Please help with this fractions question...

[Raanikeri]

The answer key says 27.

But I don't know how to obtain the answer of 27 as there is nothing in the question that tells me the amounts of anything.

How do I solve this?

 

[Dr.Peterson]

View attachment 36176

The answer key says 27.

But I don't know how to obtain the answer of 27 as there is nothing in the question that tells me the amounts of anything.

How do I solve this?

The basic idea will be (I say this without even solving it) that you will get a fraction such that, if the fraction is exact, the total number of students must be a multiple of the denominator.

Now, to solve it: What fraction of all students are female? What fraction of all students are right-handed females? What is the denominator of this fraction?

 

[blamocur]

Additional hint: the number of students in each group must be integer.

 

[Raanikeri]

The basic idea will be (I say this without even solving it) that you will get a fraction such that, if the fraction is exact, the total number of students must be a multiple of the denominator.

Now, to solve it: What fraction of all students are female? What fraction of all students are right-handed females? What is the denominator of this fraction?

Thank you, I am able to solve it with what you said in your last sentence, but I mean... I can only do it without understanding why it works deep down.

The fraction of all students who are female is 4/9 - still, I don't know how many students that is in actual number.
5/6 of them are right handed.
So if I do 5/6 x 4/9, I get 10/27
That 27 denominator right there is the answer in the answer key.

I still don't understand why just dealing with fractions- in this case, multiplying one fraction of an amount by the other fraction of an amount - gives me the actual number of students in the denominator.

(Sorry I had to edit my answer so many times as I didn't have my working in front of me and kept jumbling up the fractions... this is the final edit and is what I actually did on my working)

 

[Dr.Peterson]

Thank you, I am able to solve it with what you said in your last sentence, but I mean... I can only do it without understanding why it works deep down.

The fraction of all students who are female is 4/9 - still, I don't know how many students that is in actual number.
5/6 of them are right handed.
So if I do 4/9 x 5/6, I get 10/27
That 27 denominator right there is the answer in the answer key.

I still don't understand why just dealing with fractions- in this case, multiplying one fraction of an amount by the other fraction of an amount - gives me the actual number of students in the denominator.

Good! Now we can explore more deeply!

We know that exactly 5/9 of all students are male; so to get the number of male students if we knew the total number, we would multiply by 5/9. For instance, if there were 100 students, there would be $\frac{5}{9}\times100=\frac{500}{9}=55\frac{5}{9}$ male students. There's probably some blood pooling around that last one (or maybe he's only partly male).

Of course, we know the number of males has to be a whole number, so it's not that there is a partial male; rather, there can't be 100 students. The total number of students must be a number we can multiply by 5/9 and get a whole number. That means we have to be able to divide by 9 and get a whole number; can you see that? So the smallest number of students there could be is the smallest positive number that is a multiple of 9, which is 9 itself.

Now you've found that 10/27 of the students are right-handed females. Do you see that by the same reasoning, the number must be a multiple of 27, and the smallest such number is 27?

If not, there's more we can discuss.

By the way, in real life we might well say that [about] 5/9 of 100 students are male if there were, say, 56 males. That's why I've emphasized that in the problem, because this is a math class, it is assumed that the fraction is exact. In real life, none of this reasoning would work, because we use approximations all the time. If I wrote the problem, I would use the word "exactly" in the problem, to avoid confusing students who understand real life better than math.

 

[Raanikeri]

Good! Now we can explore more deeply!

We know that exactly 5/9 of all students are male; so to get the number of male students if we knew the total number, we would multiply by 5/9. For instance, if there were 100 students, there would be $\frac{5}{9}\times100=\frac{500}{9}=55\frac{5}{9}$ male students. There's probably some blood pooling around that last one (or maybe he's only partly male).

Of course, we know the number of males has to be a whole number, so it's not that there is a partial male; rather, there can't be 100 students. The total number of students must be a number we can multiply by 5/9 and get a whole number. That means we have to be able to divide by 9 and get a whole number; can you see that? So the smallest number of students there could be is the smallest positive number that is a multiple of 9, which is 9 itself.

Now you've found that 10/27 of the students are right-handed females. Do you see that by the same reasoning, the number must be a multiple of 27, and the smallest such number is 27?

If not, there's more we can discuss.

By the way, in real life we might well say that [about] 5/9 of 100 students are male if there were, say, 56 males. That's why I've emphasized that in the problem, because this is a math class, it is assumed that the fraction is exact. In real life, none of this reasoning would work, because we use approximations all the time. If I wrote the problem, I would use the word "exactly" in the problem, to avoid confusing students who understand real life better than math.

Thank you Dr Peterson, I understand it better now.