Given a fraction:
$$\frac{a}{b}$$
I now add a number $n$ to both numerator and denominator in the following fashion:
$$\frac{a+n}{b+n}$$
The basic property is that the second fraction is suppose to closer to $1$ than the first one. My question is how can we prove that?
What I have tried:
I know $\frac{n}{n} = 1$ so now adding numbers $a$ and $b$ to it would actually "move it away" from $1$. But I cannot understand why $\frac{a}{b}$ is actually farther away from $1$ than $\frac{a+n}{b+n}$.
Why is that? What does it mean to add a number to both the numerator and denominator?
Best Answer
There's a very simple way to see this. Just take the difference between the two fractions and 1. You want to show that this is smaller in modulus for the second fraction.
You get $$ \frac{a}{b} - 1 = \frac{a-b}{b} $$ and $$ \frac{a+n}{b+n} -1 = \frac{a-b}{b+n} $$
So the second is smaller in modulus (provided $b$ and $n$ are positive, although I supposed it also works if both are negative) because it has same numerator and larger (modulus) denominator, QED.