Hello,
I have two questions from the Flashcards Quant 1-4 Quiz, specifically questions 13 and 16.
For question 13, I’m struggling to understand it completely. I tried testing various random numbers, which led me to choose option 4 as the answer, which is wrong.
Regarding question 16, I noticed that the answers for both quantities are the same, yet I’m confused as to why Quantity A is considered greater.
Can you help clarify these points?
Can you show your work
Q13 is quite tricky actually.
First let’s consider \frac{x}{16}. We know that the remainder is an element of the set R =\{0,1,\dots, 15\}. For every number r \in R in this set, the term \frac{r}{16} is rational with a finite number of decimal points (seems to be 4 at most).
Now consider \frac{y}{21}. We know that y is not a multiple of 7, which implies that y is also not a multiple of 21 so 21 does not evenly divide y. This means we can rule out 0 as the remainder, so R = \{1,2,\dots, 20\}. Now for any number r in this set, we can think of the remainder term (which determine the decimal points) as r \frac{1}{21}, which are just scaled versions of r\frac{1}{21}\propto\frac{1}{7}. There is a bizarre phenomenon where the reciprocal of 7 has infinitely many repeating digits. Therefore \frac{y}{21} is some integer plus a remainder which is a scaled version of \frac{1}{7}, and so there are infinitely many decimal points.
For Q16, they aren’t the same. You can do a prime factorization:
\sqrt{80} = \sqrt{5* 4^2} = 4\sqrt{5}
\sqrt[3]{320} = \sqrt[3]{5 *64} = 4\sqrt[3]{5}
with 5^{1/3} < 5^{1/2} of course.
show what?
Thank you for your explanation! However, I’m still having trouble understanding question 13. Could you simplify it further for me?
As for question 16, I appreciate your help. I’ve got it now! I forgot about the cube root!
Can someone please help me with the question 13??!!
Hey @user1454 ,
for question 13, I think it seems complicated but it is a conceptual question because if you see x/16 here 16 will always be multiple of 2 and as per the rule taught by Greg if the denominator has only 2s or 5s as powers that will always be a terminating decimal but for y/21 case you see the question has already mentioned that y is not a multiple of 7 which means it will be in decimals for a lot of cases and in the cases of those decimals it will be non terminating because it is solely not a composition of 2s or 5s in the denominator (21 has 3 and 7 as its factors so that is why decimals will not be terminating)
That is why B is greater. Please confirm if that is the correct answer?
Hope that helps! Let me know if it is still unclear.
For your reference, here is Greg’s explanation- Does It Terminate? - GregMat
Thank you very much! I believe I now understand it clearly. However, as you requested, could someone from GregMat please confirm whether this explanation is accurate?
I think I am sure about this explanation. I requested for review on another question. Also, Ganesh liked the answer which means they sorta agree with it.
Oh, I didn’t notice Ganesh liked it! It looks like everything is fine now:) Thank you!
