Ramp Slope and Unit Conversion
A school is installing a wheelchair ramp at an entrance where the door threshold is 30 inches above the sidewalk. The available horizontal run is 30 feet. A student calculates the slope as rise/run = 30/30 = 1 and concludes that the ramp is far too steep, with a 100% grade. The student recommends a 60-foot run because then the slope would be 30/60 = 0.5. A second student says the units are inconsistent and that the 30-inch rise must be converted to feet before dividing. The district requires a slope no steeper than 1:12. The principal wants to know whether the existing 30-foot run is compliant, and how much run would be needed if the threshold were raised to 36 inches. The class is asked to evaluate both students' reasoning, use a common unit, and explain the result in terms of accessibility rather than only the numerical slope.
1. Evaluate both students' slope calculations. Is the 30-foot run compliant with the 1:12 requirement? Show the conversion and compare the slopes.
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The first student divided 30 inches by 30 feet, mixing units. Convert 30 inches to feet: 30 / 12 = 2.5 ft. Correct slope = 2.5 / 30 = 1/12, or in inches 30 / 360 = 1/12. The requirement is maximum 1:12, so a 1:12 slope is exactly at the limit and is compliant, assuming the rule is 'no steeper than.' The first student's slope of 1 is not a valid comparison. The second student is right about units. The 60-foot recommendation would give 2.5 / 60 = 1/24, which is gentler than required; it is not necessary for compliance, though it may be desirable.
2. If the threshold is raised to 36 inches, what minimum horizontal run is needed to keep the slope at or below 1:12? How many additional feet beyond the existing 30-foot run is that?
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36 inches = 3 ft. For slope 1:12, run = 12 × rise = 12 × 3 = 36 ft. In inches: 36 × 12 = 432 in = 36 ft. The existing 30 ft is short by 6 ft. Equivalently, the maximum allowable rise on 30 ft is 30 / 12 = 2.5 ft = 30 in; a 36-in rise exceeds that by 6 in, requiring 6 ft more run at 1:12.
3. A contractor argues that a 1:10 slope is 'close enough' to 1:12 because the difference is only 2. Use percent grade and relative comparison to evaluate this argument in an accessibility context.
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1:10 means 1/10 = 0.10 = 10% grade. 1:12 means 1/12, about 0.0833 = 8.33% grade. The difference is 1.67 percentage points, but relative to the maximum allowed grade the 1:10 ramp is 0.10 / 0.0833, or about 1.20, meaning it is 20% steeper. For a person using a wheelchair, that extra steepness increases effort and risk; 'close enough' is not a reasonable standard when a code maximum exists. The 1:10 slope also requires more force and could make stopping or turning harder. It should not be approved as compliant.