<p>Each student in a group of 30 students studies German, Italian or both. The total number of students studying German is three more than the total number of students studying Italian. If the number of students that study both subjects is the same as the number of students that study exactly one subject, how many students in the group study only Italian
A. 6
B.9
C.15
D.21
E.24</p>
<p>The question itself is really easy. They’ve just disguised it rather fiendishly with a whole lot of words, and they’ve put the useful facts backwards from the order in which you want to use them.</p>
<p>There are 30 students. Half of them study only one language (“If the number of students that study both subjects is the same as the number of students that study exactly one subject…”); that’s 15. Three more students study German than study Italian; that makes 9 German students and 6 Italian students.</p>
<p>So, how many students study only Italian? Six.</p>
<p>Here is an algebraic solution:</p>
<p>Let x be the number of students that study ONLY German, y the number of students that study BOTH, and z the number of students that study ONLY Italian. Than:</p>
<p>x+y+z=30
x+y=y+z+3
y=x+z</p>
<p>Bringing all the variables to one sides gives us:</p>
<p>x+y+z=30
x-z=3
x-y+z=0</p>
<p>Adding the first and 3rd equation gives 2x+2z=30, or x+z=15</p>
<p>x-z=3
x+z=15</p>
<p>Adding these gives
2x=18 or x=9. So z=6.</p>
<p>Remarks: (1) Sikorsky’s method is the way you should do it on the SAT, but if you’re going for an 800 it’s good to understand an algebraic solution as well.</p>
<p>(2) With the algebraic solution, once you have the following system of equations</p>
<p>x+y+z=30
x-z=3
x-y+z=0</p>
<p>you can use the matrix and rref features on your graphing calculator to quickly get the solution without actually doing any algebra yourself.</p>
<p>
</p>
<p>I totally agree with Dr. Steve that it’s good to be able to do this problem algebraically. I have no problem with the way he’s done the problem, either, but I don’t think the algebra necessarily has to be quite so advanced. You could use substitution until you can solve a single equation in a single variable, using only skills you were taught in Algebra I.</p>
<p>Using equation 3, substitute (x+z) for y in equation 1.
x+(x+z)+z=30, or
2x+2z=30. Divide both sides by 2.
x+z=15.</p>
<p>Now use equation 2.
x+y=y+z+3; subtract y on both sides.
x=z+3.
Substitute (z+3) for x into the modified equation 1 above.</p>
<p>(z+3)+z=15
2z+3=15
2z=12
z=6.</p>
<p>And Z is the number of students who study only Italian, which is what the question asked you to find.</p>
<p>The answer is 9 not 6. Just plug in the values and you’ll see. Btw it is not stated in the question that half of the students offer each subject</p>
<p>9jagurl, if there are 15 people taking only Italian or German: x+(x+3)=15
Because 3 more people take German than Italian, where x is the students taking Italian
2x=12
X=6</p>
<p>9jagurl96- what equation are you asking us to plug the numbers into? If you use the equations given by Dr.Steve, you get 6, just like he said. The original question does say that half of the students study both languages, meaning 15 students study both German and Italian, and the remaining students must either study German OR Italian. Since we know that there are 3 more students studying German, and 9+6=15 and 9-6=3 then we know the answer is 6!</p>
<p>Let the number of students taking german be represented by y, let the number taking italian be represented by x hence y=x+3. The question says the number of students taking both subjects (z) is equal to the number taking one of the subjects (not specified whether it’s german or italian) so z=x or z=y. Since there are 30 students in the class, x+y+z=30 if z=x then we have x+x+3+x=30 3x+3=30 3x=27 x=9. Dats how i think it should be solved guys.</p>
<p>Sorry, i think i misinterpreted the question, i see my mistake now thnks</p>
<p>i dont understand…where does it say that half of the kids study both language… “The question says the number of students taking both subjects (z) is equal to the number taking one of the subjects (not specified whether it’s german or italian) so z=x or z=y.” i understand that tho…some1 please help</p>
<p>“If the number of students that study both subjects is the same as the number of students that study exactly one subject,”…i have no idea where in that sentence that it says half of the students studies both subjects >.<…im so frustrated :(</p>
<p>Think about it like this, if there is a bag of 10 marbles which are all either red or blue, and the number of blue marbles equals the number of red marbles, what is the only possible solution? There are 5 of each color, 5 is half of ten. </p>
<p>The same thing applies here: there are 30 students, and the number of students who study BOTH languages is equal to the number of students who study ONLY ONE of the languages. So what is the only solution? There are 15 people who study both and 15 people who study only one. 15 is half of 30</p>