If m and n are positive integers, what is the value of 3/m + n/4 ?

when it is a multiple, the fraction wont be in its lowest term e.g . 3/12,3/9,3/6..... is not in its lowest term right? why? because 12 is a multiple of 3 and 9,6 are all multiples of 3.
lowest term for 3/12= 1/4 , 3/9=1/3 ...etc

now 3/7 ,3/8,3/5,3/13.......is in its lowest term right? why? because seven (7) aint a multiple if three (3) , we can also say 3 and 7 ,3 and 8 ........are co-prime
got it?

Hi, it should be \(\frac{12+mn}{4m}\)

Only when m is not a multiple of 3 it is in its lowest form, otherwise we could divide both numerator and denominator by 3.
Same for n. The lowest multiple is 2.

1. \(\frac{3}{m} + \frac{n}{4}\) can be simplified to \(\frac{12 + mn}{4m}\)

Option A - Insufficient
Even if we plug in \(mn = 12\), then a variable \(m\) still remains in the denominator - \(\frac{24}{4m} = \frac{6}{m}\)

Option B - Insufficient
Says that the two fractions are in the lowest terms, which means that these fractions cannot be simplified any further e.g. \(\frac{3}{4}\) or \(\frac{1}{4}\)

Combining both statements
\(m * n = 12\) can have the following values of \(m\) and \(n\)

a. 1 * 12 or 12 * 1
b. 3 * 4 or 4 * 3
c. 2 * 6 or 6 * 2

Out of the above 6 combinations a., c. and first combination from b. can be eliminated because these do not meet the second statement e.g. \(n = 12\) gives \(\frac{12}{4} = 3\) but second statement states that the fractions are in the lowest terms or cannot be simplified further.

Only b. survives the test of the second statement which gives fractions that cannot be simplified any further i.e. \(\frac{3}{m} = \frac{3}{4}\) and \(\frac{n}{4} = \frac{4}{3}\)

\(m = 4\) and \(n = 3\) can be plugged in to yield \(\frac{3}{4} + \frac{3}{4} = 1.5\)

Ans. C

Solution:

Statement One Only:
mn = 12

If m = 12 and n = 1, then 3/m + n/4 = 3/12 + 1/4 = 1/4 + 1/4 = 1/2.

On the other hand, if m = 6 and n = 2, then 3/m + n/4 = 3/6 + 2/4 = 1/2 + 1/2 = 1.

We see that statement one alone is not sufficient to answer the question.


Statement Two Only:
3/m is in lowest terms and n/4 is in lowest terms.

If m = n = 1, then 3/m + n/4 = 3/1 + 1/4 = 3 1/4 = 3.25.

On the other hand, if m = 2 and n = 3, then 3/m + n/4 = 3/2 + 3/4 = 9/4 = 2.25.

We see that statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using statement one, we can conclude that {m, n} must equal {1, 12}, {2, 6} or {3, 4}. Using statement two, we see that the only way 3/m and n/4 can both be in lowest terms is if m = 4 and n = 3. Thus, 3/m + n/4 = 3/4 + 3/4 = 3/2.

Statements one and two together are sufficient to answer the question:

Answer: C

What does lowest form exactly mean here?

Shouldn't it be the forms when 3/m and n/4 are the lowest which happens when we combine statements and take my = 12 and n = 1 and also when m = 6 and n = 2 making the sum = 0.5 and instead when we take m = 4 and n = 3 the sum is 1.5 and even the individual terms are higher in this case.

mikemcgarry Bunuel

An irreducible fraction (or fraction in lowest terms, simplest form or reduced fraction) is a fraction in which the numerator and denominator are integers that have no other common divisors than 1 (and −1, when negative numbers are considered).

So, a/b is a fraction in lowest terms when it cannot be further reduced. For example, 3/4 is in lowest terms, because you cannot reduce it further while 3/6 is NOT in lowest terms, because you can reduce it further and get 1/2, which is in lowest terms.

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