Athletic race

In a race, the second-place finisher crossed the finish line 1 1/3 minutes after the first-place finisher. The third-place finisher was 1 3/4 minutes behind the second-place finisher. The third-place finisher took 34 2/3 minutes.
How long did the first-place finisher take?

Final Answer:

a =  31 ⁷⁄₁₂ = 379/12 min

Step-by-step explanation:

$$\begin{gathered} c=34\frac{2}{3}=34+\frac{2}{3}=\frac{34\cdot 3+2}{3}=\frac{102+2}{3}=\frac{104}{3}\doteq 34.6667\text{min} \\ {t}_2=1\frac{3}{4}=1+\frac{3}{4}=\frac{1\cdot 4+3}{4}=\frac{4+3}{4}=\frac{7}{4}=1.75\text{min} \\ b=c-t_2=\frac{104}{3}-\frac{7}{4}=\frac{416}{12}-\frac{21}{12}=\frac{416-21}{12}=\frac{395}{12}\doteq 32.9167\text{min} \\ {t}_1=1\frac{1}{3}=1+\frac{1}{3}=\frac{1\cdot 3+1}{3}=\frac{3+1}{3}=\frac{4}{3}\doteq 1.3333\text{min} \\ a=b-t_1=\frac{395}{12}-\frac{4}{3}=\frac{395}{12}-\frac{16}{12}=\frac{395-16}{12}=\frac{379}{12}=\frac{379}{12}\text{min}=31\frac{7}{12}\text{min}=31.5833\text{min} \end{gathered}$$

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