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## cross multiply

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In last Friday’s Open Thread discussion, Gretchen posted this question:

My husband’s company does not provide health insurance for me and the kids, which is a \$12,000 value. In his field, there is a salary scale based on education, number of years experience, geography, etc. The salary scale assumes that the employer provides health insurance for the family. His salary is currently at 79% of the scale, and his employer wants to eventually get him up to 100%. But that doesn’t include the insurance, so it won’t really be at 100% and is not now really at 79%. But I can’t figure out which way to do the math so he can show them the actual percentage. They’re saying he’s at 79 percent. I’m saying it’s lower because they aren’t accounting for that \$12K.

All of that boils down to this: What percent of the salary scale is Gretchen’s husband actually making, given that he, and not his employer, pays the \$12,000 bill for insurance? There are two steps to this problem:

1. Find the actual salary that is at 100% of the scale.

2. Find the actual percent of Gretchen’s husband’s salary, minus the cost of insurance.

I’m going to tell you up front that we’re going to use a proportion here.  What is  proportions?  A proportion is two equal ratios.  So, if you have two fractions with an equal sign between them, you have a proportion.

And how did I know to use a proportion?  Well, my big clue was that we’re working with percents.  Percent means “per one hundred,” and per one hundred means “out of one hundred,” which just means, “put the percent value over 100.” In other words:

[pmath]79% = 79/100[/pmath]

The tricky part is figuring out what the proportions should be.

Step 1:

[pmath]salary/x = 79/100[/pmath],

where “salary” is Gretchen’s husband’s salary, and x is the top salary on the scale.

That’s because the company assumes that your husband’s salary is 79% of the scale. (Notice this: “salary” and “79″ are in the numerators — or top values of the fractions.)

To solve this proportion, we need to plug in Gretchen’s husband’s salary and then solve for x. In order to make this easy to explain, I’m going to assume that his salary is \$100,000.

substitute:   [pmath]{\$100,000}/x = 79/100[/pmath] cross multiply:   [pmath]{\$100,000*100} = 79x[/pmath] simplify:    [pmath]{\$10,000,000} = 79x[/pmath] solve for x:    [pmath]\$126,582 = x[/pmath]

So if his salary is \$100,000, the top salary on the scale is \$126,582.

Step 2:

[pmath]{\$100,000-12,000}/{126,582} = p/100[/pmath],

where p is the actual percent of the scale.

Let’s look carefully at this proportion: The first ratio is just the salary minus the cost of insurance, over the max salary in the scale.  (That’s what we found in step 1.)  The second ratio is just like the second ratio in step 1, except that we don’t know what the percent is.

Now, pay close attention to this.  Check the top numbers to be sure they match. We want to know the actual percent of the scale that Gretchen’s husband is making — and that’s what’s represented in the top number of each ration.

Check the bottom numbers to be sure they match.  Do they?  Why yes!  Yes they do!  That’s because \$126,582 is 100% of the salary scale.

(Unlike my 10-year-old daughter’s outfits, math is very matchy-matchy.  Knowing that will help you organize your problems and check to see if they’re set up properly.)

Now all we need to do is solve for p.

simplify:    [pmath]{\$88,000}/{126,582} = p/100[/pmath] cross multiply:     [pmath]{\$88,000*100} = {126,582p}[/pmath] simplify:       [pmath]{\$8,800,000} = {126,582p}[/pmath] solve for p:      [pmath] 69.5 = p[/pmath]

So what does this mean? If Gretchen’s husband makes \$100,000 a year and is paying \$12,000 for insurance, he’s earning 69.6% of the salary scale.

If you made it this far, you get a gold star!  Pat yourself on the back, and take the rest of the day off.  This is a complex problem that depends on an understanding of proportions and how to solve for a variable in an algebraic equation.

Never fear!  I’ll unravel some of these mysteries in later blog posts.  And of course, if you have a question, ask it in the comments section!