Why is $\frac{1}{2} - \frac{5}{15} = \frac{15}{30} - \frac{10}{30}$?

Question

Sorry I am a math newbie.

Why is $$\frac{1}{2} - \frac{5}{15} = \frac{15}{30} - \frac{10}{30}?$$

Answer 1

\begin{align}\frac{1}{2} - \frac{5}{15} &= \frac {15}{15}\times\left(\frac{1}{2}\right)\ -\ \, \frac22 \times \left(\frac{5}{15}\right) \\ & = \frac {15}{30} \ - \ \,\frac{10}{30}\\ & =\frac{15- 10}{30}\\ &= \frac 5{30}\\ & =\frac16 \end{align}

Answer 2

Get common denominators because multiplying a number by $1$, does not change its value so multiple the top (numerator) and the bottom (denominator) with the same number:

$\frac{1*15}{2*15}=\frac{15}{30}$ see how the value does not change as $\frac{15}{15}=1$ and multiplying by $1$ does not change value.

$\frac{5*2}{15*2}=\frac{10}{30}$ just like above, $\frac{2}{2}=1$ and multiplying by $1$ does not change value of your number.

Answer 3

This fact follows from the nature of fractions.

The short answer is that the value of a fraction doesnt change if you multiply both the nominator and the denominator with the same non-zero number.

So

$$ \frac{1}{2}=\frac{1\cdot 2}{2\cdot 2}=\frac{2}{4} $$

you could also multiply with $3$ for example

$$ \frac{1}{2}=\frac{1\cdot 3}{2\cdot 3}=\frac{3}{6} $$

More generally

$$ \frac{a}{b}=\frac{a\cdot n}{b\cdot n}, \qquad where \quad b,n\neq 0 $$

So in your exact example

$$ \frac{1}{2}=\frac{1\cdot 15}{2\cdot 15}=\frac{15}{30} $$

and

$$ \frac{5}{15}=\frac{5\cdot 2}{15\cdot 2}=\frac{10}{30}. $$

A more complicated answer would say that fractions are partitioned into equivalence classes where the equivalence relation would be

$$ \frac{a}{b}=\frac{c}{d}\iff ad=bc, \qquad where \quad b,d\neq 0 $$

and fractions being in the same equivalence class are considered having the same value (you surely not after the complicated answer I just added it for completeness sake)

Hope this helped.

Answer 4

Formally, two fractions $\frac{a}{b}$ and $\frac{c}{d}$ where $a,b,c,d$ are all integers and $b,d$ are both nonzero are defined to be equal if and only if $ad = bc$. You have as a result things like $\frac{1}{2}=\frac{15}{30}=\frac{1300}{2600}=\dots$

We can then further define subtraction of two fractions as $\frac{a}{b}-\frac{c}{d} = \frac{ad-bc}{bd}$ where the subtraction and multiplication that occur within the numerator and denominator are the usual operations as defined for integers.

In your case $\frac{1}{2}-\frac{5}{15}=\frac{1\cdot 15 - 2\cdot 5}{2\cdot 15} = \frac{15-10}{30}=\frac{5}{30}$ which can be seen to be equal to $\frac{1}{6}$

Answer 5

On the left side is $\frac{1}{2} - \frac{5}{15}$. Half of the small squares are coloured, but $5$ blocks of rectangles have been removed, where each block is one $\frac{1}{15}$th or $\frac{2}{30}$th of the whole part.

On the right side is $\frac{15}{30} - \frac{10}{30}$. $15$ of the small squares have been coloured, but $10$ of them have been removed.

The two pictures are exactly the same, hence $\frac{1}{2} - \frac{5}{15} = \frac{15}{30} - \frac{10}{30}$.