Question

Which statement is true? A.27/19<11/30 b.17/31>19/14 c.16/26>30/31 d.35/30<22/12

is this true 27/19<11/30

Answer 1

In order to check if a comparison such as this is true, you can use this logical equivalence:

`(a)/(b) > (c)/(d) \iff ad > bc,\quad (a)/(b) < (c)/(d) \iff ad < bc`

Anyway, it's not always necessary to make this check. We know that every proper fraction represents a number between 0 and 1, and any improper fraction represents a number greater than 1. So, every proper fraction is less than every improper fraction.

So, the first two options are surely wrong, because they ask for a proper fraction to be more than an improper fraction.

The third option compares two proper fractions, so we actually have to check:

`(16)/(26)>(30)/(31) \iff 16 \cdot 31 > 26 \cdot 30 \iff 496 > 780`

which is false.

Last option:

`(35)/(30)<(22)/(12) \iff 35 \cdot 12 < 30 \cdot 22 \iff 420 < 660`

Which is true.