Question 1199476: In my class the total number of students who take Geotechnical only, Theory only and Hydraulics only is 12. The total number of students who takes Hydraulics only is twice the number of students who takes Geotechnical only and the number of students who take Theory only is thrice the number of student who takes Geotechnical only. The number of students who take Theory, Geotechnical, and Hydraulics is 3, the number of students who take Theory and Geotechnical is 5, the number of students who take Theory and Hydraulics is 8, and the number of student who take Geotechnical and Hydraulics is 7. Determine the total number of students who take Hydraulics
Found 3 solutions by Edwin McCravy, ikleyn, AbeAlgebraGenius:
Answer by Edwin McCravy(20064)   (Show Source):
You can put this solution on YOUR website!
You inadvertently botched the first two sentences:
In my class the total number of students who take Geotechnical only, Theory only and Hydraulics only is 12.
We can't understand what you're saying there. Please repost your problem and
this time go slow, and be careful to get it EXACTLY right.
Edwin
Answer by ikleyn(52879)  (Show Source):
You can put this solution on YOUR website! .
In my class the total number of students who take Geotechnical only, Theory only and Hydraulics only is 12.
The total number of students who takes Hydraulics only is twice the number of students who takes Geotechnical only
and the number of students who take Theory only is thrice the number of student who takes Geotechnical only.
The number of students who take Theory, Geotechnical, and Hydraulics is 3,
the number of students who take Theory and Geotechnical is 5,
the number of students who take Theory and Hydraulics is 8,
and the number of student who take Geotechnical and Hydraulics is 7.
Determine the total number of students who take Hydraulics.
~~~~~~~~~~~~~~~~~~
Obviously, the number of students who take Hydraulics is this aggregate
(Hydraulic only) + (Theory and Hydraulics) + (Geotechnical and Hydraulics) - (Theory, Geotechnical and Hydraulics).
In this aggregate, we are given all the components in parentheses, except of very first component,
so we can write
the number of students who take Hydraulics = (Hydraulic only) + 8 + 7 - 3. (1)
Thus, only the number (Hydraulic only) is unknown now.
I will find this number, using the other part of the problem.
Let x = the number of students who takes Geotechnical only.
Then the number of students who takes Hydraulic only is 2x,
and the number of students who takes Theory only is 3x.
Next, as we read we problem, we can write this equation
x + 2x + 3x = 12,
from which we get 6x = 12, x = 12/6 = 2. So, the number of students who takes Hydraulic only is 2x = 2*2 = 4.
Finally, from (1), we get the answer to the problem's question
the number of students who take Hydraulics = 4 + 8 + 7 - 3 = 16.
ANSWER. The number of students who take Hydraulics is 16.
Solved.
Answer by AbeAlgebraGenius(2)  (Show Source):
You can put this solution on YOUR website! This problem is solved using Venn diagrams. I will break down the problem step by step and sentence by sentence as shown below.
|
The total number of students who take Geotechnical (g) only, Theory (t) only, and Hydraulics (h) only is 12.
---> E1: g + t + h = 12
|
The total number of students who take Hydraulics (h) only is twice the number of students who take Geotechnical (g) only.
---> E2: h = 2g
|
The number of students who take Theory (t) only is thrice the number of students who take Geotechnical (g) only.
---> E3: t = 3g
|
Substituting “3g” for “t”, and “2g” for “h” in equation E1, we have: g + 3g + 2g = 12 = 6g.
Thus, from Equation E1, g = 2. From equation E2, h = 4. From equation E3, t = 6.
| See the Venn diagram picture for reference.
The number of students who take Theory, Geotechnical, and Hydraulics is 3.
This is the total number of students in the innermost region of our diagram. Next, we work our way out.
|
The number of students who take Theory and Geotechnical is 5.
Let x = the number of students who take Theory and Geotechnical, but not Hydraulics.
Then, we have: x + 3 = 5. Thus, x = 2.
|
The number of students who take Theory and Hydraulics is 8.
Let z = the number of students who take Theory and Hydraulics, but not Geotechnical.
Then, we have: z + 3 = 8. Thus, z = 5.
|
The number of students who take Geotechnical and Hydraulics is 7.
Let y = the number of students who take Geotechnical and Hydraulics, but not Theory.
Then, we have: y + 3 = 7. Thus y = 4.
|
Therefore, the total number of students who take Hydraulics = y + 3 + z + h = 4 + 3 + 5 + 4 = 16.
|
|
|
