Kevin B. answered • 10/25/22
A Specialist in Math and Physics
This question is checking your ability to process relationship given in language form into equations. The math for this question is pretty simple, but extracting the math from the words can be tough; there are many different ways to say things in written language, so it takes a lot of practice to pick of on all the cues.
Your first step with this kind of problem should always be to assign variable names to the important quantities. It sounds like the important quantities are "number of students who caught fish", "number of students who ate fish but did not catch any", "number of students who caught fish but did not eat any", "number of students who neither caught nor ate fish", and "number of students who both caught and ate fish". We also know the total number of students. Let's give some variable names to these:
T: Total number of students
C: Number of students who caught fish
Co: Number of students who caught fish but did not eat any
Ao: Number of students who ate fish but did not catch any
N: Number of students who neither caught nor ate fish
B: Number of students who both caught and ate fish
There is one more-- the number this question is asking us to find is: "the number of students who ate fish". Let's give it a name:
A: Number of students who ate fish
NOTE: Based on your title, you need to draw a Venn diagram in order to get full credit for this problem. Hint: Draw a big circle and label it T to represent the whole class. Then, draw smaller circles (and label them) for each group we named above inside of T, and overlap the ones which have students that can belong to each group. You should absolutely do this even if you don't think you need to, because it is perfect for visualizing how the students are sorted into different groups, especially the students that belong to more than one group.
The next step is to start writing the mathematical relationships between these variables. Nothing like B = 3A is given in the problem, but the statements that are made can be translated into equations like that, and a few more can be inferred by simple logic. Let's look at them one-by-one:
"...the 24 students..." just means
T= 24
"9 students caught salmon." just means
C = 9
"The numbers who ate but did not catch salmon was 4 time the number who caught but did not eat salmon." Sounds a little tricky at first. But "was" just means '=' and "4 time" just means '×4'. So we can write this as
Ao = 4 Co
"The number who neither caught nor ate salmon was half the number who did both." Here we notice "half the number" which just means '÷2'. So:
N = B / 2
That's all of the statements. There is a problem, though... We are looking for A (how many students ate fish) to give as the answer to this question, but none of these equations use A. That's because we need to infer relationships by thinking about the simple logic of the problem.
When we think about A, which is the number of students who ate fish, we realize it is made up of a few smaller groups added together. (Aha, it is completely possible for 1 student to be counted in more than one group at once!)
If a student ate fish but did not catch any, should they be counted in the group called "students who ate fish"? Yes! If a student ate a fish AND caught a fish, should they be counted in the group called "students who at fish"? Yes! And finally, are there any other groups that should be counted in "students who at fish"? Nope, because all the remaining groups are not counting only students who caught fish.
Knowing this, we can infer this relationship:
A = Ao + B
Another relationship that follows exactly the same logic is that we can count all the student who caught fish by counting everyone who caught but did not eat fish, as well as those who both caught and ate fish:
C = Co + B
There is one more relationship we can infer. Adding up the groups that includes every student exactly once (these groups are called "mutually exclusive") should be our total number of students.
Hint: Use your Venn diagram to make sure you aren't choosing groups that count the same students twice! None of the groups should overlap in your diagram, but should include all groups in T. When you count everyone in your class to find out how many students there are, you don't count anyone twice because then you would count too high.
To count everyone in the class exactly once using the groups we named, we can combine the groups for "students who caught fish but didn't eat any", "students who ate fish but didn't catch any", "students who neither caught nor ate fish", and "students who both caught and ate fish". This means we can write the following equation:
T = Co + Ao + N + B
Now let's bring all our relationships into one place for easy reading (I also numbered the equations):
T= 24
C = 9
1: Ao = 4 Co
2: N = B / 2
3: A = Ao + B
4: C = Co + B
5: T = Co + Ao + N + B
Looks like a lot... but really, the hard part is done. Commend yourself for getting this far-- if you are good at solving simultaneous sets of equations, you might already know what to do.
I wrote instructions for solving for A, but I went over the character limit. I will leave it to you to solve for A... Perhaps another tutor can follow up on my answer to take you through the final part of the solution.
Hint: You need to reduce the number of variables in any equation to 1 variable so you can find its value. To do this in this problem, you need to plug in the values you know, then solve an equation in terms of another variable so that expression can be substituted into another equation, thus eliminating one variable after you simplify. Start by substituting 4Co for Ao in equation 5 (you can do this because this is the relationship given by equation 1), and you will see what I mean.
