Rate-time problems are one of the biggest headaches on the math section of the GRE. You know the type– they always start out something like: “If Jane can bake four cakes in five hours and Jane and Sam together can bake 8 cakes in twelve hours…” These problems can seem quite confusing, but in fact the math behind them is very simple: once you’ve learned to translate these problems into equations, you’ll have no trouble getting them right. To find out how, read on.

What is a rate?

A rate is a number that tells you how fast a machine or person can convert one kind of thing into another kind of thing. For example:
If a car gets 40 miles per gallon, it converts 1 gallon into 40 miles of travel.
If Jim builds three houses per week, he converts 1 week into 3 houses.
If Sheila installs 4 faucets in 5 hours, she converts 5 hours into 4 faucets.

So a rate is always a fraction: it is the output per input, or the output divided by the input. If I can kill 3 bears in 12 hours, my rate is output / input = dead bears / hours = 3/12. I can reduce this fraction: 3/12 = 1/4. So I kill one bear in four hours.

A rate often shows how much time it takes to do something, but not always! For example, miles per gallon tells you how much output (miles) you get out of the input (gallons).

How do rate equations work?

The basic form of a rate equation is: input x rate = output. This is easy to remember: the rate is what gets you from input to output! So if my car gets 40 miles per gallon, that means (G gallons) x (40 miles / gallon) = (M miles).

A rate equation is an equation with three numbers; if you have any two numbers you can get the other one.

Example 1: Very often a GRE rate question will begin: “If Jane can make 8 doilies in 3 hours…” Here it looks like we have only two numbers: 8 doilies and 3 hours. But implicit in this problem is a rate, the rate at which hours turn into doilies. You should immediately convert this into a rate equation.

input x rate = output

3 hours x rate = 8 doilies

Given this information, we can find out the rate at which Jane produces doilies:

Rate = output / input = 8 doilies / 3 hours = (8/3) doilies per hour. Jane produces 2 and 2/3 doilies each hour

Example 2: By the same token, if we’re given a rate plus an input we can find an output. So if we’re told: “John can produce 5 scarves per hour. If he produces 32 scarves, how long has he been working?”

input x rate = output
hours x (5 scarves / hour) = 32 scarves

Dividing both sides by five, we get:
hours = 32 / 5 = 6.4 hours.

John has been working 6.4 hours.

Check your work! Is 6.4 hours a reasonable time? Since John produces 5 scarves in 1 hour he can produce 30 scarves in 6 hours, so 6.4 hours is a reasonable time for 32 scarves.

Rates and Units

If you get confused on a rate problem, just remember: the units should work out. So if I have a rate (scarves / hour) and I multiply it by hours (hours x (scarves/hour)) the hours cancel and I get scarves. If I have a number of scarves and I divide it by the rate (scarves / hour), I get (scarves / (scarves / hour)) = (scarves x (hours / scarf)) = hours. Keep track of the units– which ones you have, which ones you’re looking for– and you can do a rate problem even without understanding it.

Practice Problems:

1) If Jenna can factor 40 numbers in 12 minutes, what is the rate at which she can factor numbers?
2) If my car uses 60 gallons of gas when I travel 80 miles, how many gallons of gas will it use when I travel 250 miles?

How can something so simple get so complicated? Why am I still confused about rate problems?

Glad you asked! There are two kinds of rate problems that seem a lot harder than the ones I described above.

Harder Problems 1: Man-hours.

Often you’ll see a problem on the GRE like:

Example: If it takes 30 men 40 hours to build 5 machines, how many machines can 30 men build in 15 hours?

As you can see, we now have three numbers, none of which are rates! How can we fit all this into our rate equation? The secret here is to rethink our concept of “input.” Instead of thinking in terms of years, we’ll think in terms of fanatic-years: the amount of work a man can do toward producing a machine in 1 hour. If one man can do x amount of work in 1 hour, 5 men can do 5x work in one hour, right? So to get the “man-hours” we’ll multiply time x workers and use that as our input. Our rate will then be in machines / man-hour.

input = 30 men x 40 hours = 1200 man-hours.
input x rate = output
1200 man-hours x rate = 5 machines
rate = 5 machines / 1200 man-hours = (5 / 1200) machines / man-hour
Now we know the rate. To solve the problem we set up a new rate equation conforming to the second situation:
input x rate = output
men x hours x rate = machines
men = 30, hours = 15, rate = 5/1200, machines = ?

30 men x 15 hours x (5/1200 machines / man-hour) = 1.875 machines

So remember, if you get a work problem where both the time and the number of workers vary, use people-time as your unit for input.

Practice problems:
1) If it takes 80 workers 10 days to build 12 cabins, how many workers will it take to build 16 cabins in 5 days?
2) If it takes 20 frat boys 60 minutes to finish 12 six-packs, how many frat-boys will it take to finish 28 six-packs in 80 minutes?

Harder Problems 2: Different Rates

Another tough rate problem looks like this:

Example: If Alice cleans 10 dishes in 15 minutes and Bob cleans 8 dishes in 6 minutes, how many dishes can they clean in 40 minutes if they work together?

Here we have two different rates. Alice does 10/15 dishes per minute, while bob does 8/6 dishes per minute. How can we fit these together?

Think about it: Each of them will do the same amount of work while they’re working together as they would have done alone. We’re looking for the total number of dishes, but that’s just the sum of the dishes Alice cleans and the dishes Bob cleans.

Total = Alice’s dishes + Bob’s dishes
Total output = output 1 + output 2

So if we can find the output for each of them, we can find the total output by adding together. Can we?

Input (minutes) x Alice’s rate (dishes/minute) = Alice’s output (dishes)
40 minutes x (10 / 15 dishes / minute) = 26.6666 dishes

Input (minutes) x Bob’s rate (dishes/minute) = Bob’s output (dishes)
40 minutes x (8/6 dishes/ minute) = 53.33 dishes

Alice’s output + Bob’s output = 80 dishes.

Whatever you do, don’t take the average of the two rates! Just use two separate equations.

Practice Problems:

1) If Factory A produces 80 shirts per hour and Factory B produces 50 shirts per hour, how long will it take them working together to produce 300 shirts?

2) If it takes a physics major 3 hours to do a calculus problem and it takes a math major 1 hour to do a calculus problem, how long will they take to do six problems if they work together?

Next Step Test Preparation offers complete packages of one-on-one GRE tutoring for less than the price of a packed prep course.  For more information, see our GRE tutoring page, contact info@nextsteptestprep.com or call 888-530-NEXT.

Photo credit vladythephotogeek under a Creative Commons license.

You have to guess on every single GRE problem– you won’t be able to continue with the test until you’ve chosen an answer. So you might as well learn to guess well. On Quantitative Comparison questions, as I’ll show, this is especially, almost scandalously easy.

Let’s take a made-up QC question:
-1 < x < y < 1
x does not equal 0.
y does not equal 0.

Column A: x / y
Column B: y / x

A. Column A is larger.
B. Column B is larger.
C. The two quantities are equal.
D. Cannot be determined from the information given.

You have four choices; assuming you knew no math whatsoever, your odds of getting the question right would be 25%. But we can up those odds substantially with a simple trick:

Plug in Numbers

Let’s choose two numbers that satisfy the conditions. How about:
x = -0.5
y = 0.5

Now, let’s test them out. We get:

Column A: x / y = -0.5 / 0.5 = -1
Column B: y / x = 0.5 / -0.5 = -1

What we have discovered is that for the number we chose, the two quantities are equal. Now, we can’t be sure just from this that the answer is C: it may be that the two quantities are sometimes equal and sometimes not, which would indicate answer choice D. But we can be sure that neither Column A nor Column B is always larger, so we can eliminate both A and B. Just like that, without so much as thinking about it, we have narrowed the choices down to C and D and doubled our odds of guessing correctly; we now have a 50% chance.

If you’re unsure of how to attack a quantitative comparison question, the first thing you should do is plug in numbers. You will always be able to eliminate some answer choices very quickly this way.

Next Step Test Preparation offers complete packages of one-on-one GRE tutoring for less than the price of a packed prep course.  For more information, see our GRE tutoring page, contact info@nextsteptestprep.com or call 888-530-NEXT.

Photo credit Eleaf under a Creative Commons license.

The GRE is an important part of your admissions process, so it’s worth devoting the blood, sweat and tears to prepare thoroughly for it. But how should you do so? You have three options: self-study, a class, and tutoring. In this post I’ll discuss the pros and cons of each.
Self-Study
You could just buy a book, set aside some time and study for the GRE by yourself. How hard could it be, right?
Pros: It’s cheap, you can do it on your own schedule, and you’ll only work on the skills you need to improve.
Cons: Self-study is tough for a couple of reasons. First of all, anyone who has tried to teach herself something knows that it’s tough to stay organized. Most people who decide to study on their own do a good job for the first week or so, then lose interest and procrastinate until a week before the test. The other problem is that it’s tough to teach yourself what you don’t already know: You’ll keep making the same mistakes without being able to figure out how to fix them.
Take a ClassDozens of companies offer GRE prep classes, and this is the option most students choose.
Pros: A class will keep you organized, and you’ll get lots of inside information on test-taking strategies if you go with a reputable test-prep company.
Cons: A class is aimed at the average student, which means that it’s not aimed at any particular student. If you’re doing okay on the verbal section but need help with math, you’ll waste some time listening to stuff you don’t need to know about the verbal section and then feel like the math curriculum moves way too fast for you. Every test-taker has different strengths and weaknesses, but a class can’t address all of them.
Hire a TutorMany students hire a tutor to prep them for the GRE one-on-one.
Pros: In addition to organizing your study, a tutor provides education addressed specifically to your needs, goals, strengths and weaknesses. For this reason, tutoring provides the best results in terms of score improvement. If you feel you need to significantly up your score to get admitted to the programs you’d like to attend, you should consider hiring a tutor.
Cons: Tutoring can be quite expensive– some companies charge more than a hundred dollars an hour for one-on-one sessions. Next Step, however, offers GRE tutoring at a far more affordable price.

Next Step Test Preparation offers complete packages of one-on-one GRE tutoring for less than the price of a packed prep course.  For more information, see our GRE tutoring page, contact info@nextsteptestprep.com or call 888-530-NEXT.
Photo credit Newton Free Library under a Creative Commons license.

Let’s say you’re faced with the following quantitative comparison problem:

Column A: 20 / sqrt(10)
Column B: 10 / sqrt(5)

The first thing to note here is that since there aren’t any variables in this situation, you can eliminate D right off the bat. You can be sure that one of these numbers is always bigger than the other, and if you had infinite time you could figure out which. But you don’t have infinite time. What are you going to do?

You probably don’t know what the square root of 5 or 10 is off the top of your head, even approximately. And you won’t have a calculator to figure it out with. You could probably ballpark these, but that would take time. What are you going to do?

Think about it: if the number in A is bigger than the number in B, squaring the number in A will give a result larger than squaring the number in B. 3 is bigger than two, and 3 squared = 9 is bigger than 2 squared = 4. So if you square the numbers in both columns, then compare the squares, you’ll be able to infer which column is bigger. Let’s try it.

Column A: ( 20 / sqrt(10) ) squared = 400 / 10 = 40.
Column B: ( 10 / sqrt(5) ) squared = 100 / 5 = 20.

Since Column A squared turns out to be bigger than column B squared, Column A must have been bigger than Column B. So the answer is A.

There are many quantitative comparison questions where the quickest way to solve the problem is by doing the same thing to both sides. Be careful, though: If you’re working with negative numbers, you may run into trouble. Multiplying both columns by a negative number will reverse their relationship. For example, we know that
4 < 5

But if we multiply both 4 and 5 by -2, we get:

-8 > -10. The inequality has reversed. So this technique is best used when you’re sure you’re working with positive numbers.

Next Step Test Preparation offers complete packages of one-on-one GRE tutoring for less than the price of a packed prep course.  For more information, see our GRE tutoring page, contact info@nextsteptestprep.com or call 888-530-NEXT.

Photo credit 24oranges under a Creative Commons license.

The authors of GRE Reading Comprehension passages are, generally speaking, inexplicably addicted to describing other peoples’ views on various matters. Whatever the topic– Puerto Rican poets, the mating habits of bats, the Chilean electoral system or the French Revolution– odds are the GRE passage on the subject quotes about eight different scholars, all saying different but not necessarily contradictory things. To answer questions on these passages correctly, you’ll have to figure out who’s saying what and why and try to find some order in the chaos. How is this possible? Well, it gets a lot easier if you keep track of the author.

Keeping track of the author means focusing on what the author of the passage thinks and why he’s telling you what he’s telling you. After all, the author isn’t just quoting or spouting facts at random– he always has a point to make, even if he goes about it oddly. The author’s intention provides the overall structure of the passage; if you can figure it out, you’ll have a much easier time keeping track of all the facts and opinions included in the passage.
But how do you go about keeping track of the author? Here are some tips.
1. Always know who’s speaking. Sometimes the author of the passage will quote or summarize someone’s views only to refute them later. If you get the impression that the views he’s quoting are his own, you’ll think he’s contradicting himself. So always keep track of which opinions are the author’s own and which he attributes to someone else.

2. Look for transition words. If a paragraph starts with “However,” or “Despite Johnson’s claims,” or “On the other hand,” then the point the author is making has changed too! When you’re reading quickly you may be tempted to skip over phrases like this in order to get to the “meat” of the paragraph– the facts, dates and names. But resist the temptation! These words tell you the place of a paragraph in the overall structure of the passage, and that’s much more important than knowing any particular detail.

3. Look for opinion markers. There’s a difference between writing “In his book, Johnson demonstrates that…” and saying “In his book, Johnson asserts that…” Generally speaking, you’ll write the former if you agree with Johnson and the latter if you disagree with him. If he “demonstrated” it, it’s true; if he just “asserted” it, you probably think it’s false. It’s a subtle difference, but it’s important: Clues like this can tell you what the author is thinking even if he’s less than explicit about it.

Next Step Test Preparation offers complete packages of one-on-one GRE tutoring for less than the price of a packed prep course.  For more information, see our GRE tutoring page, contact info@nextsteptestprep.com or call 888-530-NEXT.

Photo credit Sasja Lund under a Creative Commons license.