Question 227229: Allison has more than 2 but less than 3 hours to spend on her homework. She has work in math, english, and social studies. She plans to spend equal amounts of time studying English and studying social studies, and to spend twice as much time studying math as in studying english. a. What is the minimum number of minutes she can spend on english homework? b. What is the maximum number of minutes she can spend on social studies? c. What is the maximum number of minutes she can devote to math?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! let m = number of minutes spent on math.
let e = number of minutes spent on english.
let s = number of minutes spent on social studies.
e = s (spend as much time on english as on social studies)
m = 2*e (spend twice much time on math as on english)
since e = s, we can say m = 2*e or m = 2*s or m = e+s. they would all mean the same thing.
let t = total number of minutes allison can study.
2 hours = 120 minutes
3 hours = 180 minutes
120 < t < 180 (number of minutes allows is more than 2 hours * 60 and less than 3 hours * 60)
we want to find the minimum e.
we want to find the maximum s.
we want to find the maximum m.
our constraints are:
t = m + s + e
120 < t < 180
s = e
m = 2*e = 2*s = s+e
we have 2e = m
we have s = e
we have 120 < t < 180 which means that 120 < e + s + m < 180
since m = 2e and s = e, this equation becomes:
120 < e + e + 2e < 180 which becomes:
120 < 4e < 180
we divide all sides of this equation by 4 to get:
30 < e < 45
the minimum e is > 30
the maximum e is < 45
since m = 2e, then:
the minimum m is > 60
the maximum m is < 90
the minimum and maximum s is the same as the minimum and maximum e.
let's see how this works.
if m and e and s are minimized they would have to be > 60 + 30 + 30 = 120 minutes which is the minimum total time allowed.
if m and e and s are maximized they would have to be < 90 + 45 + 45 = 180 minutes which is the maximum amount of time allowed.
the questions were:
a. What is the minimum number of minutes she can spend on english homework? b. What is the maximum number of minutes she can spend on social studies? c. What is the maximum number of minutes she can devote to math?
the answer to a is that she can spend a minimum of 30 minutes on english.
the answer to b is that she can spend a maximum of 45 minutes on social studies.
the answer to c is that she can spend a maximum of 90 minutes on math.
assume 30 minutes on english.
this means 30 minutes on social studies.
this means 60 minutes on math.
total time spent is 120 minutes.
assume 45 minutes on social studies.
this means 45 minutes on english.
this means 90 minutes on math.
total time spent is 180 minutes.
technically the problem states > 120 minutes and < 180 minutes.
minimum of 30 minutes or maximum of 45 minutes for english would violate this becdause then the problem should have been stated as >= 120 minutes and <= 180 minutes.
this is picking hairs but mathematically you could not state a minimum of 30 minutes. you would have to state a minimum of more than 30 minutes. likewise for the rest.
hopefully this is not an issue but i'm just mentioning it in case it comes up.
since there is a fixed relationship between m and e and s, a minimum of any one of them automatically implies a minimum of the others and a maximum of any one of them automatically implies a maximum of the others.
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