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
Mohan can do a piece of work in 10 days and Sohan in 15 days. They started working together, but after 3 days Mohan left the work . What time will Sohan take to finish the work?
A tank is filled in 8 hours by three taps A, B and C. The tap C is thrice as fast as B and B is twice as fast as A. How much time will pipe B alone take to fill the tank?
Had been one menless, then the number of days required to do a piece of work would have been one more. If the number of Man. Days required to complete the work is 56, how many workers were there?
A can do a piece of work in 16 hours, B and C can do it in 8 hours while A and C can do it 12 hours. How long will B alone take to do it?
- lf 12 men can do a work in 20 days, in how many days will the work be done by 15 men-
