Problema Solution
a disc jokey must play 14 commercial spots during 1 hour of a radio show. each commercial is either 30 seconds or 60 seconds long. if the total commercial time during 1 hour is 11 min, how many 30-second commercials were played that hour? how many 60-second commercials?
Answer provided by our tutors
Lets call the 30-second commercials A
Lets call the 60-second commercials B
The length of a 30-second commercial is .5A minutes (example, one 30-second commercial .5minutes)
The length of a 60-second commercial is 1B (example, one 60-second commercial is 1 minute)
Now that we have the variables named and the minute-time calculated, we set up a system of equations using A and B:
A+B=14 (this equation shows that the number of 30-second commercials added to the number of 60-second commercials is 14)
.5A+B=11 (this equation shows that the total airtime of 30-second long and 60-second long commercials is 11 minutes)
So, without the descriptions, our system of equations looks like this:
A+B=14
.5A+B=11
Now lets multiply the bottom equation by -1. This will leave a +B on the top and a -B on the bottom, so the B's will cancel out, leaving an equation using just A's:
A+B=14
-.5A-B= -11
Now lets add the two equations:
A+B=14
+ -.5A-B= -11
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.5A=3
A=6
Using A+B=14, we get that 6+B=14, so B=8
There are 6 30-second long commercials and 8 60-second long commercials.
CHECK: Lets check that 6 and 8 satisfy our inital equations:
6+8=14 ~~~ CHECK
6*.5+8=11 ~~~ CHECK
The equations check out, so there are indeed 6 30-second long commercials and 8 60-second long commercials.