Work and Time Calculation Example

This example demonstrates how to calculate the number of days needed to complete a job when factors like the number of people and their efficiency change.

Work Formula Explained

The fundamental relationship in work problems is:

Work = Number of Persons $\times$ Number of Days $\times$ Efficiency

Let $W$ represent the total work, $P$ the number of persons, $D$ the number of days, and $E$ the efficiency. The total work ($W$) required for a specific job is constant.

Initial Scenario: Work Calculation

Given information for the first case:

• Number of Persons ($P_1$) = 15
• Number of Days ($D_1$) = 30
• Efficiency ($E_1$) = Let's denote it as $E_1$

The total work ($W$) can be calculated as:

$W = P_1 \times D_1 \times E_1 = 15 \times 30 \times E_1 = 450 E_1$

New Scenario: Days Calculation

Information for the second case:

• Number of Persons ($P_2$) = 30
• Efficiency ($E_2$) = $2 \times E_1$ (twice the initial efficiency)
• Number of Days ($D_2$) = ? (This is what we need to find)

Using the same work formula for the second scenario:

$W = P_2 \times D_2 \times E_2 = 30 \times D_2 \times (2 E_1) = 60 D_2 E_1$

Final Answer Derivation

Since the total work ($W$) is constant, we equate the work done in both scenarios:

$450 E_1 = 60 D_2 E_1$

Divide both sides by $E_1$ (assuming $E_1 \neq 0$):

$450 = 60 D_2$

Solve for $D_2$:

$D_2 = \frac{450}{60}$

$D_2 = \frac{45}{6}$

$D_2 = 7.5$

Thus, the job can be completed in 7.5 days.