Answer

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**Hint:**In this problem, we have to find the multiplication and division of the given mixed fractions. We should first know that an improper fraction is a fraction, where the numerator is greater than or equal to the denominator. Here we can convert the mixed fraction into its improper fraction by multiplying the whole number term and the denominator and adding it to the numerator. We can then multiply and divide them to get the required solution.

**Complete step-by-step solution:**

1. Divide \[5\dfrac{3}{5}\div 2\dfrac{13}{15}\]

We can now convert the given mixed fractions to its improper fraction.

We can first take the whole number part of the first term, multiply the whole number part 5 and the denominator 5 and add the number 25 to the numerator 3. Write the result 28 in the numerator and the denominator remains the same there.

We can now write the mixed fraction, we get

\[\Rightarrow \dfrac{28}{5}\]

Similarly, we can convert the second term, we get

\[\Rightarrow \dfrac{43}{15}\]

We can now divide the terms, we get

\[\Rightarrow \dfrac{28}{5}\div \dfrac{43}{15}\]

Which can be written as,

\[\Rightarrow \dfrac{\dfrac{28}{5}}{\dfrac{43}{15}}=\dfrac{28}{5}\times \dfrac{15}{43}\]

We can now simplify the above step, we get

\[\Rightarrow \dfrac{28}{5}\times \dfrac{15}{43}=\dfrac{84}{43}\]

Therefore, \[5\dfrac{3}{5}\div 2\dfrac{13}{15}=\dfrac{84}{43}\]

2. Multiply \[1\dfrac{7}{8}\times \dfrac{16}{21}\times 1\dfrac{13}{15}\]

We can now convert the above mixed fraction into improper fraction, as we done in the above problem, we get

\[\Rightarrow \dfrac{15}{8}\times \dfrac{16}{21}\times \dfrac{28}{15}\]

We can now multiply the above terms, we get

\[\Rightarrow \dfrac{56}{21}\]

Therefore, \[1\dfrac{7}{8}\times \dfrac{16}{21}\times 1\dfrac{13}{15}=\dfrac{56}{21}\]

3. We can now find the smallest number between \[\dfrac{1}{20}\div \dfrac{1}{5}\] or \[\dfrac{1}{5}\div \dfrac{1}{20}\]

We can now divide the given fraction and reciprocal the fraction, by multiplying the denominator inversely to the numerator, we get

\[\Rightarrow \dfrac{\dfrac{1}{20}}{\dfrac{1}{5}}=\dfrac{1}{20}\times \dfrac{5}{1}=\dfrac{1}{4}=0.25\]……. (1)

Similarly, we can write the next terms as,

\[\Rightarrow \dfrac{\dfrac{1}{5}}{\dfrac{1}{20}}=\dfrac{1}{5}\times \dfrac{20}{1}=4\]….. (2)

Now by comparing (1) and (2), we can say that (1) is smaller.

**Therefore, \[\dfrac{1}{20}\div \dfrac{1}{5}\] is smaller than \[\dfrac{1}{5}\div \dfrac{1}{20}\].**

**Note:**Students make mistakes while converting the mixed fractions into its improper fraction, by multiplying the whole number part to the denominator and adding it to the numerator and finally writing the result in the numerator. We should concentrate while cross multiplying the numbers.

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