Step 1: Understand the law in play.
A fixed amount of a metal always combines with chemically equivalent amounts of different elements. So the chlorine that joins the metal and the oxygen that joins the same metal must be equivalent to each other.
Step 2: Find the chlorine that combined.
The metal chloride weighs $26.7$ g and the metal in it is $5.4$ g. So the chlorine part is \[ 26.7 - 5.4 = 21.3 \ \text{g}. \] Thus $5.4$ g of metal binds $21.3$ g of chlorine.
Step 3: Use equivalent masses of Cl and O.
Chlorine and oxygen combine in equivalent ratio $35.5$ g of Cl to $8$ g of O. So $21.3$ g of chlorine is equivalent to \[ \frac{8\times 21.3}{35.5} = 4.8 \ \text{g of oxygen}. \]
Step 4: State the metal to oxygen link.
This means $5.4$ g of the metal combines with $4.8$ g of oxygen, since the metal binds equivalent amounts of either element.
Step 5: Scale up to 48 g of oxygen.
Use simple proportion. If $4.8$ g oxygen needs $5.4$ g metal, then $48$ g oxygen needs \[ x = \frac{48 \times 5.4}{4.8}. \]
Step 6: Compute and conclude.
\[ x = \frac{259.2}{4.8} = 54 \ \text{g}. \] So the mass of metal that reacts with $48$ g of oxygen is \[ \boxed{54 \ \text{g}} \]