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Let's work through this problem step-by-step.

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### **Question Summary:**
- Ben and Carla share a sum of money equally.
- Ben spends **1/6** of his share.
- Carla spends **$45** of her share.
- After their spending, **3/4** of the total original amount remains.
- We need to find **the original sum of money shared** by Ben and Carla.

---

### **Solution:**

Let the original total amount be **$ 2x $** (since the money was split equally between Ben and Carla, each receives $ x $ dollars).

#### Step 1: How much Ben and Carla have after spending
- Ben's initial share is $ x $. He spends $ \frac{1}{6} $ of his share:
  \[
  \frac{1}{6} \times x = \frac{x}{6}.
  \]
  So, Ben has:
  \[
  x - \frac{x}{6} = \frac{5x}{6}.
  \]

- Carla spends **$45** from her share of $ x $. Thus, Carla now has:
  \[
  x - 45.
  \]

#### Step 2: Find the remaining amount
The total amount left with both Ben and Carla after their spending is:
\[
\frac{5x}{6} + (x - 45).
\]
Simplify:
\[
\frac{5x}{6} + \frac{6x}{6} - 45 = \frac{11x}{6} - 45.
\]

#### Step 3: Use the fact that **3/4 of the original amount remains**
Since **3/4** of the original amount $ 2x $ is left, we have:
\[
\frac{3}{4} \times 2x = \frac{3x}{2}.
\]
This is the amount left, so:
\[
\frac{11x}{6} - 45 = \frac{3x}{2}.
\]

#### Step 4: Solve for $ x $
First, clear the fractions by multiplying the entire equation by 6:
\[
11x - 270 = 9x.
\]
Rearrange the equation:
\[
11x - 9x = 270 \quad \Rightarrow \quad 2x = 270 \quad \Rightarrow \quad x = 135.
\]

#### Step 5: Find the original total amount
Since the total amount was $ 2x $:
\[
2 \times 135 = 270.
\]

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### **Answer:**
The original sum of money shared by Ben and Carla is **$270**.

Thus, the correct option is **(3) $270**.

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Would you like further details or clarifications?

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### **Related Questions:**
1. If Carla spent $10 more, how much would be left with both of them?
2. What fraction of the original amount did Carla spend?
3. How much would remain if Ben spent 1/2 of his share instead of 1/6?
4. What percentage of their individual shares did Ben and Carla each spend?
5. What would the original sum be if only 1/2 of the original amount was left?

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**Tip:** Always keep track of shared amounts by representing them with variables!

Ben shared a sum of money equally with Carla. To buy a present for their mother, Ben spent 1/6 of his money and Carla spent $45 of her money. In the end, 3/4 of the original sum of money was left. What was the original sum of money shared by Ben and Carla?

Learn how to solve a real-life money-sharing problem where Ben and Carla split a sum equally, spend part of it, and 3/4 of the original amount remains. This problem covers algebra, fractions, and linear equations, perfect for Grades 9-12 students.

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