Let a=1/x, b=1/y, c=1/z. The ratios sum to 10, so we can represent a+b+c=10n where n is a number.
Therefore a=2/(10n), b=3/(10n), c=5/(10n). The ratio a:b:c=2:3:5.
x=1/a=5n, y=1/b=10n/3, z=1/c=2n.
If we let n=3: x=15, y=10, z=6.
1/15:1/10:1/6 should be 2:3:5. Multiply by 30: 2:3:5, so the ratios are correct.
x, y, z do not have absolute values, only relative values. If we had set n=6 we would have had:
x=30, y=20, z=12; 1/30:1/20:1/12 multiplied by 60: 2:3:5 is the same ratio.