## Fractional Steps

#### Problem

A set of steps leading up to a platform are to be constructed. The platform is 3' 8" above the ground and each tread has a fixed thickness of 1^{1}/_{8}". For safety and comfort each riser should measure between 5 and 7 inches and practically cannot be cut more accurate than ^{1}/_{8}".

How many risers should be used and what size should they be?

#### Solution

From the bottom to the top each step comprises of a riser and a tread. However, the top step will have no tread as this is where a riser meets with the platform. To make calculations easier, and so as to work with a whole number of complete steps, we shall imagine that the top step includes a tread. Working in inches, 3' 8" = 44", so the total drop shall be considered to be 45^{1}/_{8}".

Given the requirement that each riser must be between 5 and 7 inches, the height of a single step must be between 6^{1}/_{8} and 8^{1}/_{8} inches.

To simplify calculations further, and to include the constraint that risers cannot be cut any more accurate than ^{1}/_{8}", we shall use ^{1}/_{8}" as our base unit.

Drop = 45^{1}/_{8}" 45 8 + 1 = 361 units

Minimum step height = 6^{1}/_{8}" 49 units

Maximum step height = 8^{1}/_{8}" 65 units

For practical purposes most carpenters attempt to make each step the same height and custom cut the last riser *in situ* to accomodate any errors in previous cuts. The standard method is to divide the drop by the number of steps to determine the height of each step. This value is rounded and the overall gain/loss of using this rounded value is calculated. The carptenter can then distribute this gain/loss among the steps. However, if this difference is no more than ^{1}/_{2}" it is common to allow the top or bottom step to make up the difference.

Working with the minimum height step, 361/49 7.4, which is an incomplete number of steps. However, if we used eight steps then each step would be smaller and would fall below the minimum height requirement. So the maximum number of steps we can use is seven.

Similarly, working with the maximum height step, 361/65 5.6, which means that if we tried to use 5 steps then each would be taller than the maximum requirement. So the minimum number of steps we can use is six.

6 Steps: 361/6 60.2 and 6 60 = 360, which is 1 unit short, so we could make the top step 61 units tall.

7 Steps: 361/7 51.6 and 7 52 = 364, which is 3 units over, so we could make the top step 49 units tall. But as this is at the limit of code it leaves no room for error. So using 7 51 = 357, which is 4 units short, and we could then make the top step 55 units tall.

Subtracting 9 units from these measurements (for the 1^{1}/_{8}" tread) and converting back to inches we have two practical options:

__Option 1:__ 6 steps comprising of 5 risers @ 6^{3}/_{8}" and 1 @ 6^{1}/_{2}".

__Option 2:__ 7 steps comprising of 6 risers @ 5^{1}/_{4}" and 1 @ 5^{3}/_{4}".