I am studying the book "Algebra" by Israel Gelfand and am having trouble with this problem:
Which of the fractions and is bigger?
I know I could calculate it by hand by finding a common denominator, but I'm fairly certain the book is not interested in rote calculation of this magnitude, so there must be a simpler way. For example, the previous problem was to determine whether or was bigger. By noting that both fractions are very close to one, but that the latter is the closest, you can easily determine which is larger. Is there something similar you can do for this one?
since the values added are in the same proportion as in the original fraction.
Hence if the proportions of the additional values are different, the fraction will change
Hence, for your fraction, the numerator is < the denominator.
If we add values that are equal, then their proportion is greater than the proportions
of the original numerator and denominator,
hence the fraction increases.