# Rates

## Arithmetic: Rates

Read the following SAT test question and then select the correct answer.

Use the same method with each math question to avoid making mistakes.  Start by reading carefully and identifying the bottom line.  What question must you answer?  Then assess your options for answering the question, choosing the most time efficient method to attack the problem.  When you have an answer, loop back to verify that your answer matches the bottom line.

Machine X, working at a constant rate, can produce x bolts per hour. Machine Y, working at a constant rate, can produce x + 6 bolts per hour. In terms of x, how many bolts can both machines working together at their respective rates produce in 4 hours?

Bottom line: #bolts in 4 hr = ?

Assess your Options:  You could choose numbers for x and y and then see which of your answer choices matches the answer that you get, but you will still have to write an equation.  It will be much faster to leave the variable in the problem and write an equation to find the answer.

Attack the Problem:  You know that you have two machines, X and Y.  You know how much each of these machines produces in an hour.  Find out the total that they can produce in one hour.

X + Y (both machines)= x + x + 6          Combine like terms.
X + Y (both machines)= 2x + 6

In one hour you can produce 2x + 6 bolts.  However, your bottom line requires you to find the number of bolts that can be produced in 4 hours.  Multiply 2x + 6 by 4.

4(2x + 6)          Distribute the 4.
8x + 24

Loop Back:  You solved for 4 hours rather than just 1 hr, so you are ready to look at the answer choices.

(A) 4x + 12
(B) 4x + 24
(C) 6x + 30
(D) 8x + 24
(E) 8x + 36

On sat.collegeboard.org, 59% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

# Rates

## Arithmetic: Rates

Read the following SAT test question and then select the correct answer.

Use the same process for every math problem so that you are not intimidated by any question.  (1) Read the question carefully.  (2) Identify the bottom line – what is the question asking?  (3) Take the time to assess your options – which methods can you use to solve this problem most efficiently?  (4) Attack the problem and work though it logically.  (5) Loop back to make sure that your answer matches the bottom line – did you complete every step of the problem?

A train traveling 60 miles per hour for 1 hour covers the same distance as a train traveling 30 miles per hour for how many hours?

Bottom line: Make a quick note that you are solving for hours: hrs = ?

Assess Your Options:  You could try to use logic for this problem by thinking that if a train goes more slowly, it must take longer to go the same distance as it did at a faster speed.  Unfortunately, logic will not eliminate all of your answer choices.  Use the distance formula to solve this problem.

Attack the problem:  The distance formula is distance is equal to rate(speed) times time:  D = R × T.  Start with the first train and multiply the rate (60 m/hr) by the time (1 hr) to get the distance:

60 × 1 = 60

The first train traveled 60 miles.  You know that both trains traveled the same distance, so plug in 60 as the distance for the second train. You also know that the rate is 30 and the time is unknown.  That should look like:

30 × T = 60
30T = 60
T = 2

Note:  If you are good at balancing equations, there is an even faster way to do this problem.  Look at the distance equation:  D = R × T.  If the distance for a problem stays the same, but you increase the speed (rate), then you must decrease the time by the reciprocal of the speed increase.  That keeps the equation balanced.  Ex:  If you double the speed, you must halve the time.  In this particular problem you halve the speed (from 60 to 30), so you must double the time.  2 × 1 hour = 2 hours.  This reciprocal rule will always work!

Loop Back:  You solved for the time of the second train, which is already in hours, so you are ready to look at your answer choices!

(A) 3
(B) 2
(C) 1
(D)
(E)

On sat.collegeboard.org, 78% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

# Rates

## Arithmetic: Rates

Use the same method for every math problem on the SAT.  Read the problem carefully, identify the bottom line, and assess your options for solving the problem.  Choose the most efficient method to attack the problem.  Often there will be multiple steps to a single problem, so when you have an answer, be sure to loop back and verify that it matches the bottom line.

A machine can insert letters in envelopes at the rate of 120 per minute. Another machine can stamp the envelopes at the rate of 3 per second. How many such stamping machines are needed to keep up with 18 inserting machines of this kind?

Bottom Line:  # stamping machines = ?

Assess your Options:  You could try to work backwards from the answers, but there is no need.  It will be faster just to solve the problem.

Attack the Problem:  You have been given two different units of time: minutes and seconds.  There are 60 seconds in a minute.  Changing the minutes to seconds will be easiest, so the letter inserting machine works at a rate of 120 letters per 60 seconds.  120 divided by 60 is 2 letters in envelopes per second.  If there are 18 letter inserting machines, then together they will insert 36 letters in envelopes per second (2 × 18 = 36).

You don’t know how many stamping machines you need, so use x to represent that number.  Stamping machines have a rate of 3 envelopes per second, so each machine will finish 3 envelopes in a second.  You know that the stamping machines must keep pace with the 18 letter inserting machines that finish 36 envelopes per second, so the outcome must be 36.  Write 3x = 36.  When you solve for x,  x = 12.

Loop Back:  You used x to represent the number of stamping machines, your bottom line, so you are ready to look at the answer choices.

(A) 12
(B) 16
(C) 20
(D) 22
(E) 24

On sat.collegeboard.org, 58% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

# Rates

## Arithmetic: Rates

Read the following SAT test question and then select the correct answer.

Using the same method with every math problem to minimize mistakes.  Read the question carefully.  Identify the bottom line and assess your options for finding it.  Choose the most efficient method to attack the problem.  Once you have an answer, loop back to make sure it addresses the bottom line.

A woman drove to work at an average speed of 40 miles per hour and returned along the same route at 30 miles per hour. If her total traveling time was 1 hour, what was the total number of miles in the round trip?

Bottom Line: Td = ?  (Total distance)

Assess your Options: Remember that speed is really a rate.  There are 4 key rate scenarios: separation, overtake, round trip, and meet in between--this one is a round trip.  You can figure all of these out by using the distance formula (rate × time = distance), but it can be difficult to keep track of which scenario you have unless you treat all of them the same way.  Knowsys recommends that you use a chart to quickly organize your thoughts so that you can be sure that you accounted for all of the information in the problem. (Spoiler: many students make mistakes on these types of problems!  You do not get any extra points for ignoring the chart, so use it!)

Attack the Problem:  Here is the chart that you should use with all rate scenarios:

 1 2 Total Rate Time Distance

Start filling in the information that you know.  The first trip was at a rate of 40 miles per hour and the second trip was at a rate of 30 miles per hour.  The total time was 1 hour.

 Trip 1 Trip 2 Total Rate 40 30 Time 1 Distance

If you don’t know the time for the first trip, choose a variable to represent the unknown.  Put an “x” in that box.  You know that the time for the trips together must total 1 hour (x + ? = 1).  Therefore, the second trip is equal to 1 minus x

 Trip 1 Trip 2 Total Rate 40 30 Time x 1 – x 1 Distance

You already know that rate × time = distance, so multiply the two columns representing the trips.

 Trip 1 Trip 2 Total Rate 40 30 Time x 1 – x 1 Distance 40x 30(1 – x)

Before you start worrying about the total number of miles, remember that this person is using the same route each time.  That means the distance traveled each time is an equal length.  Set the distances equal to each other.

40x = 30(1 – x)
40x = 30 – 30x
70x = 30
$x=\frac{3}{7}$

If you know x, you can now find a number value for each part of your chart.  What was the bottom line?  You need to find the total number of hours.  You could plug x into both distances and add them up; however, there is an even faster method.  Take the first distance and multiply it by 2.  (Remember that the distances are the same.)

$2\times40\times \frac{3}{7}=Total\; distance$

$\frac{240}{7}=Total\; distance$

$34\frac{2}{7}=Total\; distance$

(A) 30
(B) $30\frac{1}{7}$
(C) $34\frac{2}{7}$
(D) 35
(E) 40

On sat.collegeboard.org, 38% of the responses were correct.

For more help with SAT math, visit www.myknowsys.com!

# Ratios, Rates, and Proportions

George Mason University's History News Network is an unusual news site that puts current events in a broad historical context. Normal news stories focus only on what has happened recently, but HNN strives to connect current events to the history that created them.

## 5/12 Ratios, Rates, and Proportions

The c cars in a car service use a total of g gallons of gasoline per week. If each of the cars uses the same amount of gasoline, then, at this rate, which of the following represents the number of gallons used by 5 of the cars in 2 weeks?

First, note the bottom line.

5 cars 2 weeks = ?

Next, assess your options. Since the problem gives so much information about the cars using words rather than numbers, a good place to start is to translate its question into mathematical terms.

c = total number of cars

g = total gallons of gas per week

The gas used in two weeks is easy to find: 2g. The tricky part involves determining how much gas is used by only 5 cars. It is tricky rather than difficult because if you know the trick, this problem is easy. Simply find the gas used by one car over the course of a week and multiply that by 5 cars.

$\frac{g}{c}$ = gas per week for 1 car

$\frac{5g}{c}$ = gas per week for 5 cars

Since the question asks how much gas will be used in 2 weeks, multiply this term by 2. This incorporates the 2g you identified earlier.

$\frac{10g}{c}$ = gas for 5 cars for 2 weeks

Now look at the answer choices.

A) $10cg$

B) $\frac{2g}{5c}$

C) $\frac{5g}{2c}$

D) $\frac{g}{10c}$

E) $\frac{10g}{c}$

On sat.collegeboard.org, 32% of responses were correct.

For more help with math, visit www.myknowsys.com!

# Rates

College.gov has your starting place for financial aid. One of the major concerns for prospective college students is money; college is expensive, and rising tuition costs combined with less government funding show no end in sight. However, there is money available to help students who need it, and the FAFSA is the best place to start.

## 4/15 Rates

Always remember to follow the Knowsys Method--even in your math classes. Thinking strategically and logically will help you be more efficient far beyond the SAT. First, read carefully to see what the question is actually asking. Then assess your options and select the best one. Attack the problem efficiently, then loop back to make sure that the answer you found matches the question that was originally asked.

A train traveling 60 miles per hour for 1 hour covers the same distance as a train traveling 30 miles per hour for how many hours?

First, note the bottom line.

train 2 hours= ?

Next, look back at the question to determine how you could solve it. You could determine the total distance traveled by train 1 and then calculate the time it would take train 2 to travel the same distance. You could also calculate the times relative to one another. That might sounds odd, but it is actually more efficient than the first method.

The first step is to set up a ratio of  the two rates given. Put the "new" rate, that of train 2, on top because it is the variable you are trying to find. Always reduce ratios to lowest terms.

$\frac{train 2}{train1}=\frac{30}{60}=\frac{1}{2}$

You now have a ratio of the two rates. Here's the cool part: simply flip it over to find a ratio of the two times.

If a car or train travels at twice the planned speed, the trip will take half as long as projected. If it travels at half the planned speed, the trip will take twice as long. This rule applies when traveling 2/3, 5/4, or any fraction of the original rate; the ratio of the times will be the reciprocal of that fraction.

$\frac{train2}{train1}=\frac{2}{1}$

This means that train 2 spent twice as much time as train 1 covering the same amount of ground.

train2 = 2(train1)

Since train 1 traveled for 1 hour, train 2 traveled for 2 hours. Loop back to make sure you answered the right question.

train 2 hours = 2

Good job! Now look at the answer choices.

A) 3

B) 2

C) 1

D) $\frac{1}{2}$

E) $\frac{1}{3}$

On sat.collegeboard.org, 81% of responses were correct.

Want more help with math? Visit www.myknowsys.com!

# Rates

The Industrial Revolution is well-known as a time of explosive economic growth and invention. Here are five top inventions--and why they're important! Any of them would make a noteworthy Excellent Example in your SAT essay.

## 4/6 Rates

Always stop and read carefully before you do anything else. Make sure that you mark the bottom line and carefully label each step of your scratch work. It's easy to let confidence trick you into thinking it's safe to skip steps, but on a high-stakes test like the SAT, writing down each step and making sure that every answer is right can be the difference between acceptance and rejection at the school of your dreams. Once you find an answer, loop back to make sure that you have answered what the question asked. Then, and only then, should you select your answer from among the answer choices.

A machine can insert letters in envelopes at the rate of 120 per minute. Another machine can stamp the envelopes at the rate of 3 per second. How many stamping machines are needed to keep up with 18 inserting machines of this kind?

When you read this question carefully, one thing that should jump out at you is the fact that you have two different units of measurement. Immediately think, "I could convert both of these to seconds." Look for the bottom line: How many stamping machines do you need?

s = ?

Next, focus on your options. What could I do? What should I do? You could, as previously mentioned, convert one rate so that they use the same unit of measurement. But what could you do after that? You could guess, if you feel that this problem is too hard and that your time is better spent elsewhere, but guessing is only allowed on the test. No guessing during practice! The next option is to write an equation to describe what is happening in this problem, and then solve for s.

Let's start by converting the inserting machine's rate into seconds.

i: 120/60 = 2

Each inserting machine can stuff 2 envelopes per second.

Next, set up a formula to compare the number of envelopes that the machines can prepare. Each inserter can stuff 2 envelopes each second, and each stamper can stamp 3 envelopes per second. Plug the 18 inserting machines n for i.

2i = 3s

2(18) = 3s

Now solve for s.

$\frac{2(18)}{3}=s$

2(6) = s

12 = s

Loop back to make sure that what you found was the correct answer. s represents the number of stamping machines needed to prepare an equal number of envelopes as i inserting machines. Each inserting machine completes 2 envelopes per second, while each stamping machine finishes 3. Logically, you need fewer stamping machines than inserting machines. Look at the answer choices:

A) 12

B) 16

C) 20

D) 22

E) 24