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She can eat 21 hot dogs in 6 minutes.
She would eat 35 hot dogs in m minutes.

The proportion can be:
[tex]\frac{21}{35}=\frac{6}{m}[/tex] or [tex]\frac{21}{6}=\frac{35}{m}[/tex]

The second option is correct.

Solution:
[tex]\frac{21}{6}=\frac{35}{m} \ \ \ \ \ \ \ \ |\times m \\ \frac{21}{6}m=35 \ \ \ \ \ \ |\times \frac{6}{21} \\ m=35 \times \frac{6}{21} \\ m=\frac{210}{21} \\ m=10[/tex]

Mika would need 10 minutes to eat 35 hot dogs.
Winkdude
Mika can eat 21 hot dogs in 6 minutes. So Mika eats 21 hot dogs/6 minutes
So Mika eats 35 hot dogs/m minutes. The proportion is set up like this:
21/6 = 35/m Then, cross-multiply:
6·35 = 21m
m = 210/21
m = 10 minutes