$ to base six:

1. **Determine the largest power of six less than or equal to 116:**
   $$
   6^3 = 216 \quad (\text{too large}) \
   6^2 = 36
   $$

2. **Calculate the coefficients:**
   $$
   116 \div 36 = 3 \quad \text{with a remainder of} \quad 116 - (3 \times 36) = 8 \
   8 \div 6 = 1 \quad \text{with a remainder of} \quad 8 - (1 \times 6) = 2
   $$

3. **Write the base six number:**
   $$
   3 \times 6^2 + 1 \times 6^1 + 2 \times 6^0 = 312_{\text{six}}
   $$

### **Answer:**  
**C. $$ 312_{\text{six}} $$**
The answer is $$ 312_{\text{six}} $$, choice C.

Now answer the following questions.
Evaluate:
A.
B.
C.
D.

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