Which of the following fractions is the largest?

Although unlikely to be a general method, I found inverting the fractions and finding the smallest to be useful and fast.

\(\text{(A) }\frac{1}{6} = \frac{3}{18} = 0.1\bar{6}\\\\
\text{(B) }\frac{3}{14}\\\\
\text{(C) }\frac{3}{15}\\\\
\text{(D) }\frac{4}{21}\\\\
\text{(E) }\frac{6}{35} = \frac{3}{17.5}\)

By setting the numerator to 3, we can eliminate all but one \(\Big(\frac{3}{18} < \frac{3}{17.5} < \frac{3}{15} < \frac{3}{14}\Big)\).

We are left with \(\frac{4}{21}\text{ and }\frac{1}{6}\). \(\frac{1}{6} = \frac{4}{24}\).

Therefore \(\frac{1}{6}\) is the smallest. \(\frac{5}{6}\) is therefore the largest.

A) 5/6 = 25/30 = 30/36 = 35/42

B) 11/14 - knowing fractions (adding one to each numerator and denominator) this is known to be less than choice C and can be eliminated

C) 12/15 = 24/30 so less than choice A

D) 17/21 = 34/42 so also less than choice A

E) 29/35 - same as choice B, we know this is less than 30/36 which is choice A

Therefore the answer is A.

another way is by taking LCM of denominators(2*3*5*7)
(a) 175 (as lcm is 2*3*5*7 where 2*3 part is already in denominator just multiply 5*7 to numerator) ,similarly
(b) 165
(c)168
(d)170
(e)174

clearly (a) is largest..

Ans A

Rohit8865,

I also used your method. I wonder, however, if it is the fastest way to tackle this problem.

i find multiplication faster than divisiblity and more accurate and confident to answer
i would also love to know any faster method ,if any....

the fastest way IMO is- to compare each with others such as
5/6 and 11/14 = 5*14>11*6
5/6 and 12/15 = 5*15>6*12
5/6 and 17/21= 5*21>17*6
5/6 and 29/35= 5*35>6*29

hence 5/6 is the greatest

Convert the fractions to percentages and then compare.

LCM of denominators of all the fractions i.e LCM of 6, 14, 15, 21, 35 = 210

As a next stem we need to make all denominators equal to the LCM

A. \(\frac{5}{6} => \frac{5}{6}*\frac{35}{35}=\frac{175}{210}\)

B. \(\frac{11}{14} =>\frac{11}{14}*\frac{15}{15}=\frac{165}{210}\)

C. \(\frac{12}{15} =>\frac{12}{15}*\frac{14}{14}=\frac{168}{210}\)

D. \(\frac{17}{21} =>\frac{17}{21}*\frac{10}{10}=\frac{170}{210}\)

E. \(\frac{29}{35} => \frac{29}{35}*\frac{6}{6}=\frac{174}{210}\)

Clearly option A is the greatest

Answer A

In order to compare fractions, take LCM of all denominators, 6,14,15,21,35.
LCM=210
A) \(\frac{5}{6}\) = \(\frac{5*35}{6*35}\)=\(\frac{175}{210}\)
B) \(\frac{11}{14}\) = \(\frac{11*15}{14*15}\)=\(\frac{165}{210}\)
C) \(\frac{12}{15}\)= \(\frac{12*14}{15*14}\)= \(\frac{166}{210}\)
D) \(\frac{17}{21}\) = \(\frac{17*10}{21*10}\) = \(\frac{170}{210}\)
E) \(\frac{29}{35}\) = \(\frac{29*6}{35*6}\)= \(\frac{174}{210}\)

We see that A is the largest.
Answer A

hi..

the fractions are such that you can do away with LCM and thus some sort of calculations..

Remember - when we add SIMILAR quantity to BOTH numerator and denominator the resulting number is BIGGER
example \(1/2 <\frac{1+1}{2+1}<\frac{1+3}{2+3}\)

now lets see the choices..

Compare B and C as both have a difference of 3 in numerator and denominator.. 11/14 and 12/15 \(\frac{12}{15}= \frac{11+1}{14+1}\), so 12/15 is bigger
Now \(\frac{5}{6}= \frac{5*3}{6*3}=\frac{15}{18}\)... similarly 15/18>12/15
so A is the greatest in A,B and C..

lets check A with D and E...

D 17/21..... difference of 4 in numerator and denominator, so multiply 5/6 with 4/4.... \(\frac{5*4}{6*4}=\frac{20}{24}=\frac{17+3}{21+3}>\frac{17}{21}\).... so A is greater

compare with E.
E 29/35...... \(\frac{5}{6} = \frac{30}{36}=\frac{29+1}{35+1}>\frac{29}{35}\)
so A is greater

ans A

Check the difference between the numerator and denominator.

Fraction with least difference with be the highest fraction.

Hi nehasri,

This logic will not be useful when the difference between numerator and denominator will be the same for different fractions
E.g. 1/2 & 2/3

In this question also if option E had been 30/35 which is 6/7 then the approach would have to be changed

Posted from my mobile device

A simple and fast way is to determine quickly an approximate value of each an every fractions: 5/6=0.8, which won't be reached by any of the other fractions which are all below 0.8, either 0.7 or 0.6! Hope this helped you, however I definitely suck at quant...

Can use Bowtie Method to quickly crunch comparison calculations between the fractions.

\(\frac{5}{6}\) compared to \(\frac{11}{14}\) --> 5 * 14 = 70 compared to 6 * 11 = 66 --> \(\frac{5}{6}\) is larger, cross out \(\frac{11}{14}\). Remember denominator of opposing fraction times numerator is the multiplied value for the fraction with that numerator.

Repeat for all other fractions.
\(\frac{5}{6}\) vs. \(\frac{12}{15}\) --> 5 * 15 = 75 vs. 6 * 12 = 72 --> \(\frac{5}{6}\) is larger, cross out \(\frac{12}{15}\)
\(\frac{5}{6}\) vs. \(\frac{17}{21}\) --> 5 * 21 = 105 vs. 6 * 17 = 102 --> \(\frac{5}{6}\) is larger, cross out \(\frac{17}{21}\)
\(\frac{5}{6}\) vs. \(\frac{29}{35}\) --> 5 * 35 = 175 vs. 6 * 29 = 174 --> \(\frac{5}{6}\) is larger, cross out \(\frac{29}{35}\)

A. \(\frac{5}{6}\) is the largest fraction.

I tried timing myself with multiple different solution methodologies including finding LCM for numerator, LCM for denominator, converting to decimals, and finding the LCM of the numerator of the inverted fractions for each of these, and for me personally just using the Bowtie Method ended up being the fastest.

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