If a fraction is multiplied by itself and then divided by the reciprocal of…

Concept

For a non-zero fraction x, dividing by its reciprocal is the same as multiplying by x.

Therefore, multiplying x by itself and then dividing by 1/x produces x3.

Application

1. Let the required fraction be \(x\).

2. Convert the mixed number: \(49\frac{8}{27}=\frac{49\times27+8}{27}=\frac{1331}{27}\).

3. The stated operation gives x × x ÷ (1/x) = x3, so

   x3 = \(\frac{1331}{27}\).

4. Since \(1331=11\times11\times11\) and \(27=3\times3\times3\), \(x=\frac{11}{3}=3\frac{2}{3}\).

Cross-check

For \(3\frac{2}{3}=\frac{11}{3}\), the operation gives \(\frac{11}{3}\times\frac{11}{3}\div\frac{3}{11}=\frac{1331}{27}=49\frac{8}{27}\), reproducing the given result.

Result

The required fraction is \(3\frac{2}{3}\).