1. **Stating the problem:** Mark and Tony had the same weight before vacation. Mark's weight increased by $\frac{1}{16}$ and Tony's by $\frac{1}{20}$. Together, their combined weight after vacation is 101.4 kg. We need to find their original weights. 2. **Define variables:** Let $x$ be the original weight of each friend (since they had the same weight). 3. **Express weights after vacation:** - Mark's weight after vacation: $x + \frac{1}{16}x = x\left(1 + \frac{1}{16}\right) = x\frac{17}{16}$ - Tony's weight after vacation: $x + \frac{1}{20}x = x\left(1 + \frac{1}{20}\right) = x\frac{21}{20}$ 4. **Write equation for combined weight:** $$ x\frac{17}{16} + x\frac{21}{20} = 101.4 $$ 5. **Find common denominator and combine:** The common denominator of 16 and 20 is 80. $$ x\frac{17}{16} = x\frac{17 \times 5}{80} = x\frac{85}{80} $$ $$ x\frac{21}{20} = x\frac{21 \times 4}{80} = x\frac{84}{80} $$ So, $$ x\frac{85}{80} + x\frac{84}{80} = x\frac{169}{80} = 101.4 $$ 6. **Solve for $x$:** $$ x = 101.4 \times \frac{80}{169} $$ 7. **Calculate $x$:** $$ x = \frac{101.4 \times 80}{169} = \frac{8112}{169} \approx 48.0 $$ 8. **Conclusion:** Each friend weighed approximately 48.0 kg before the vacation.