x, y - the numbers
The ratio of x to y is 2:3.
\(\frac{x}{y}=\frac{2}{3} \\
3x=2y \\
x=\frac{2}{3}y\)
Their product is 600.
\(xy=600 \\ \\
\hbox{subsitute } \frac{2}{3}y \hbox{ for x and solve for y:} \\
\frac{2}{3}y \times y=600 \\
\frac{2}{3}y^2=600 \\
y^2=600 \times \frac{3}{2} \\
y^2=900 \\
y=\pm \sqrt{900} \\
y=-30 \ \lor \ y=30 \\ \\
\hbox{the numbers are positive so y=30} \\
xy=600 \\
x \times 30=600 \\
x=\frac{600}{30} \\
x=20\)
The numbers are 20 and 30.
