Solving Typing Rate Problems: A Step-by-Step Guide

Typing rate problems are common in various scenarios, from workplace efficiency to competitive examinations. In this article, we will walk through a detailed solution for a specific problem to help you understand the concepts and methods involved.

Introduction to Typing Rate Problems

Typing rate problems often involve calculating the work done by individuals or groups in a given time. These problems are essential for understanding efficiency and productivity in different contexts. This article will solve a typical typing rate problem to illustrate the process step-by-step.

Problem Statement

A can type 75 pages in 25 hours. A and B working together can type 135 pages in 27 hours. What is the time needed by B to type 100 pages?

Step-by-Step Solution

To solve this problem, we need to determine the typing rates of both A and B. Let's start by breaking down the problem and solving it step-by-step.

Step 1: Calculate A's Typing Rate

A can type 75 pages in 25 hours. Therefore, A's typing rate is:

Rate of A 75 pages / 25 hours 3 pages per hour

This means that A can type 3 pages in one hour.

Step 2: Calculate the Combined Typing Rate of A and B

A and B together can type 135 pages in 27 hours. Therefore, their combined typing rate is:

Combined Rate of A and B 135 pages / 27 hours 5 pages per hour

This means that together, A and B can type 5 pages in one hour.

Step 3: Calculate B's Typing Rate

Let ( r_B ) be the typing rate of B. The combined rate of A and B is the sum of their individual rates:

Rate of A Rate of B Combined Rate

Substituting the known values:

3 pages/hour ( r_B ) 5 pages/hour

Solving for ( r_B ):

( r_B ) 5 - 3 2 pages per hour

This means that B can type 2 pages in one hour.

Step 4: Calculate the Time Needed by B to Type 100 Pages

Now that we know B's typing rate, we can find out how long it will take B to type 100 pages:

Time Pages / Rate 100 pages / 2 pages/hour 50 hours

This means that B will need 50 hours to type 100 pages.

Conclusion

The solution to the problem is that B will need 50 hours to type 100 pages. This approach can be applied to similar typing rate problems, providing a clear and systematic method for solving them.

Reference Solutions:

Another method to solve this problem, based on the solutions provided:

Given that 75 pages / 25 hours 3 pages per hour for A, and A and B together typing 135 pages in 27 hours gives a combined rate of 5 pages per hour. Solving 3 ( r_B ) 5 for ( r_B ) yields 2 pages per hour for B. Therefore, the time needed for B to type 100 pages is 100 pages / 2 pages per hour 50 hours.

A's typing rate is 75/25 3 pages per hour. In 27 hours, A types 81 pages, leaving B to type 54 pages in 27 hours, which means B's rate is 2 pages per hour. Hence, 100 pages will take 50 hours.

Mathematical Expressions:

The problems can also be solved using the following mathematical expressions:

1. ( frac{75 , pg}{25 , h} - frac{42 , pg}{b , hrs} frac{135 , pg}{27 , h} Rightarrow 3 - frac{42}{b} 5 Rightarrow frac{b}{42} frac{1}{2} Rightarrow b frac{42}{2} 21 , hrs )

2. Let A can write 1 page in x hours: ( frac{75}{25} 3 ) pages per hour; In 27 hours, A had written 81 pages, leaving B to write 54 pages in 27 hours, making B's rate 2 pages per hour for 42 pages. Hence, B will take 21 hours.

3. A can write 75 pages in 25 hours, so in 27 hours, A can write 81 pages, leaving B to write 135 - 81 54 pages in 27 hours, making B's rate 2 pages per hour. Therefore, 42 pages will take ( frac{42}{2} 21 ) hours.