Could Someone Explain These Steps? Especially the Each Ratio Part.

Question

Could you explain the steps, especially the each ratio= sum of numerators/sum of denominators part.

Answer 1

This is because is,

$$\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$$

then,

$$\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{a+c+e}{b+d+f}$$

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This can be proved as follows.

Let $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = r$$

$$\implies a = rb, c = rd, e = rf$$

$$\frac{a+c+e}{b+d+f} = \frac{r(b+d+f)}{b+d+f} = r$$

So, $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{a+c+e}{b+d+f}$$

Here, $$r = \frac{y+x+y+y}{x-z+z+y} = \frac{2(x+y)}{x+y} = 2, (x+y\not=0)$$

Compare $r$ with every fraction and obtain $x:y:z$