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.
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.
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.
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?
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