$\frac{a}{b} = \frac{c}{d} $ if $ad = cb$, how to intuitively understand this?

Solution 1:

Would a picture help?

Note that $\frac{a}{b} = \frac{c}{d}$ by similar triangles. The blue and green rectangle has area $ad$ while the green and yellow rectangle has area $bc$. These are equal, namely fraction $\frac{a}{b} = \frac{c}{d}$ of the area of the big rectangle.

Solution 2:

HINT $\ $ It boils down to putting the two fractions over the common denominator $\rm\:b\:d\:,\:$ or, equivalently, changing the "unit" of measurement on your ruler from $1$ to $\rm\:1/(b\:d)\:.$

On the new ruler $\rm\ \dfrac{1}b\ $ has measure $\rm\ d\ $ since $\rm\ \dfrac{1}b\: =\ d\:\dfrac{1}{b\:d}\ $ hence $\rm\ a\:\dfrac{1}b\ $ has measure $\rm\ a\:d\:.$

Similarly $\rm\ c\dfrac{1}d\ $ has measure $\rm\:c\:b\:.$

Analogously, you can use this ruler to compare any fractions whose denominator divides $\rm\:b\:d\:.$

Solution 3:

If $\frac{a}{b}$ represents "the solution to the equation $bx=a$", then saying that $\frac{c}{d}=\frac{a}{b}$ means that any solution to $bx=a$ is a solution to $dy=c$, and vice-versa. So if $x$ is a solution to $bx=a$, then multiplying by $d$ we have $ad = dbx = b(dx)$. But since $x$ is also a solution to $dy=c$, that means that $dx=c$, so $ad=b(dx) = bc$.

So if $\frac{a}{b}=\frac{c}{d}$, then $ad=bc$.

Conversely, if $ad=bc$, and $x$ is a solution to $bx=a$, then it is also a solution to $dbx = da=bc$. Since $b\neq 0$, $dbx = bc$ if and only if $dx=c$, so $x$ is a solution to $bx=a$ if and only if it is a solution to $cy=d$.

In short, the equations $bx=a$ and $cy=d$, with $a,b,c,d$ integers, $b$ and $d$ nonzero, have the same solution if and only if $ad=bc$. So if $\frac{r}{s}$ for integers $r,s$, $s\neq 0$, represents "the solution to $sx=r$", then for integers $a,b,c,d$, $b\neq 0$, $d\neq 0$, $$\frac{a}{b}=\frac{c}{d}\text{ if and only if }ad=bc.$$

Solution 4:

Lately, I've been thinking about it this way:

If $\frac{a}{b} = \frac{c}{d} $ then $ \frac{c}{d} $ must be equal to $\frac{ka}{kb} $. Their ratio is the same if and only if $a$ and $b$ are scaled by a constant $k$.

Therefore $\frac{a}{b} = \frac{c}{d} $ can be rewritten as $\frac{a}{b} = \frac{ka}{kb} $.

If the constant $k$ is the same, which is the condition of equality, we can simply cancel it out, which will leave us with $\frac{a}{b} = \frac{a}{b} $ which is evidently true.

If we didn't want to cancel out the constant $k$, we can try the $ad = bc$ method. Let's move $kb$ and $b$ to the other sides of the equation:

$akb = kab$ -> $akb = akb$

which says that if the constant in the numerator and the denominator is different, the equality fails. Is this also a way of looking at this?