Multi Step Solutions

These are different staircases in different lighthouses. Can you tell which one is taken from the top?
Either way, that's a lot of crazy looking steps!

When building staircases, a carpenter must be extremely careful of many things.
One detail we will consider is baluster (or post) spacing.

Most building codes require that a 4" radius sphere cannot pass through the gaps between balusters.
Can you imagine getting your head stuck in a staircase?

Let's try our hand at a handrail layout and see if we can take care of some details a carpenter would need to know.

Suppose the distance between two large anchor posts (newel posts) is 38 ¾ inches. The balusters are each 1 ¼ inches wide and without curves. This handrail system is in a 4" ball rule zone. How many balusters are needed and how big should the gap be between them?

NOTE: There are 2 variables in this problem. Can you find them? One of them is called a discrete variable while the other is continuous.

Let n = the number of balusters needed. Let g = the gap measurement (in inches) between balusters. Since n is a discrete variable, we'll try to find it first. (It's discrete since only whole numbers can be values for n.) Because it is discrete, it will be easier to find than g. Consider a drawing to help what equation would be helpful. Note there is one more space than there are balusters. Each baluster is 1 ¼ inch wide. That's 1.25 inches. Since there are n balusters, the total width is 1.25n for all the balusters. Each gap (or space) measurement is unknown, but we're calling it g. There are n + 1 spaces, so the total space in the gaps is g(n + 1). Since the total width of balusters plus gaps is 38 ¾ inches, our equation must be 1.25n + g(n + 1) = 38.75 Recall that n is discrete. So we can simply guess and check reasonable values for n. Here's a table listing values of n and what our equation would be: n 1.25n + g(n + 1) = 38.75 5 1.25*5 + g(5 + 1) = 38.75 6 1.25*6 + g(6 + 1) = 38.75 7 1.25*7 + g(7 + 1) = 38.75 8 1.25*8 + g(8 + 1) = 38.75 9 1.25*9 + g(9 + 1) = 38.75 Solving for g in each of those rows would take a long time! Which one could be the right one? Since the gaps are not to exceed 4 inches, we could see what happens if g = 4. 1.25n + g(n + 1) = 38.75 1.25n + 4(n + 1) = 38.75 1.25n + 4n + 4 = 38.75 5.25n + 4 = 38.75 5.25n = 38.75 − 4 5.25n = 34.75 n = 34.75 ÷ 5.25 n = 6.619 Of course n cannot be a decimal. So the next largest number, 7, should work! Taking the row from our table above when n = 7, 1.25*7 + g(7 + 1) = 38.75 8.75 + g(8) = 38.75 8.75 + 8g = 38.75 8g = 38.75 − 8.75 8g = 30 g = 30 ÷ 8 g = 3.75 That's it! When n = 7 and g = 3.75 the spacing should work! Let's check. If n = 7, then the total for all the balusters is 1.25*7 = 8.75. If g = 3.75, then the total for all the gaps is 3.75*8 = 30. The total of baluster and gap widths is then 8.75 + 30 which is 30.75. Yes! (Note: if more than 7 balusters are used, then the gap decreases until the point at which there's no more gap! This can be expressed as 0 < g ≤ 4. )

Wow, that was a lot of crazy steps! Let's practice the algebra in the middle of that last example.

Here are some other examples where multiple steps of algebra are used to solve the equation.

Problem	Solve 1.25*9 + g(9 + 1) = 38.75
Algebra Solution	Algebra steps Reasoning 1.25*9 + g(9 + 1) = 38.75 given 11.25 + g(10) = 38.75 simplifying 11.25 + 10g = 38.75 rewriting 10g = 38.75 − 11.25 subtracting 11.25 from both sides to undo the add by 11.25 10g = 27.5 simplifying g = 27.5 ÷ 10 dividing both sides by 10 to undo the mutliply by 10 g = 2.75 simplifying

 

Problem	Solve 2.25n + 5(n + 1) = 63
Algebra Solution	Algebra steps Reasoning 2.25n + 5(n + 1) = 63 given 2.25n + 5n + 5 = 63 distributing 7.25n + 5 = 63 combining like terms 7.25n = 63 − 5 subtracting 5 from both sides to undo the add by 5 7.25n = 58 simplifying n = 58 ÷ 7.25 dividing both sides by 7.25 to undo the mutliply by 7.25 n = 8 simplifying