Question 1: Divide the number 1316 into 3 parts that are inversely proportional to \(\frac{2}{3};\frac{5}{4};2\). The biggest part is:

Call the three parts to be searched as x, y, z (x, y, z > 0)
Since x , y , z are inversely proportional to \(\frac{2}{3};\frac{5}{4};2\) so we have: \(\frac{2}{3}x = \frac{5}{4}y = 2z\)

Therefore \(\frac{{2x}}{3} = \frac{{5y}}{4} = \frac{{2z}}{1} \Leftrightarrow \frac{{2x}}{{3.10}} = \ frac{{5y}}{{4.10}} = \frac{{2z}}{{1.10}} \Leftrightarrow \frac{x}{{15}} = \frac{y}{8} = \frac{z }{5}\)

The sum of the three parts is 1316, so we have x + y + z = 1316

According to the property of the series of equal ratios

\(\frac{x}{{15}} = \frac{y}{8} = \frac{z}{z} = \frac{{x + y + z}}{{15 + 8 + 5} } = \frac{{1316}}{{28}} = 47\)

So x = 15.47 = 705; y = 8.47 = 376; z = 235

So the largest part is 705