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.

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

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.