Question

Are these fractions equivalent or nonequivalent?

Answer 1

To compare whether or not two fractions are equivalent, they must have the same denominator. 

What is a common denominator you can make between $\frac{4m}{5n}$ and $\frac{8m}{10n}$? Some possible options are 5n or 10n.

I'll make both denominators 5n:
1) $\frac{4m}{5n}$ already has 5n in the denominator.
2) To get 5n in the denominator for $\frac{8m}{10n}$, you want to divide both the numerator and denominator by 2 (same as multiplying top and bottom by $\frac{1}{2}$). Since you're doing the same thing to the numerator and denominator, you are not changing the value of the fraction:
$\frac{8m}{10n} \times \frac{ \frac{1}{2}}{\frac{1}{2}} \\ = \frac{ \frac{8m}{2}}{\frac{10n}{2}}\\ = \frac{4m}{5n}$

Now compare the two fractions. Since $\frac{4m}{5n} = \frac{4m}{5n}$, your fractions are equivalent!

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Answer: They are equivalent