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x, y - the numbers

The sum of two numbers is 150.
[tex]x+y=150 \ \ \ |-y \\ x=150-y[/tex]

The ratio of the two numbers is 3:2.
[tex]\frac{x}{y}=\frac{3}{2} \ \ \ |\times y \\ x=\frac{3}{2}y[/tex]

Set both expressions equal to each other:
[tex]x=x \\ 150-y=\frac{3}{2}y \ \ \ |+y \\ 150=\frac{3}{2}y+y \\ 150=\frac{3}{2}y+\frac{2}{2}y \\ 150=\frac{5}{2}y \ \ \ |\times \frac{2}{5} \\ 150 \times \frac{2}{5}=y \\ 30 \times 2=y \\ y=60 \\ \\ x=150-y=150-60=90[/tex]

The numbers are 60 and 90.
The bigger number is 90.

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