Rate Problems: Draw the Table EVERY Single Time

Here's a scene I watch play out every single week. Say you're eight questions into a GRE Quant section. Two trains leave two cities. You read it once, nod, and start juggling numbers in your head. Sixty miles an hour, three hours, the other train left later... You get an answer. It isn't one of the choices. So you read the problem again. And again. Two minutes are gone and you still don't have it.

My verdict: that wasn't a math problem. You can multiply. You can divide. A 12-year-old can do every calculation in a rate problem. The question fell apart because you tried to hold five numbers in your head at once while a clock was running. The fix takes about ten seconds, and it works EVERY single time. Draw the table.

Rate problems are one equation wearing a costume

A rate is how much of something happens in one unit of time: miles per hour, pages per minute, gallons per hour. Every rate problem on the GRE runs on one equation, whether it's about trains, pipes, printers or people painting fences:

Rate × Time = Amount

That's it. Travel problems call the amount "distance." Work problems call it "jobs" or "pages." Same equation. The test just changes the costume so you don't notice.

So why do these questions eat so much time? Stop reading and think about it. The last time you missed one, what actually went wrong?

I'll bet it was one of these: you mixed up whose rate was whose, you mixed minutes with hours, or you averaged two numbers you had no business averaging. None of those is a math mistake. They're ORGANIZATION mistakes. And organization mistakes get fixed with paper.

The table

At the test center you get scratch paper. Use it. Before you calculate ANYTHING, draw three columns: Rate, Time, Amount. Give each person, machine or leg of a trip its own row. If things work together, or the question asks about a whole trip, add one more row for that.

Then follow three rules:

• Across any row, rate × time = amount. Know two boxes and you can fill the third.
• For legs of a trip done one after another, the times add and the amounts add. The rates do NOT.
• For things working at the same time, the rates add. That's the only time rates add.

Fill in boxes as you read. Put the units in the column headers. Circle the box the question asks for. Now the problem isn't in your head anymore. It's on the paper, and the empty boxes tell you exactly what to do next.

Try it on a work problem

Notice what the on-screen calculator could and couldn't do there. It can divide 300 by 50. It has no idea which numbers belong together. The table does that part.

Try it on a Quantitative Comparison

A Quantitative Comparison question gives you two quantities, A and B, and you decide which is bigger. The four answer choices are always the same. It's one of the question types you'll see in the GRE Quant section, along with regular multiple choice and Numeric Entry, where you type in your own number.

The mistakes I see every week

1. Doing it in your head. You feel fast for twenty seconds, then you reread the problem three times. INSTEAD: draw the three columns before you touch a number. You get two Quant sections of 20 questions in 35 minutes each, which is an average of 1 minute 45 seconds per question. A ten-second table that saves you one reread pays for itself.
2. Averaging speeds or times. It feels fair. It's almost always wrong. INSTEAD: build a Total row and divide total amount by total time.
3. Mixing units. A rate in miles per hour and a time in minutes will quietly wreck your answer. INSTEAD: write the units in the column headers and convert BEFORE a number goes in a box. 45 minutes goes in as 3/4 of an hour.
4. Adding rates that shouldn't be added. Outbound and return rates never add. INSTEAD: add rates only in a "together" row, when things work at the same time.
5. Panicking at percent changes. "Machine B works 25% faster than Machine A" freezes people. INSTEAD: turn the percent into a fraction and change the rate box. 25% is 1/4, so the new rate is 5/4 of the old one. If A prints 20 pages per minute, B prints 20 × 5/4 = 25. Percents are just fractions in disguise, and that's never more useful than inside a rate box.

There's a sixth one hiding under all five: thinking "this one's easy, I don't need the table." That's pride talking. The easy ones are exactly where the silly mistakes live, and silly mistakes on gettable questions cost you more points than the monster questions ever will. You have to decide what you want MORE: a higher Quant score, or the feeling of not needing scratch paper.

Your assignment for today

Pull out every rate problem you've missed in your practice. Redo each one, table first, before you calculate anything. Time yourself. I'll bet most of them go FASTER than your first try, and I'll bet you get them right. Then tomorrow, do the same with the rate problems you got right but felt shaky on. Draw the table every single time, even when it feels too easy. ESPECIALLY then.