Question

In the previous question (number 33 ) if Zahira starts first then in how many days, the work will be completed? \(\begin{array}{ll}\text { (a) } 14 \frac{1}{5} & \text { (b) } 14\end{array}\) (c) \(13 \frac{1}{5}\) (d) \(13 \frac{2}{5}\)

Short answer

Answer: 3 days

Step-by-step solution

Calculate combined work per day

Calculate the combined work per day for Zahira and Farooq by summing their individual work completion rates: \(W_{combined} = \frac{1}{5} + \frac{1}{3}\) Find the lowest common denominator (LCD) and add the fractions: \(W_{combined} = \frac{3}{15} + \frac{5}{15} = \frac{8}{15}\) Thus, they can complete \(\frac{8}{15}\) of the work when working together in one day.

Create an equation for the number of days to complete the work

Zahira works alone on the first day and after that, they work together. Let's represent the number of days it will take for Zahira to complete the work with x. On the first day, Zahira will complete \(\frac{1}{5}\) of the work. In the following x days, both of them will work together completing \(\frac{8}{15}\) of the work per day. To represent the total work, we create the following equation: \(\frac{1}{5} + x \cdot \frac{8}{15} = 1\)

Solve the equation

Solve the equation to find the number of days it will take for them to complete the work: \(\frac{1}{5} + x \cdot \frac{8}{15} = 1\) Subtract \(\frac{1}{5}\) from both sides: \(x \cdot \frac{8}{15} = \frac{4}{5}\) Divide both sides by \(\frac{8}{15}\) to find x: \(x = \frac{4}{5} \cdot \frac{15}{8} = \frac{6}{4}\) Simplify the fraction: \(x = \frac{3}{2}\) So, x is equal to \(1 \frac{1}{2}.\)

Calculate the total number of days

Since Zahira works alone on the first day and they work together for the next x days, the total number of days taken to complete the work will be: Total days = 1 day (Zahira works alone) + \(1 \frac{1}{2}\) days (working together) Total days = \(2 \frac{1}{2}\) days However, since both days and people are in whole numbers, we can not use half a day. Thus, if we round this up to the nearest whole number, it will take 3 days for them to complete the work when Zahira starts first. In this case, none of the given answer options matches the correct result of 3 days. Therefore, the correct answer is not among the given options.

Key concepts

Fraction Addition

Fraction addition is a key element in many time and work problems. It involves summing up fractions to combine rates, which allows us to determine the total work done together by different parties. To add fractions like \(\frac{1}{5}\) and \(\frac{1}{3}\), you first need to identify the least common denominator (LCD). In this case, the LCD for 5 and 3 is 15.

• Convert \(\frac{1}{5}\) by multiplying both the numerator and the denominator by 3 to get \(\frac{3}{15}\).
• Convert \(\frac{1}{3}\) by multiplying both the numerator and denominator by 5, resulting in \(\frac{5}{15}\).

After finding equivalent fractions, you can add them together: \(\frac{3}{15} + \frac{5}{15} = \frac{8}{15}\). This final fraction gives you the combined rate of work, which can be used in further calculations.

Equation Solving

To find out how long it will take Zahira and Farooq to complete the entire work, you need to set up and solve an equation. This equation represents the situation described in the problem. In our case, Zahira starts the work alone and then both of them continue together.
Thus, the equation is: \[\frac{1}{5} + x \cdot \frac{8}{15} = 1\]The term \(\frac{1}{5}\) represents the work Zahira does alone on the first day. \(x \cdot \frac{8}{15}\) stands for the work done together in \(x\) days. The right side, 1, indicates the completion of the whole work.
Subtract \(\frac{1}{5}\) from both sides to isolate the terms involving \(x\):\[x \cdot \frac{8}{15} = \frac{4}{5}\]You solve for \(x\) by dividing both sides by \(\frac{8}{15}\), which results in:\[x = \frac{4}{5} \cdot \frac{15}{8}\]

Work Rate Combination

Combining work rates is crucial in scenarios where multiple people collaborate on a task. Each person or group has an individual work rate represented as a fraction of the work completed per day. By combining these rates, as we did with Zahira and Farooq, you can determine how quickly the partnership finishes the task.

• Zahira's rate: \(\frac{1}{5}\) per day
• Farooq's rate: \(\frac{1}{3}\) per day
• Combined rate: \(\frac{8}{15}\) per day

This method allows us to examine the efficiency of their combined efforts. By knowing that they complete \(\frac{8}{15}\) of the work together, we can use this fraction to break the work into manageable parts and calculate how many days or what portion of each day they need to complete it.

Problem Solving Steps

To effectively solve time and work problems, follow a structured approach:

• Identify the individual work rates and convert them into fractions if necessary.
• Find the combined work rate by calculating the least common denominator and adding the fractions.
• Set up an equation representing the entire task, which often involves combining individual work with joint work efforts.
• Solve the equation step by step, isolating the unknown variable to find the solution.

In our example, these steps led us to realize that the combined work rate for Zahira and Farooq was \(\frac{8}{15}\) per day, and by carefully setting up and solving the equation, we determined Zahira and Farooq could finish in approximately 3 days. Always address any discrepancies such as rounding when necessary to match real-world constraints, like working whole days.