X can type 8 pages in 3 minutes, Y can type 3 pages in 5 minutes. Working together, how many pages can they type in 15 minutes?
49
Solving Typing Speed Problems: Working Together
This problem involves calculating the combined typing speed of two individuals, X and Y, and then determining how many pages they can type together in a specific amount of time.
To solve this, we first need to find the individual typing rates of X and Y per minute.
Calculating Individual Typing Rates
The typing rate is the number of pages typed per unit of time (in this case, per minute).
- X's Typing Rate:
- X types 8 pages in 3 minutes.
- X's rate = $\frac{\text{Number of pages}}{\text{Time taken}} = \frac{8 \text{ pages}}{3 \text{ minutes}} = \frac{8}{3}$ pages per minute.
- Y's Typing Rate:
- Y types 3 pages in 5 minutes.
- Y's rate = $\frac{\text{Number of pages}}{\text{Time taken}} = \frac{3 \text{ pages}}{5 \text{ minutes}} = \frac{3}{5}$ pages per minute.
Calculating Combined Typing Rate
When X and Y work together, their typing rates add up. The combined rate is the sum of their individual rates per minute.
- Combined Rate = X's Rate + Y's Rate
- Combined Rate = $\frac{8}{3} + \frac{3}{5}$ pages per minute.
To add these fractions, we need a common denominator. The least common multiple of 3 and 5 is 15.
- Convert X's rate: $\frac{8}{3} = \frac{8 \times 5}{3 \times 5} = \frac{40}{15}$
- Convert Y's rate: $\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}$
Now, add the converted rates:
- Combined Rate = $\frac{40}{15} + \frac{9}{15} = \frac{40 + 9}{15} = \frac{49}{15}$ pages per minute.
So, together, X and Y can type $\frac{49}{15}$ pages every minute.
| Person | Pages Typed | Time (minutes) | Rate (pages/minute) |
|---|---|---|---|
| X | 8 | 3 | $\frac{8}{3}$ |
| Y | 3 | 5 | $\frac{3}{5}$ |
| X & Y Together | ? | 1 | $\frac{49}{15}$ |
Total Pages Typed in 15 Minutes
To find out how many pages they can type together in 15 minutes, we multiply their combined rate by the total time.
- Total Pages = Combined Rate $\times$ Time
- Total Pages = $\frac{49}{15} \text{ pages/minute} \times 15 \text{ minutes}$
- Total Pages = $49$ pages.
Therefore, working together, X and Y can type 49 pages in 15 minutes.
Revision Table: Key Concepts
| Concept | Formula/Explanation |
|---|---|
| Work Rate | Work Done / Time Taken |
| Total Work | Rate × Time |
| Combined Rate | Sum of individual rates (when working together) |
Additional Information on Work and Time Problems
Work and time problems often involve calculating how long it takes for one or more individuals or machines to complete a task, or how much of a task can be completed in a given time.
- The basic principle is that Work = Rate $\times$ Time.
- If a person completes a task in 'T' units of time, their rate is $\frac{1}{T}$ of the task per unit of time.
- When multiple people work together, their rates are typically added to find the combined rate, assuming they work independently but simultaneously on the same task.
- Efficiency is sometimes used interchangeably with rate; higher efficiency means a higher rate of work.
Understanding how to calculate individual rates and combine them is crucial for solving these types of problems.
Important Questions from Time and Work
A and B can do a piece of work in 18 days. B and C together can do it in 30 days. If A is twice as good a workman as C, find in how many days B alone can do the work?
Two men and 7 women can complete a work in 28 days, whereas 6 men and 16 women can do the same work in 11 days. In how many days will 5 men and 4 women, working together, complete the same work?
A man and a woman, working together can do a work in 66 days. The ratio of their working efficiencies is 3 ∶ 2. In how many days 6 men and 2 women together can do the same work?
Each one of five men independently can complete a work in 20 days. The work is started by one person. Next day one more person joins and every next day one more person joins. From the fifth day, five persons continued working as a team. In how many days, will the work be completed?
Numan does half the work as Gagan in 4/5 of the time. If together they take 16 days to complete a piece of work, then how long will it take Gagan to complete the work?
