Three diggers dug a ditch of 324 m deep in six days working simultaneously. During one shift, the third digger digs as many metres more than the second as the second digs more than the first. The third digger’s work in 10 days is equal to the first digger’s work in 14 days. How many metres does the first digger dig per shift?

Options

A. 15 m
B. 18 m
C. 21 m
D. 27 m

Answer

To solve the problem step by step, we will break it down into manageable parts.

### Step 1: Understand the Problem
We have three diggers (let's call them A, B, and C) who together dig a ditch of 324 meters deep in 6 days. We need to find out how many meters the first digger (A) digs per shift.

### Step 2: Calculate Total Work Done Per Day
First, we calculate how much work is done per day by all three diggers combined.
\[ \text{Total work} = 324 \text{ meters} \]
\[ \text{Total days} = 6 \text{ days} \]
\[ \text{Work done per day} = \frac{324 \text{ meters}}{6 \text{ days}} = 54 \text{ meters/day} \]

### Step 3: Set Up the Variables
Let:
- \( x \) = work done by Digger A (first digger) in meters per day,
- \( y \) = work done by Digger B (second digger) in meters per day,
- \( z \) = work done by Digger C (third digger) in meters per day.

According to the problem, we know:
1. \( x + y + z = 54 \) (total work done by all three diggers per day).

### Step 4: Understand the Relationships
From the problem, we have the following relationships:
- The third digger digs as many meters more than the second as the second digs more than the first:
\[ z - y = y - x \implies z = 2y - x \]

- The third digger’s work in 10 days is equal to the first digger’s work in 14 days:
\[ 10z = 14x \implies z = \frac{14}{10}x = 1.4x \]

### Step 5: Substitute and Solve
Now we can substitute \( z \) in the first equation:
\[ x + y + 1.4x = 54 \]
\[ 2.4x + y = 54 \quad \text{(1)} \]

Substituting \( z = 2y - x \) into the equation:
\[ 2y - x = 1.4x \]
\[ 2y = 2.4x \implies y = 1.2x \quad \text{(2)} \]

### Step 6: Substitute Equation (2) into Equation (1)
Now substitute \( y = 1.2x \) into equation (1):
\[ 2.4x + 1.2x = 54 \]
\[ 3.6x = 54 \]
\[ x = \frac{54}{3.6} = 15 \]

### Step 7: Conclusion
Thus, the first digger (A) digs **15 meters per shift**.