If difference between the \(\frac{4}{5}\) of \(\frac{3}{4}\) of a number and \(\frac{2}{5}\) of \(\frac{1}{6}\) of the same number is 648 , then number is

To find the number based on the given condition, let’s denote the number as \(N\). The condition states:

\[
\frac{4}{5} \times \frac{3}{4} \times N – \frac{2}{5} \times \frac{1}{6} \times N = 648
\]

Let’s simplify the equation step by step:

\[
\left(\frac{4}{5} \times \frac{3}{4}\right)N – \left(\frac{2}{5} \times \frac{1}{6}\right)N = 648
\]

Multiplying the fractions:

\[
\left(\frac{12}{20}\right)N – \left(\frac{2}{30}\right)N = 648
\]

Simplifying the fractions:

\[
\left(\frac{3}{5}\right)N – \left(\frac{1}{15}\right)N = 648
\]

Finding a common denominator to combine the fractions:

\[
\left(\frac{9}{15} – \frac{1}{15}\right)N = 648
\]

Subtracting the fractions:

\[
\frac{8}{15}N = 648
\]

Solving for \(N\):

\[
N = \frac{648 \times 15}{8}
\]

\[
N = 81 \times 15
\]

\[
N = 1215
\]

Therefore, the number is \(\boldsymbol{1215}\).