SOLUTION: to use a certain computer data base, the charge is $30 per hour during the day and $10.50 per hour at night. if a research company paid $411 for 28 hours of use, find the number of

Question 584664: to use a certain computer data base, the charge is $30 per hour during the day and $10.50 per hour at night. if a research company paid $411 for 28 hours of use, find the number of hours charged at the daytime reate and at the nighttime rate

Answer by bucky(2189) (Show Source): You can put this solution on YOUR website!
You have two unknowns, the number of daytime hours (call them D) and the number of nighttime hours (call them N). This means that you will need two independent equations to solve for these two unknowns. So let's go through the problem and see if we can find two such equations.
.
First, we know that D plus N is the total number of hours that were spent using the computer. The problem tells you that this total was 28 hours. Therefore, we can write one of the equations as follows:
.
D + N = 28
.
Next the problem tells you that the bill for using the computer was $411. But this bill came from using the computer both during the day and also during the night. Each hour during the day that the computer was used cost $30. So the total cost for daytime use can be found by multiplying $30 times D, the number of hours it spent on daytime use. This can be written as 30D. Similarly, the total spent on using the computer at night was $10.50 per hour times N, the number of hour of nighttime use. This can be written as 10.5N. The addition of these two costs (daytime plus nighttime) equals the total $411 cost of using the computer. So we can write the second equation as:
.
30D + 10.5N = 411
.
Now all that needs to be done is to solve this set of two equations to find the values of D and N. One way to do this is by using substitution. Look at the first equation, and let's solve it for one of the variables in terms of the other. For example, let's subtract D from both sides of this equation. This will get rid of the D on the left side of the equation and a minus D will appear on the right side. Therefore, this equation becomes:
.
N = 28 - D
.
Now we know that N equals 28 - D, we can go to the second equation and replace N in it with its equivalent 28 - D. When we do that, the second equation becomes:
.
30D + 10.5*(28 - D) = 411
.
Do the distributed multiplication on the left side by multiplying 10.5 times each of the two terms in the parentheses. This results in the equation becoming:
.
30D + 294 - 10.5D = 411
.
On the left side combine the two terms that contain D by subtracting 10.5D from 30D to get:
.
19.5D + 294 = 411
.
Next, get rid of the 294 on the left side by subtracting 294 from both sides. The resulting equation is:
.
19.5D = 117
.
Finally, solve for D by dividing both sides by 19.5 to get:
.
D = 117/19.5 = 6
.
So we now know that 6 hours was spent using the computer during daytime hours. And since the first equation tells us that a total of 28 hours was spent using the computer, the remaining 22 hours must have been for nighttime use. Let's check that by finding what the cost of such usage would be.
.
For 6 hours of daytime use at $30 per hour the cost would be 6*$30 = $180. And for 22 hours of nighttime use at $10.50 per hour the cost would be 22*$10.50 = $231. So the total cost would be $180 + $231 and this equals $411 which is what the problem says it should be.
.
The answer therefore is that the computer was used 6 hours during the day and 22 hours at night.
.
Hope that this discussion helps you to understand better how to determine the number of equations you need for problems such as this one (the number of equations needed equals the number of unknown variables), and once you have the equations, one way that you can solve for the unknown variables.
.
RELATED QUESTIONS
Please help me solve this. A city council voted to conduct a study on inner-city... (answered by ikleyn)

Please help me solve this. A city council voted to conduct a study on inner-city... (answered by ikleyn)

Please help me solve this. A city council voted to conduct a study on inner-city... (answered by Theo,ikleyn)

During summer vacation, you charge people $10 per hour to teach their children math and (answered by josmiceli)

Youre a plumber and youll charge a certain amount just for showing up at a job and youll... (answered by josgarithmetic)

A computer repair service charges $50 to come out for a service call which is applied... (answered by checkley71)

A company pays 15,40 euro/ hour during the daytime and 19,60 euro/hour during night... (answered by checkley71)

Please help me to solve this question!! : one third of a certain journey was covered... (answered by josgarithmetic,Solver92311,ikleyn)

The cost to rent a computer is Php 10.00 for the first hour or less plus Php 3.00 per... (answered by ikleyn)