A ratio compares two quantities of the same kind, written a : b. Multiplying or dividing both terms by the same non-zero number does not change a ratio, which is why 6 : 10 and 3 : 5 are identical.
To divide an amount in a given ratio, add the ratio terms and give each share its fraction. Dividing 64 in the ratio 3 : 5 means eight parts in total, so the shares are 24 and 40.
In a proportion a : b = c : d the product of the extremes equals the product of the means, so ad = bc. This cross-multiplication is the workhorse of most proportion questions.