Source: Married to the Sea.

This started out as an Old MacDonald’s farm question. No wait, I thought to myself, not depressing enough.

The prize this week: You’ll get the satisfaction of knowing that you probably solved this problem in less time than I spent staring at my computer screen trying to come up with a clever prize for this week. I swear I used to be more creative.

In Wendell’s house, the ratio of unopened credit card offers to out-of-date phone books is 9 to 5. The ratio of magazines to crushed loose cigarettes is 25 to 7, and the ratio of McDonald’s Happy Meal toys to rotting, half-eaten pizzas is 3 to 2. There are 6 used-up batteries lying around for each broken VCR. The ratio of crushed loose cigarettes to McDonald’s Happy Meal toys is 5 to 8, and the ratio of used-up batteries to out-of-date phone books is 5 to 7. There are 30 magazines for each 4 broken VCRs. If there were 108 unopened credit card offers, how many rotting, half-eaten pizzas would Wendell have in his house?

Please note that, as usual for the weekend challenge, I’m taking a concept that SAT has been known to test, and extending it to an extreme to which the SAT would not go (not only in subject matter, but in scope as well). All of this is to say that these weekend challenges are meant to be fun for you, not to freak you out if you can’t get them. They’re usually a degree or two harder than what the SAT will throw your way.
Put your answers in the comments. I’ll post a solution Monday (probably late in the day). Good luck!UPDATE: Nice work, Chris. And special thanks to Catherine from kitchen table math, who pointed out that my original wording (“If there are…how many does…”) made non-integer quantities (fractional batteries??) quite unsavory. I’ve changed the wording of the last sentence a bit now for the benefit of people who find this in the archives and want to torture themselves with this question later.

Solution below the cut.

The SAT would never throw such a complex question at you, but the solution I advocate is one that might help you on the harder ratio questions the SAT will toss your way. Remember that on ratio questions, units are paramount. When you’re presented with ratios of more than two things and asked to suss out the relationship between just two of those things, the best and most elegant solution is to line up all the fractions and multiply them together, eliminating unwanted units along the way. Let me show you what I mean:

What we’re given: a bunch of ratios relating together the following things (in order of appearance):

credit card offers (CCO)
phone books (PBK)
magazines (MGZ)
cigarettes (CIG)
Happy Meal toys (HMT)
half-eaten pizzas (HEP)
batteries (BAT)
VCRs (same, duh.)

What we want: the ratio of unopened credit card offers (CCO) to rotting, half-eaten pizzas (HEP). Once we have that ratio, then we’ll deal with the 108 CCO.

How we get there: Start by listing the ratios that contain the units you want. Make sure to put CCO on top, and HEP on bottom.

Simplify…

We need to get rid of PBK and HMT, so let’s find some ratios we can use to do so, one at a time.

Simplify…

And then we just keep going. This might get monotonous (it’s a challenge question), but it’s really just the same procedure over and over again until the desired result.

Simplify…

Repeat:

Simplify…

Repeat:

Simplify…

Repeat:

Gasp…Deep breath…Simplify…

That wasn’t so bad now, was it?

Now set up a proportion to see how many HEP will exist if there are 108 CCO:
Note: Although it will get gnarly with things like fractional batteries, if you’re good with your calculator you can also do this by plugging in 108 for CCO and solving consecutive ratios until you end at 16 HEP. I’m not going to spell that way out here. For some more realistic SAT-style questions that can be solved similarly, look here.