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Please see these fractions:

(A) $\frac{33}{128}$ (B) $\frac{45}{138}$ (C) $\frac{53}{216}$ (D) $\frac{83}{324}$ (E) $\frac{15}{59}$.

I need to find out quickly (in about a minute) the smallest of these fractions. I am not allowed to use a calculator, though a little rough calculation is allowed.

As I see the problem, without a calculator, converting these to decimal values is not an option. Moreover, since the numbers involved are fairly large, finding the least common denominator and changing each fraction to make their denominators the same as the least common denominator is again next to impossible without a calculator. For the same reason, we can not try the method of making the numerators identical.

If we try the approximate method of changing the numerators and denominators to easy numbers, we can get something like this:

(A) $\frac{30}{120}$ (B) $\frac{45}{135}$ (C) $\frac{50}{200}$ (D) $\frac{80}{320}$ (E) $\frac{15}{60}$.

Unfortunately, this makes all of the fractions approximately equivalent to $\frac{1}{4}$ except B, which becomes $\frac{1}{3}$.

This is the point where I can not proceed any further.

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  • $\begingroup$ Compare each of A), C), D), and E) to 1/4 by cross multiplying $\endgroup$ – sharding4 Jul 26 '17 at 0:43
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    $\begingroup$ As you note, three values are near $\frac 14$. It's not hard to see that only one is less than $\frac 14$. $\endgroup$ – lulu Jul 26 '17 at 0:44
  • $\begingroup$ one way is to compare the ratio of the numerators to the denominators. 3 of the 5 options for the numerators, divide by 3 ( as does at least one part of each fraction) so you can get exact ratios the denominators must adhere to to be equal or less than. $\endgroup$ – user451844 Jul 26 '17 at 0:47
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    $\begingroup$ there is also a book by Art Benjamin, which i hear is quite good (secrets of mental maths or such), not sure how useful that might be to you, but worth a read $\endgroup$ – mdave16 Jul 26 '17 at 1:12
  • $\begingroup$ @mdave16 Thanks a lot. Actually, this problem is for my son who is trying university admission. This problem appeared in an old admission examination paper. $\endgroup$ – Masroor Jul 26 '17 at 4:16
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Reciprocate/flip them, divide (long division), and see which of these is the biggest. The biggest reciprocal should correspond to the smallest number.

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All fractions are greater than $1/4$ except for $\frac{53}{216}$

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    $\begingroup$ I think this is the intended solution. $\endgroup$ – Ross Millikan Jul 26 '17 at 5:04
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If you have $\frac{a}{b}$ and $\frac cd$ then $\frac ab > \frac cd \iff ad> bc$. Multiplication isn't too slow.

Now pick your favourite fraction, compare it against another, discarding the larger. To ensure you are working with smaller numbers, you can discard any common factors between $ab$ and $cd$. You need 4 comparisons to check which is smallest. Each comparison requires 2 multiplications. I understand that 7.5 seconds is not enough to do a multiplication, but it does not seem too bad sometimes.

As both @lulu and @roddy point out, there are more tricks for these specific fractions, which speed up the process quite a bit. Practicing your multiplication as well as exploiting these tricks should make this problem easy.

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  • $\begingroup$ they can also reduce a lot of them in the comparison as at least one part of each fraction divides by 3. $\endgroup$ – user451844 Jul 26 '17 at 0:49

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