A Rates and Speed Problems Shortcut – Now That’s Sweet!

For anyone who ate too much chocolate last week and is looking for a sweet treat that won’t add to his or her waistline, look no further: How would you like a quick way to solve rate & speed problems that have lots of variables? It probably doesn’t sound quite as good as a box of truffles does, but truffles unfortunately aren’t going to get you into grad school.

The beautiful thing about rate problem is that they virtually always are written as a fraction. Think about it: Any rate – miles/gallon, cookies/jar, heartbeats/minute – is just one term divided by another term. So when you’re setting up a rate problem in which you have variables without real numbers, figuring out where each variable goes in the fraction – the numerator or the denominator – will often get you directly to the correct answer. Let’s look at an example to demonstrate this:

Sweetheart Candies sells boxes holding p pieces of chocolate each. The boxes are shipped in crates, each holding b boxes. What is the price charged per piece of chocolate, in cents, if Sweetheart Candies charges m dollars for each crate?

(A) 100bp/m

(B) 100m/bp

(C) bp/100m

(D) m/bp

(E) bp/m

What’s this question asking us for? We’re looking for the price per piece of chocolate, in cents. The keyword “per”, along with the fact that the answer choices are all fractions, clue us into the fact that this is a rate problem. We have three variables, so let’s see where we can put them in the fraction.

Anything that comes before the “per” will go in the fraction’s numerator. In this case, that’s the price. m is the only variable that has anything to do with money, so it must be on top. If we glance back up at the answer choices right now, which answer choices can we eliminate because they don’t correctly place m? That’s right: (A), (C), and (E) are now gone.

The only difference between (B) and (D), the two remaining choices, is that (B) multiplies m by 100. Is that extra 100 a necessary component of the rate that we’ve been asked to set up? Let’s think: The problem asks for the price, in cents, but m represents a number of dollars. So yes, we absolutely need to multiply m by 100 to get from dollars to cents. And voila – choice (D) is out, and we’ve determined that (B ) is the correct answer.

The next time that you’re confronted with a tough rate problem that involves multiple variables, don’t despair – just figure out whether each variable belongs on the top of the bottom of the fraction. On Test Day, you’ll have nailed points and moved on before your competition is even done reading the problem!