SOLUTION: A machine produces 30 articles in t hours. If the machine were to produce 5 more articles each hour, it would take half an hour less to produce 30 articles. Find t.

Question 271852: A machine produces 30 articles in t hours. If the machine were to produce 5 more articles each hour, it would take half an hour less to produce 30 articles. Find t.
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
The basic fact we need to understand to solve this problem is that if
n = number of articles produced
r = number of articles produced per hour (rate)
t = number of hours of production
then
n = rt

We have two sets of data. First it says 30 articles are produced in t hours. So
30 = rt

Then it says that it would take 30 minutes less time to produce 30 if the rate was 5 more articles per hour. So

(Note the use of 1/2. The time is supposed to be in hours so 30 minutes was converted to 1/2 hour.) We now have a system of two equations in two variables. Since these are not linear equations we'll use the Substitution Method to solve the system. Solving the first equation for r we get:

Substituting this into the second equation we get:

Now we solve this one variable equation. Multiplying out the right side we get:

Subtracting 30 from each side we get:

Next we'll get rid of the fractions by multiplying both sides by the Lowest Common Denominator (LCD). The LCD here is 2t:

On the right side we need to use the Distributive Property:

which simplifies to:

This is a quadratic equation. To solve it we'll first get it in the correct order:

Now we'll factor it (or use the Quadratic Formula):

From the Zero Product Property we know that this product is zero only if one of the factors is zero. 5 can never be zero. But (t+2) and (t-3) could be zero with the "right" values of y. So we solve the following:
or
or
or
Since t is a number of hours we will reject the negative solution. So the only practical, real-life solution for t is 3.
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