Pick out the greatest fraction $8\dfrac{1}{4},2\dfrac{9}{{13}},4\dfrac{1}{2},1\dfrac{3}{6}.$

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Hint: The value of mixed fraction $$a\dfrac{b}{c} = \dfrac{{(a \times c) + b}}{c}$$

Complete step-by-step answer:
The given mixed fractions are $8\dfrac{1}{4},2\dfrac{9}{{13}},4\dfrac{1}{2},1\dfrac{3}{6}.$
Converting these mixed fraction into fractions

$8\dfrac{1}{4} = \dfrac{{\left( {8 \times 4} \right) + 1}}{4} = \dfrac{{33}}{4}$

$2\dfrac{9}{{13}} = \dfrac{{\left( {13 \times 2} \right) + 9}}{{13}} = \dfrac{{35}}{{13}}$

$4\dfrac{1}{2} = \dfrac{{\left( {2 \times 4} \right) + 1}}{2} = \dfrac{9}{2}$

$$1\dfrac{3}{6} = \dfrac{{\left( {6 \times 1} \right) + 3}}{6} = \dfrac{9}{6} = \dfrac{3}{2}$$

The values of these fractions are

$$\dfrac{{33}}{4} = 8.25,\;\dfrac{{35}}{{13}} \approx 2.69,\;\dfrac{9}{2} = 4.5,\;\dfrac{3}{2} = 1.5$$

If we compare the values, we get $$\dfrac{3}{2} < \dfrac{{35}}{{13}} < \dfrac{9}{2} < \dfrac{{33}}{4}$$

$$ \Rightarrow \dfrac{{33}}{4}$$ is the greatest of the four fractions.

Corresponding mixed fraction for $$\dfrac{{33}}{4} = 8\dfrac{1}{4}.$$

$$\therefore $$ The greatest fraction of the four fractions is $8\dfrac{1}{4}.$

Note: We have to find the greatest mixed fraction of the given mixed fractions. We need to know the values of these mixed fractions to find the greatest fraction. So we converted mixed fractions to rational numbers to find their values easily.
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