A and B complete a work in 6 days. If A alone can do it in 9 days, then B alone can do $\frac{1}{3}$ of the same work in ________ (days).
Understanding the Work and Time Problem
This problem involves calculating the time taken by an individual (B) to complete a portion of a task, given information about the combined time taken by two people (A and B) and the time taken by one of them (A) individually.
Calculating Work Rates
We need to determine how much work A and B do per day, both together and individually.
- Let the total amount of work be represented by 1 unit.
- A and B complete the work together in 6 days. This means their combined work rate is $\frac{1}{6}$ of the work per day.
- A alone can complete the work in 9 days. So, A's individual work rate is $\frac{1}{9}$ of the work per day.
Finding B's Individual Work Rate
To find B's work rate, we subtract A's rate from the combined rate:
B's work rate = (Combined work rate of A and B) - (A's work rate)
B's work rate = $\frac{1}{6} - \frac{1}{9}$
To subtract these fractions, we find a common denominator, which is 18:
B's work rate = $\frac{3}{18} - \frac{2}{18} = \frac{1}{18}$ of the work per day.
Calculating Time for B to Complete Full Work
Since B completes $\frac{1}{18}$ of the work each day, the total time B would take to complete the entire work (1 unit) is the reciprocal of B's work rate:
Time for B (full work) = $\frac{1}{\text{B's work rate}} = \frac{1}{1/18} = 18$ days.
Calculating Time for B to Complete 1/3 of the Work
The question asks for the time B takes to complete $\frac{1}{3}$ of the same work. We can calculate this by multiplying the time needed for the full work by $\frac{1}{3}$:
Time for B ($\frac{1}{3}$ work) = (Time for B (full work)) $\times \frac{1}{3}$
Time for B ($\frac{1}{3}$ work) = $18 \times \frac{1}{3}$
Time for B ($\frac{1}{3}$ work) = $\frac{18}{3} = 6$ days.
Conclusion
Therefore, B alone can do $\frac{1}{3}$ of the same work in 6 days.
Important Questions from Time & Work (Notes)
- A fresh water tap fills a fish tank in 40 minutes. The same tank is filled by a salt water tap in 120 minutes. If both the taps are open, how many minutes will it take to fill the tank?
- A completes $\frac{7}{10}$ of a work in 15 days and then he completes the remaining work with the help of B in 5 days. In how many days can A and B together complete the entire work?
Aman can do 50% of the job in 16 days, and Bhanu can do 25% of the job in 24 days. In how many days can they do $\frac{1}{4}^{th}$ of the job working together ?
- X can finish a job in $141$ days. He worked for $57$ days alone and the remaining work was completed by Y, in $84$ days. How many days would both together take to complete the entire job?
Ravina, Sujata, and Saroj can complete a work of painting separately in 32, 48, and 64 hours, respectively. They started working together, but Saroj left after 5 hours. From the 6th hour, Ravina and Sujata decided to work on alternate hours starting with Ravina. In how much time will the entire work of painting be completed?
