Ordering an even and odd fraction that are close

Question

We know that $1/4 < 5/11 < 1/2$.

I did it this way from small to large:

$$\frac{1 \cdot 3}{4 \cdot 3} = \frac{3}{12}$$

$$\frac{5}{11}$$

$$\frac{1 \cdot 6}{2 \cdot 6} = \frac{6}{12}$$

It is hard for me to decide between 5/11 and 6/12 without a calculator. Is there an easier way to compare them?

Answer 1

This answer is very similar to multiplying all three fractions by the least common denominator, but simply write each fraction so they all have a common denominator!

The least common multiple of 4, 11, and 2 is 44 so multiply the first fraction by 11 over 11, multiply the second fraction by 4 over 4, and multiply the third fraction by 22 over 22. Now you are specifically still comparing the actual fractions (via equivalent fractions).

$\frac{1}{4}\times\frac{11}{11}=\frac{11}{44}$

$\frac{5}{11}\times\frac{4}{4}=\frac{20}{44}$

$\frac{1}{2}\times\frac{22}{22}=\frac{22}{44}$

Answer 2

Multiply all the numbers by the least common multiple of the denominators - $44$, and you get the inequality:

$$11<20<22$$

Answer 3

You want to compare $\frac{a}{b}$ and $\frac{a+1}{b+1}$ where $a=5$ and $b=11$.

This is a good test of how good your algebra is.

Here's how this goes:

$\begin{array}\\ \frac{a}{b}-\frac{a+1}{b+1} &=\frac{a(b+1)}{b(b+1)}-\frac{b(a+1)}{b(b+1)}\\ &=\frac{a(b+1)-b(a+1)}{b(b+1)}\\ &=\frac{ab+a-(ba+b)}{b(b+1)}\\ &=\frac{a-b}{b(b+1)}\\ \end{array} $

Therefore, if $b > 0$ then the sign of $\frac{a}{b}-\frac{a+1}{b+1}$ is the same as the sign of $a-b$.

Therefore, if $b > 0$, then $\frac{a}{b}>\frac{a+1}{b+1}$ if $\frac{a}{b} > 1$ and $\frac{a}{b}<\frac{a+1}{b+1}$ if $\frac{a}{b} < 1$.