The amount of work in a leather factory is increased by `50%`. By what percent is it necessary to increase the number of workers to complete the new amount of work in previously planned time , if the productivity of the new labour is `25%` more.

Options

A. `20%`
B. `66.66%`
C. `40%`
D. `33.33%`

Answer

To solve the problem step by step, we need to analyze the changes in work, productivity, and the number of workers required.

### Step 1: Determine the increase in work
The initial amount of work is considered as 100% (let's denote it as W). If the work is increased by 50%, the new amount of work (W') can be calculated as:
\[ W' = W + 0.5W = 1.5W \]

**Hint:** When you increase something by a percentage, you can express it as the original amount plus that percentage of the original amount.

### Step 2: Determine the new productivity of workers
The productivity of the new labor is 25% more than the original productivity. If we denote the original productivity as P, the new productivity (P') can be calculated as:
\[ P' = P + 0.25P = 1.25P \]

**Hint:** To find a percentage increase, multiply the original value by (1 + the percentage increase expressed as a decimal).

### Step 3: Relate work, productivity, and number of workers
The relationship between work, productivity, and the number of workers (N) can be expressed as:
\[ \text{Work} = \text{Number of Workers} \times \text{Productivity} \times \text{Time} \]
Let’s denote the original number of workers as N and the time as T. For the original scenario:
\[ W = N \times P \times T \]
For the new scenario with increased work and productivity:
\[ W' = N' \times P' \times T \]
Where N' is the new number of workers needed.

### Step 4: Set up the equation with the new values
Substituting the values we found:
\[ 1.5W = N' \times 1.25P \times T \]
Since \( W = N \times P \times T \), we can substitute W in the equation:
\[ 1.5(N \times P \times T) = N' \times 1.25P \times T \]
Now, we can cancel out T and P from both sides (assuming they are not zero):
\[ 1.5N = 1.25N' \]

### Step 5: Solve for the new number of workers (N')
Rearranging the equation to find N':
\[ N' = \frac{1.5N}{1.25} \]
Calculating this gives:
\[ N' = 1.2N \]

### Step 6: Calculate the percentage increase in the number of workers
To find the percentage increase in the number of workers:
\[ \text{Percentage Increase} = \left( \frac{N' - N}{N} \right) \times 100 \]
Substituting N' into the formula:
\[ \text{Percentage Increase} = \left( \frac{1.2N - N}{N} \right) \times 100 = (0.2) \times 100 = 20\% \]

### Final Answer
The number of workers needs to be increased by **20%** to complete the new amount of work in the previously planned time.

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