For those of you who have your TIR from April: can you tell me why D is not also a correct answer? The correct answer is B, but any number that is a factor of “a” is also a factor of “c”.
@loriyacht If you don’t mind, what was the question? I presume it’s safe to discuss it as the April 2016 date has the TIR service.
Consider the fractions 1/a, 1/b and 1/c, where a and b are distinct prime numbers greater than 3 and c = 3a. Suppose that (a)(b)© is used at the common denominator when finding the sum of these fractions. In order for the sum to be in lowest terms, its numerator and denominator must be reduced by a factor of which of the following?
A. 3
B. a
C. b
D. c
E. ab
So, say a= 7 and b = 5. Then c = 21. So the common denominator if you multiply them is 735. The LCD for those is 7 x 5 x 3 = 105. So you would reduce the numerators and denominators by 7. Which is a factor of both a and c. Answers B and D.
What am I not seeing here?
@loriyacht oh I think I see it now. “Reduced by a factor of 7” is supposed to mean “reduced sevenfold,” not reduced by some arbitrary number that happens to be a factor of 7. Interesting wording…
For this question, you cannot use example values to solve the question. Use the variables.
1/a + 1/b +1/c = 1/a + 1/b + 1/3a = 3ab/3a^2b + 3a^2/3a^2b + ab/3a^2b;
the only common factor is a, not 3a (which would be c).
But the question reads, exactly: In order for the sum to be in lowest terms, its numerator and denominator must be reduced by a factor of which of the following?
They did not ask for the common factor. In your example, @theofrelord , a is a factor of a and it is also a factor of 3a.
@MITer94 : If you are correct, and they just mean “a-fold” or, in my example, “seven-fold”, then it is not interesting wording, but very misleading wording. They should have said something like: what is the largest number that they will be reduced by? Or what will everything be divided by?
Thoughts?
@loriyacht the “common factor” theofrelord is referring to is the number or variable that we reduce the fraction by here. The sum 1/a + 1/b + 1/c is equal to (4ab + 3a^2)/(3a^2 b), and the GCF of the numerator and denominator is a. So the fraction is equal to (4b + 3a)/(3ab).
Furthermore, that fraction is in simplest form for all pairs of distinct primes a,b > 3. If “a” is some prime greater than 3, then 3ab is divisible by that prime (and so is 3a) but 4b cannot possibly be divisible by a.
These two wordings are more confusing. But even in common-speak, when we say “reduced by a factor of 7” we mean “reduced seven-fold.”
I see your point. But this is not common-speak, this is a mathematics section, where math terms have certain definitions. Where factor means “a number that divides evenly into…”.
Thanks so much for your reply and the interesting conversation. BTW, I didn’t take this test, but saw the problem when one of my students brought it to my attention. I still think that the ACT should give the point to anyone who answered B or D.
In my humble opinion, the question should have read: “… its numerator and denominator must be reduced by which of the following?”, leaving out the words “a factor of”. Because a is factor of a and also a factor of c. Then, the only correct answer would have been B.
@loriyacht I find this wording more confusing, since “reduced by x” often entails “subtract x.”
Lots of words in math have different meanings depending on context.
Also, “a” is a factor of “ab.”
True and true. OK, I see it. Thank you 