Sumi can complete a job working 5 hours per day in 2 days. If she doubles her working hours per day, then in how many days will she complete the work?
1
Understanding the Work Problem
The question asks us to determine how many days Sumi will take to complete a specific job if she changes her daily working hours. The key idea here is that the total amount of work required for the job remains constant.
The total work done can be thought of as the product of the rate of work (hours per day) and the time taken (number of days).
Initially, Sumi works 5 hours per day and completes the job in 2 days.
So, the total work required for the job is:
\$\text{Total Work} = \text{Hours per day} \times \text{Number of days}\$
\$\text{Total Work} = 5 \text{ hours/day} \times 2 \text{ days}\$
\$\text{Total Work} = 10 \text{ hours}\$
Calculating the New Working Rate
The problem states that Sumi doubles her working hours per day. Her original working hours per day were 5 hours.
Her new working hours per day will be:
\$\text{New Hours per day} = 2 \times \text{Original Hours per day}\$
\$\text{New Hours per day} = 2 \times 5 \text{ hours/day}\$
\$\text{New Hours per day} = 10 \text{ hours/day}\$
Finding the New Time to Complete the Work
Now we know the total work required (10 hours) and Sumi's new working rate (10 hours per day). We can use the same formula for total work to find the new number of days she will take:
\$\text{Total Work} = \text{New Hours per day} \times \text{New Number of days}\$
We need to find the New Number of days. Rearranging the formula:
\$\text{New Number of days} = \frac{\text{Total Work}}{\text{New Hours per day}}\$
Plugging in the values:
\$\text{New Number of days} = \frac{10 \text{ hours}}{10 \text{ hours/day}}\$
\$\text{New Number of days} = 1 \text{ day}\$
Summary of Calculations
Let's summarize the steps and findings:
- Original working hours: 5 hours/day
- Original days to complete: 2 days
- Total work = \$5 \text{ hours/day} \times 2 \text{ days} = 10 \text{ hours}\$
- New working hours: Doubles the original hours = \$2 \times 5 \text{ hours/day} = 10 \text{ hours/day}\$
- New days to complete = \$\frac{\text{Total Work}}{\text{New Hours per day}} = \frac{10 \text{ hours}}{10 \text{ hours/day}} = 1 \text{ day}\$
So, if Sumi doubles her working hours to 10 hours per day, she will complete the work in 1 day.
| Parameter | Original Scenario | New Scenario |
|---|---|---|
| Working Hours per Day | 5 hours | \$5 \times 2 = 10\$ hours |
| Days to Complete Work | 2 days | ? |
| Total Work | \$5 \text{ hours/day} \times 2 \text{ days} = 10 \text{ hours}\$ | 10 hours (Constant) |
Using the formula: \$\text{Days} = \frac{\text{Total Work}}{\text{Hours per day}}\$
New Days = \$\frac{10 \text{ hours}}{10 \text{ hours/day}} = 1 \text{ day}\$
The new number of days required is 1 day.
Revision Table: Understanding Work and Time
| Concept | Explanation | Formula Relationship |
|---|---|---|
| Work | The total amount of task to be completed. Assumed constant for the same job. | Work = Rate \$\times\$ Time |
| Rate | The amount of work done per unit of time (e.g., hours per day, units per hour). | Rate = \$\frac{\text{Work}}{\text{Time}}\$ |
| Time | The duration taken to complete the work (e.g., days, hours). | Time = \$\frac{\text{Work}}{\text{Rate}}\$ |
| Inverse Proportion | If Work is constant, Rate and Time are inversely proportional. Doubling the rate halves the time, tripling the rate takes one-third the time, and so on. | Rate \$\propto \frac{1}{\text{Time}}\$ (when Work is constant) |
Additional Information: Inverse Proportion in Work Problems
This problem is a classic example of inverse proportion. When the total amount of work is fixed, increasing the rate of work will decrease the time taken to complete it, and vice-versa.
In this case:
- Original Rate = 5 hours/day
- New Rate = 10 hours/day
The rate has been doubled (\$\times 2\$).
Since Work is constant, the time taken should be halved (\$\times \frac{1}{2}\$).
- Original Time = 2 days
- New Time = \$\frac{1}{2} \times 2 \text{ days} = 1 \text{ day}\$
This confirms our calculation. Understanding this inverse relationship can often help quickly estimate or check answers for work-related problems.
Important Questions from Work Efficiency
Five men and 2 boys can do in 30 days as much work as 7 men and 10 boys can do in 15 days. How many boys should join 40 men to do the same work in 4 days?
A man completes 7/8 of a job in 21 days. How many more days will it take him to finish the job if quantum of work further increased by 50%?
24 men and 12 women can do a piece of work in 30 days. In how many days can 12 men and 24 women do the same piece of work?
3 men working 7 hours a day can complete a piece of work in 45 days. In how many days will 9 men working 6 hours a day complete the same work?
A and B together can do a piece of work in 4 days, B and C can do it in 6 days, A and C can do it in 8 days. Then A, B and C together can do the same work in :-
