A contractor deployed some men to plant 1800 trees in a certain no. of days. But in 1/3rd of the planned time 120 plants could be less planted so to fulfill the target for the rest of the days everyday 20 more plants were planted. Thus it saved one day out of the initially planned no. of days. How many plants he planned to plant each day initially?

Options

A. A)180
B. B)100
C. C)120
D. D)160

Answer

To solve the problem step by step, we will break down the information given and use it to find out how many plants the contractor initially planned to plant each day.

### Step 1: Define Variables
Let \( x \) be the number of trees the contractor planned to plant each day initially. Let \( d \) be the total number of days planned to plant 1800 trees.

### Step 2: Calculate Total Trees Planned
The total number of trees to be planted is 1800. Therefore, the equation for the total trees planted in \( d \) days is:
\[ x \cdot d = 1800 \]

### Step 3: Determine Trees Planted in One-Third of Planned Time
In one-third of the planned time, the contractor would have worked for \( \frac{d}{3} \) days. The number of trees that should have been planted in this time is:
\[ \text{Trees planted} = x \cdot \frac{d}{3} \]
However, it is given that 120 trees less were planted, so the actual number of trees planted is:
\[ x \cdot \frac{d}{3} - 120 \]

### Step 4: Calculate Remaining Trees
The remaining trees to be planted after one-third of the time is:
\[ 1800 - \left( x \cdot \frac{d}{3} - 120 \right) = 1800 - x \cdot \frac{d}{3} + 120 \]
This simplifies to:
\[ 1800 + 120 - x \cdot \frac{d}{3} = 1920 - x \cdot \frac{d}{3} \]

### Step 5: Determine Remaining Days
The remaining time after one-third of the planned time is \( \frac{2d}{3} \) days. However, since the contractor saves one day, the effective remaining days are:
\[ \frac{2d}{3} - 1 \]

### Step 6: Calculate New Daily Planting Rate
To meet the target, the contractor plants 20 more trees each day than initially planned. Therefore, the new daily planting rate is:
\[ x + 20 \]
The total trees planted in the remaining days is:
\[ (x + 20) \left( \frac{2d}{3} - 1 \right) \]

### Step 7: Set Up the Equation
Now we can set the equation for the remaining trees:
\[ 1920 - x \cdot \frac{d}{3} = (x + 20) \left( \frac{2d}{3} - 1 \right) \]

### Step 8: Expand and Simplify the Equation
Expanding the right side:
\[ 1920 - x \cdot \frac{d}{3} = (x + 20) \cdot \frac{2d}{3} - (x + 20) \]
This gives:
\[ 1920 - x \cdot \frac{d}{3} = \frac{2xd}{3} + \frac{40d}{3} - x - 20 \]

### Step 9: Rearranging the Equation
Rearranging the equation to isolate \( x \):
\[ 1920 + 20 = \frac{2xd}{3} + x + \frac{40d}{3} + x \cdot \frac{d}{3} \]
This simplifies to:
\[ 1940 = \frac{2xd + 3xd + 40d}{3} \]

### Step 10: Solve for \( x \)
To solve for \( x \), we can substitute \( d \) from the first equation \( x \cdot d = 1800 \):
\[ d = \frac{1800}{x} \]
Substituting this back into the equation will allow us to solve for \( x \).

### Final Calculation
By solving the equations step by step, we can find the value of \( x \) which represents the number of trees the contractor planned to plant each day initially.