US options once quoted in sixteenths. Express 11/16 as a percentage instantly, and give the general method for any denominator that is a power of two.

US options once quoted in sixteenths. Express 11/16 as a percentage instantly, and give the general method for any denominator that is a power of two.

Approach: Build the table by halving 100 repeatedly, then multiply the single-unit value by the numerator using a quarter-of-25 rewrite.

68.75%. One sixteenth is 6.25%, which comes from halving 100 four times: 50, 25, 12.5, 6.25. So 11/16 as a percentage is 11 * 6.25, and the multiplication is easier written as 11 * 25/4 = 275/4 = 68.75. Any denominator that is a power of two converts exactly, because 100 can be halved indefinitely in decimal, and the whole table is worth holding: 3/8 is 6/16 or 37.5%, 5/8 is 62.5%, 7/16 is 43.75% and 13/32 is 40.625%. A denominator carrying a factor of 3 or 7 never terminates, which is the quick test for whether the conversion is exact or an approximation.

Follow-up: A market is quoted 3/16 wide around a mid of $4.50. What is the half-spread as a percentage of the mid?

Key concepts: sixteenths, halving, percentage conversion.