Understanding the Time and Work Problem

This question is about time and work, involving two people, Genelia and Jessica, with different efficiencies. Jessica's work rate is given indirectly by the time she takes to complete the work alone. Genelia's efficiency is related to Jessica's. Genelia starts the work alone, and Jessica joins later for the final few days. We need to find out how many days Genelia worked by herself.

Calculating Individual Work Rates and Time

First, let's figure out the work rate of each person. Work rate is the amount of work done per day.

• Jessica completes the work in 25 days. So, Jessica's daily work rate is $\frac{1}{25}$ of the total work.
• Genelia's efficiency is 25% more than Jessica's. This means Genelia's work rate is $125\%$ of Jessica's work rate.
• Genelia's daily work rate = $125\%$ of $\frac{1}{25}$ = $\frac{125}{100} \times \frac{1}{25}$ = $\frac{5}{4} \times \frac{1}{25}$ = $\frac{5}{100}$ = $\frac{1}{20}$ of the total work per day.
• If Genelia works alone at this rate, she can complete the entire work in 20 days (since her rate is 1/20 work per day).

Analyzing the Work Schedule

The work happens in two phases:

1. Phase 1: Genelia works alone for a certain number of days.
2. Phase 2: Genelia and Jessica work together for the last 5 days.

Let $D$ be the number of days Genelia worked alone. The total number of days the work took is $D + 5$.

Setting Up the Work Equation

The total work done is the sum of the work done in each phase, which equals the total work (represented as 1 unit).

• Work done by Genelia alone in $D$ days = Genelia's rate $\times$ $D$ = $\frac{1}{20} \times D$.
• In the last 5 days, both Genelia and Jessica worked together. Their combined work rate is the sum of their individual rates:
• Combined rate = Genelia's rate + Jessica's rate = $\frac{1}{20} + \frac{1}{25}$.
• To add these fractions, find a common denominator, which is 100.
• Combined rate = $\frac{5}{100} + \frac{4}{100} = \frac{9}{100}$ of the total work per day.
• Work done by Genelia and Jessica together in the last 5 days = Combined rate $\times$ 5 = $\frac{9}{100} \times 5 = \frac{45}{100} = \frac{9}{20}$.

The total work done is the sum of the work in phase 1 and phase 2:

Work in Phase 1 + Work in Phase 2 = Total Work

$\frac{D}{20} + \frac{9}{20} = 1$

Solving for the Number of Days Genelia Worked Alone

Now, we solve the equation for $D$:

$\frac{D + 9}{20} = 1$

Multiply both sides by 20:

$D + 9 = 20$

Subtract 9 from both sides:

$D = 20 - 9$

$D = 11$

So, Genelia worked alone for 11 days.

Summary of Work Done

• Genelia worked alone for 11 days. Work done by Genelia alone = $11 \times \frac{1}{20} = \frac{11}{20}$.
• Genelia and Jessica worked together for 5 days. Combined rate = $\frac{9}{100}$. Work done together = $5 \times \frac{9}{100} = \frac{45}{100} = \frac{9}{20}$.
• Total work done = Work alone + Work together = $\frac{11}{20} + \frac{9}{20} = \frac{11 + 9}{20} = \frac{20}{20} = 1$. The total work is completed.

The number of days Genelia worked alone is 11 days.

Revision Table: Key Concepts

Additional Information: Time and Work Principles

Time and work problems often rely on the concept that the total work is constant, usually taken as 1 unit. The relationship between time, work rate, and total work is fundamental.

• If a person completes a work in $N$ days, their daily work rate is $\frac{1}{N}$.
• If a person works for $D$ days at a rate of $R$, the total work done is $R \times D$.
• When multiple people work together, their work rates add up to find the combined rate.
• Problems can involve different people working for different durations or phases, as seen in this question. Breaking down the work into these phases is key to setting up the correct equation.
• Understanding percentage increase or decrease in efficiency is crucial for calculating relative work rates.