All of us, plus half of us and quarter of us and you alone taken together equal 71. How many are we really?

Options

A. 70 (2)
B. 60
C. 50 (4)
D. 40

Explanation

To solve the problem step by step, let's break down the information given in the question.

### Step 1: Define the Variables
Let \( x \) represent the total number of people (all of us).

### Step 2: Set Up the Equation
According to the question, we have:
- All of us = \( x \)
- Half of us = \( \frac{x}{2} \)
- Quarter of us = \( \frac{x}{4} \)
- You alone = \( 1 \)

Putting this all together, we can write the equation:
\[ x + \frac{x}{2} + \frac{x}{4} + 1 = 71 \]

### Step 3: Simplify the Equation
First, we can subtract 1 from both sides of the equation:
\[ x + \frac{x}{2} + \frac{x}{4} = 70 \]

### Step 4: Find a Common Denominator
To combine the terms on the left side, we need a common denominator. The least common multiple (LCM) of 1, 2, and 4 is 4. Therefore, we can rewrite the equation as:
\[ \frac{4x}{4} + \frac{2x}{4} + \frac{x}{4} = 70 \]
This simplifies to:
\[ \frac{4x + 2x + x}{4} = 70 \]
\[ \frac{7x}{4} = 70 \]

### Step 5: Solve for \( x \)
Now, we can solve for \( x \) by multiplying both sides by 4:
\[ 7x = 70 \times 4 \]
\[ 7x = 280 \]
Now, divide both sides by 7:
\[ x = \frac{280}{7} \]
\[ x = 40 \]

### Conclusion
Thus, the total number of people (all of us) is \( \boxed{40} \).

---