Question

A manufacturer built a machine which will address 500 envelopes in 8 minutes. He wishes to build another machine so that when both are operating together they will address 500 envelopes in 2 minutes. The equation used to find how many minutes x it would require the second machine address 500 envelopes alone is:
A. $8 - x = 2$
B. $\dfrac{1}{8} + \dfrac{1}{x} = \dfrac{1}{2}$
C. $\dfrac{{500}}{8} + \dfrac{{500}}{x} = 500$
D. $\dfrac{x}{2} + \dfrac{x}{8} = 1$
E. None of these

Answer 1

Hint:
This problem is based on linear equations. When the equation has only one variable, then this type of equation is known as a linear equation in one variable.
There are several steps to solve the word problem:
1. Read the problem carefully, and figure out what it is asking you to find.
2. Assign a variable to the quantity you are trying to find.
3. Write down what the variable represents.
4. Re-read the problem and write an equation for the quantities given in the problem.
5. Solve the equation.
6. Answer the question in the problem. Many times you will need to take the answer you get from the
    equation and use it in some other way to answer the question originally given in the problem.
7. Check your solution.
In this question determine the number of envelopes addressed in 1 minute by each machine. We have to add the equation obtained by both machines and then equate it with another machine equation which he wants to build in order to address 500 envelopes in 2 minutes.

Complete step by step solution:
Machine first addresses 500 envelopes in 8 minutes.
∴ Number of envelope addressed in 1 minute $ = \dfrac{{500}}{8}$
Again,
Machine second addresses 500 envelopes in x minutes
∴ Number of envelope addressed in 1 minute $ = \dfrac{{500}}{x}$
Now, in case of together 500 envelopes addressed in 2 minute
∴ In 1 minute total envelope addressed by both machine together is $\dfrac{{500}}{2}$
According to the question,
$\dfrac{{500}}{8} + \dfrac{{500}}{x} = \dfrac{{500}}{2}$
Taking 500 common
$
  500\left[ {\dfrac{1}{8} + \dfrac{1}{x}} \right] = \dfrac{{500}}{2} \\
  \dfrac{1}{8} + \dfrac{1}{x} = \dfrac{1}{2} \\
$
Hence it is the required equation in x.

Note:
The main step of this question is to find the number of envelopes addressed by different machines in 1 minute. Then we have to balance both statements mathematically. To solve these types of questions, reasoning must be performed based on common sense knowledge and the information provided by the source problem. Some word problems ask to find two or more numbers. We will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.