# How to avoid mixing up ratios when comparing two values?

In the following question for example:

$1$ Neptune day = $18$ hours.

$1$ Earth day = $24$ hours.

How many Neptune days = $10$ Earth days?

My intial reaction was to do the following: $$\frac{18..Hours (on.Neptune)} {24..Hours (on.Earth)}=\frac{x..Neptune.Days} {10..Earth.Days}$$

Solving for $x$ gives $7.5$

By the answer is actually given by $$\frac{24..Hours (on.Earth)} {18..Hours (on.Neptune)}=\frac{x..Neptune.Days} {10..Earth.Days}$$

Solving for $x$ giving $12.33$

The confusing part is that on the Left, it is Earth/Neptune, but on the right, it it Neptune/Earth. This lead me to write the equation wrong intially as I would have thought it would be Neptune/Earth=Neptune/Earth.

$$\frac{24..Hours (on.Earth)} {18..Hours (on.Neptune)}=\frac{x..Neptune.Days} {10..Earth.Days}$$

Or in more simple terms:

$1$ Neptune day = $18$ hours.

$1$ Earth day = $24$ hours.

In $1$ Earth day ($24$ Hours), we have $24/18$ Neptune Days. In $10$ Earth days, we have $10*24/18=12.3$ Neptune Days.

This way is quite clear and I wouldn't usually make a mistake in it. But I would like to be able to do it using the ratios to mechanise the process (as I need to do it in exam conditions where time is limited).

But I often end up getting the ratios the wrong way around.

$$\begin{eqnarray}10 \text{ Earth days} & = & 10 \text{ Earth days} \times 1 \times 1 \\ & = & 10 \text{ Earth days} \times \frac{24 \text{ hours}}{1 \text{ Earth day}} \times \frac{1 \text{ Neptune day}}{18 \text{ hours}}\\ & = & \frac{10 \text{ Earth days}}{1 \text{ Earth day}} \times \frac{24 \text{ hours}}{18 \text{ hours}} \times 1 \text{ Neptune day}\\ & = & 12.33 \text{ Neptune days}\end{eqnarray}$$