When working with mixed numbers, it is usually easier to convert them to improper fractions, do the operations, and then to convert any resulting improper fraction back to a mixed number:
1 5/9 ÷ 1/3 = (1×9+5)/9 ÷ 1/3 = 14/9 ÷ 1/3 = 14/9 × 3/1 = (14×3)/(3×3) = 14/3 = (4×3+2)/3 = 4 2/3.
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2 over 3 divided by 1 over 5 is 3 1/3.
1 over 3 divided by 5 over 6 is 0.4.
3/5 divided by 1/7 is 21/5 or 4 and 1/5
4/5 divided by 1/2 = 1.6 or 1 3/5
The "invert and multiply" rules applies here: 1/3 divided by 4/5 = 1/3 * 5/4 = 5/12