In nation Z, 10 terble coins equal 1 galok. In nation Y, 6 barbar coins equal 1 murb. If a galok is worth 40% more than a murb, what is the raio of the value of 1 terble coin to the value of 1 barbar coin?
I derived the following equalities:
- 10 terbles = 1 galok
- 1 galok = 1.4 murbs
- 1 murb = 6 barbars
I'm getting two different solutions depending on my approach.
Approach 1
Compute the number of barbars for a single terble coin:
$1\,terble \times\,\frac{1\,galok}{10\,terble} \times\,\frac{1.4\,murbs}{1\,galok}\times\,\frac{6\,barbars}{1\,murb} = \frac{21}{25}\, barbars.$
Since the question is asking for the fraction $\frac{terbles}{barbars}$. So,
$\frac{1\,terble}{\frac{21}{25}\,barbars} = \frac{25\, terbles}{21\,barbars}.$
Approach 2
Compute terbles and barbars in terms of a third unit, such as murbs, and form a ratio in that unit.
$1\,terble \times\frac{1\,galok}{10\,terble} \times\,\frac{1.4\,murbs}{1\,galok} = \frac{14}{100}\,murbs$
$1\,barbar\times\frac{1\,murb}{6\,barbars} = \frac{1}{6}\,murbs$
$\frac{1\,terble}{1\,barbar} = \frac{\frac{14}{100}\,murbs}{\frac{1}{6}\,murbs} = \frac{21\,terbles}{25\,barbars}$
The book uses Approach 2, but I'm stumped on why Approach 1 is producing an incorrect result.
You have $1 \mathsf{terble} = \frac{21}{25}\mathsf{barbar}$ which is okay. $\color{darkgreen}\checkmark$ Then you go all awry. You just needed to divide both side by $\color{navy}{1}$ barbar.
The ratio required is clearly a dimensionless number: $\dfrac{1\mathsf{terble}}{1\mathsf{barbar}}=\dfrac {21}{25}$