Slopes and how to calculate them are something you’re going to encounter on all six ARE 5.0 exams. Topography, plumbing and sewers, code and ADA ramps all have to do with different slopes, so it’s a good idea to have a good understanding of how slope math works.
In this post I'm going to show how to solve a few typical slope problems. We'll start with a simple rise over run and then look at some more complicated examples with mixed units or missing information.
This is all taken from the PA Study Questions course. Check out that page for more info and to get a free sample, which includes Assignment 1 with all the work shown here.
And if you can't read this post because you're driving you can always watch this on YouTube. Check out the ARE 5.0 Slope Math video.
Alright, enough talk. Let's get started.
ARE 5.0 SLOPE MATH EXAMPLE 1: FINDING SLOPE PERCENTAGE
In the first one you are given a rise and run and asked to find what’s the slope percentage. 
The slope is just rise over run. The rise is the vertical and the run is the horizontal. It can be expressed as a ratio, like 1:8 or as a percentage, like 12.5%.
If you're given some unit-less numbers like in this example, you can easily rewrite the slope to a ratio, in this case 3:12. To go to a percentage you can rewrite this as 3/12, which equals 0.25, so the slope is 25%. This is very steep!
ARE 5.0 SLOPE MATH EXAMPLE 2: SOLVE FOR THE MISSING RISE
This next one is a little more tricky. You’re given the slope and you’re given the run, and you’re asked to find the rise. On the ARE a lot of times the question might be simple, but they make it more complicated by changing units or giving mixed units. In this case you are given a run of 15 feet and a slope of 5 percent, but you’re asked to find the rise in inches. 
Just like we saw before, rise over run = slope. To find the missing variable we can rewrite it as x/15 = 5%, then solve for X. That becomes X = .05 • 15’ so X =0.75’. Remember, the question asked for rise in INCHES, so you're not done yet. 0.75 ft = 9 in because 0.75 * 12 = 9.
If you think you might not remember to do the conversion at the end, you can convert at the earliest point possible. Right at the beginning we can change X/15′ to X/180 and then convert it right away, so 15’ • 12 = 180”. So, then rewrite that as X = .05 • 180. The answer comes out to 9”.
SOLVE FOR MISSING RUN
In this example you're given a rise of 2’, you have to figure out what the run is, and you’re given a 1:12 slope. That’s just a ratio so it doesn’t have any units.
We set up our equation so 2/x = 1/12. So, you just cross multiply and divide, (x)(1) = (2)(12). X= 24, and since the only unit they gave was feet the answer is in feet, so the answer is X=24’.
FINDING THE RISE PER 100 FEET
Another way you can see the slope presented is in a value per 100’. So, in this case you have a slope of 4’ for every 100’. So that means for every group of 100’ you have you go up or down 4’.
In this question you have a run of 33’ and you are asked to find the rise in inches. The quickest way to set this one up is to figure out how many groups of 100’ you have first.
33 divided by 100 gives us .33 groups of 100’. Each of those groups I move up 4’, so .33 multiplied by 4 feet, you get 1.32’. Our answer is inches, so multiply by 12 to convert to inches and you get 15.84”. Round that up to 16.
Just be aware that in the exam you’ll be told to round. If you’re confused by that, we’ll go over rounding with more detail in the last example.
Another way to set this one up is you can always change the slope to a ratio. So 4’ in 100’ is the same as saying 1:25. So you rewrite that as your run is 33’, your rise is unknown and your slope is 1:25 Then you just set that up as before. x/33 = 1/25, cross multiply and divide and you end up with the exact same values. x=1.32’ is 15.84”. Then round up to 16”.
SLOPE 5
So the last example is probably the trickiest one. You’re given a rise of 1’-3”, you have to figure out the run and you’re given a slope of ⅛” : 12”. That’s always tricky because you might think it's 8/12, but be careful, it's actually one-eigth over 12. So, it’s actually .125 divided by 12.
The best way to set this one up, again, is a ratio. 1’3”/x = ⅛”/ 12. Let’s rewrite that into something more simple because it’s hard to do 1’-3”. 3” is .25 of a foot, so that becomes
1.25’ /x = .125/12 . When you’re dealing with two things that are the same unit, like this, the inches will cancel each other out and it just becomes a ratio. Again, you cross multiply and divide. You have .125x = 1.25’•12, and that works out to 120’.
The other way you can solve this is using the same percentage equation we did earlier, but this is an area where you have to pay attention to rounding.
One thing you might recognize is that ⅛”/12 is just about a 1% slope, but it’s not exactly a one percent. So, if we round at the beginning and said that’s a 1% slope, then 1.25’/x at a 1% slope, X would work out to 125’. However, we already proved that the answer was 120’, so if we rounded early gives us an answer that’s off by 5’.
So, if you’re a little more accurate and said that ⅛” is actually closer to . 0104 and we do the math, X works out to 120.19, which can easily be rounded down to 120’.
If you’re even more accurate and said that ⅛” is actually . 01041667, then X works out to 119.999962, which obviously rounds to 120.
You will almost always be given how to round on the exam, whether it’s to the nearest foot, inch, up, down etc. However, when I’m doing these problems, I don’t round until the end. So, if I was solving this problem mathematically, I would type in the whole value of .0141667, end up with the 119.999962 answer and round up to 120. So, don’t round too early. Regardless, make sure your ARE 5.0 Slope Math is up to code for those exams!
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