# What fraction is between 1/3 and 1/2 ?

Dec 17, 2017

$\frac{2}{5}$

#### Explanation:

$\frac{1}{2} = 0.5$

$\frac{1}{3} = 0.333333$

So

$\frac{2}{5} = 0.4$

Dec 17, 2017

There are infinitely many possibilities, but the midpoint between $\frac{1}{3}$ and $\frac{1}{2}$ is $\frac{5}{12}$

#### Explanation:

For any two numbers $a$ and $b$, the average $\frac{1}{2} \left(a + b\right)$ lies between them.

So to find a fraction between $\frac{1}{3}$ and $\frac{1}{2}$ we can form the average:

$\frac{1}{2} \left(\frac{1}{3} + \frac{1}{2}\right) = \frac{1}{2} \left(\frac{2}{6} + \frac{3}{6}\right) = \frac{1}{2} \left(\frac{5}{6}\right) = \frac{5}{12}$

Dec 17, 2017

$\frac{5}{12}$

#### Explanation:

First, make the denominators equal:

$\left(\frac{1}{2}\right) \cdot \left(\frac{3}{3}\right) = \frac{3}{6}$

$\left(\frac{1}{3}\right) \cdot \left(\frac{2}{2}\right) = \frac{2}{6}$

Now figure out the middle number between the numerators:

$\frac{2}{6} < x < \frac{3}{6}$

$x = \frac{2.5}{6}$

Since $2.5$ is a decimal, multiply both the numerator and denominator to make the $2.5$ into a whole number:

$\left(\frac{2.5}{6}\right) \cdot \left(\frac{2}{2}\right) = \frac{5}{12}$

(1/3) < (5/12) < (1/2) " "?

$\frac{1}{2}$ is the same thing as $\frac{6}{12}$

$\frac{1}{3}$ is the same thing as $\frac{4}{12}$

so

$\left(\frac{4}{12}\right) < \left(\frac{5}{12}\right) < \left(\frac{6}{12}\right)$ is true!

Dec 20, 2017

There are many ... but $\frac{5}{12}$ is exactly half-way between them.

#### Explanation:

There are many fractions between $\frac{1}{2} \mathmr{and} \frac{1}{3}$

If you use the LCD you end up with $\frac{3}{6} \mathmr{and} \frac{2}{6}$.
The values between $2 \mathmr{and} 3$ are all fractions.

However, use a larger value in the denominator:

$\frac{6}{12} \mathmr{and} \frac{4}{12}$

Now it is easy so see that a fraction exactly between them is $\frac{5}{12}$

Consider an even bigger value in the denominator:

$\frac{12}{24} \mathmr{and} \frac{8}{24}$

Now we have the fractions: $\frac{9}{24} , \frac{10}{24} , \frac{11}{24}$ lying between them.

You can continue in this using larger and larger values and each time you will find more and more fractions between $\frac{1}{2} \mathmr{and} \frac{1}{3}$

There are infinitely many equivalent fractions.

$\frac{2}{5}$ is one of them.

#### Explanation:

The fraction $\frac{a + c}{b + d}$ is between $\textcolor{red}{\frac{a}{b}}$ and $\textcolor{b l u e}{\frac{c}{d}}$ for any integers $a , c$ and positive integers $b , d$.

Proof:
Assuming that $\frac{a}{b} < \frac{c}{d}$

$\textcolor{red}{\frac{a}{b}} = \textcolor{red}{\frac{a}{b}} \cdot \frac{b + d}{b + d}$

$= \textcolor{red}{\frac{a}{b}} \cdot \frac{b}{b + d} + \textcolor{red}{\frac{a}{b}} \cdot \frac{d}{b + d} <$

$< \frac{a}{b} \cdot \frac{b}{b + d} + \frac{c}{d} \cdot \frac{d}{b + d} <$

$< \textcolor{b l u e}{\frac{c}{d}} \cdot \frac{b}{b + d} + \textcolor{b l u e}{\frac{c}{d}} \cdot \frac{d}{b + d}$

$= \textcolor{b l u e}{\frac{c}{d}} \cdot \frac{b + d}{b + d} = \textcolor{b l u e}{\frac{c}{d}}$

The middle term
$\frac{a}{\cancel{b}} \cdot \frac{\cancel{b}}{b + d} + \frac{c}{\cancel{d}} \cdot \frac{\cancel{d}}{b + d} = \frac{a}{b + d} + \frac{c}{b + d} = \frac{a + c}{b + d}$

That means
$\frac{a}{b} < \frac{a + c}{b + d} < \frac{c}{d}$

In our case the fraction between $\frac{1}{3}$ and $\frac{1}{2}$ is $\frac{1 + 1}{3 + 2} = \frac{2}{5}$

It's a nice trick if you need a fraction between fractions quick.

Ford's circles are closely related to that.