Rate Problems - GMAT Quantitative
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If it takes Sally 3 hours to drive q miles, how many hours will it take her to drive r miles at the same rate?
If it takes Sally 3 hours to drive q miles, how many hours will it take her to drive r miles at the same rate?
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If Sally drives q miles in 3 hours, her rate is 3/q miles per hour. Plug this rate into the distance equation and solve for the time:
Distance = ratetimes time
r=\frac{q}{3}$times t
t=\frac{3r}{q}$
If Sally drives q miles in 3 hours, her rate is 3/q miles per hour. Plug this rate into the distance equation and solve for the time:
Distance = ratetimes time
r=\frac{q}{3}$times t
t=\frac{3r}{q}$
A cat runs at a rate of 12 miles per hour. How far does he run in 10 minutes?
A cat runs at a rate of 12 miles per hour. How far does he run in 10 minutes?
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We need to convert hours into minutes and multiply this by the 10 minute time interval:
$\frac{12 miles}{1 hour}$x$\frac{1 hour}{60 min}$x$\frac{10 min}{1}$=\frac{120 miles}{60}$=2 miles
We need to convert hours into minutes and multiply this by the 10 minute time interval:
$\frac{12 miles}{1 hour}$x$\frac{1 hour}{60 min}$x$\frac{10 min}{1}$=\frac{120 miles}{60}$=2 miles
A group of students are making posters to advertise for a bake sale. 12 large signs and 60 small signs are needed. It takes 10 minutes to paint a small sign and 30 minutes to paint a large sign. How many students will be needed to paint all of the signs in 2 hours or less?
A group of students are making posters to advertise for a bake sale. 12 large signs and 60 small signs are needed. It takes 10 minutes to paint a small sign and 30 minutes to paint a large sign. How many students will be needed to paint all of the signs in 2 hours or less?
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In 2 hours, 1 student can paint 4 large signs or 12 small signs. Therefore, 3 students are required to paint the large signs (
) and 5 students are required to paint the small signs (
). In total, 8 students are required.
In 2 hours, 1 student can paint 4 large signs or 12 small signs. Therefore, 3 students are required to paint the large signs (
Jason is driving across the country. For the first 3 hours, he travels 60 mph. For the next 2 hours he travels 72 mph. Assuming that he has not stopped, what is his average traveling speed in miles per hour?
Jason is driving across the country. For the first 3 hours, he travels 60 mph. For the next 2 hours he travels 72 mph. Assuming that he has not stopped, what is his average traveling speed in miles per hour?
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In the first three hours, he travels 180 miles.
In the next two hours, he travels 144 miles.
for a total of 324 miles.
Divide by the total number of hours to obtain the average traveling speed.
In the first three hours, he travels 180 miles.
In the next two hours, he travels 144 miles.
for a total of 324 miles.
Divide by the total number of hours to obtain the average traveling speed.
Tom runs a 100m race in a certain amount of time. If John runs the same race, he takes 2 seconds longer. If John ran at 8m/s, approximately how fast did Tom run?
Tom runs a 100m race in a certain amount of time. If John runs the same race, he takes 2 seconds longer. If John ran at 8m/s, approximately how fast did Tom run?
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Tom runs a 100m race in a certain amount of time. If John runs the same race, he takes 2 seconds longer. If John ran at 8m/s, how fast did Tom run?
Let 
denote the amount of time that it took Tom to run the race. Then it took John 
seconds to run the same race going 8m/s. At 8m/s, it takes 12.5 seconds to finish a 100m race. This means it took Tom 10.5 seconds to finish. Running 100m in 10.5 seconds is the same as 
Tom runs a 100m race in a certain amount of time. If John runs the same race, he takes 2 seconds longer. If John ran at 8m/s, how fast did Tom run?
Let
Jim and Julia, a married couple, work in the same building.
One morning, both left at 9:00, but in different cars. Jim arrived at 10:10; Julia arrived 10 minutes later. If Jim's average speed was 54 miles per hour, what was Julia's average speed (nearest whole number)?
Jim and Julia, a married couple, work in the same building.
One morning, both left at 9:00, but in different cars. Jim arrived at 10:10; Julia arrived 10 minutes later. If Jim's average speed was 54 miles per hour, what was Julia's average speed (nearest whole number)?
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Jim arrived at the common destination in 70 minutes, or 
hours. His average speed was 54 miles per hour, so their workplace is
miles away from Jim and Julia's home.
Julia traveled those 63 miles in 80 minutes, or 
hours, so her average speed was
,
or, rounded, 47 miles per hour.
Jim arrived at the common destination in 70 minutes, or
Julia traveled those 63 miles in 80 minutes, or
or, rounded, 47 miles per hour.
Randall traveled 75 kilometers in 600 minutes. What was Randall's per hour rate?
Randall traveled 75 kilometers in 600 minutes. What was Randall's per hour rate?
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We need to pay close attention to some details here.
-
We are given time in minutes, but asked for an answer in hours.
-
A rate can be defined as distance over time.
Taking the first detail, we convert 600 minutes to 10 hours, since there are 60 minutes in one hour.
Taking the second detail, we divide 75 kilometers by 10 hours. This gives us an answer of 7.5 kilometers per hour.
We need to pay close attention to some details here.
-
We are given time in minutes, but asked for an answer in hours.
-
A rate can be defined as distance over time.
Taking the first detail, we convert 600 minutes to 10 hours, since there are 60 minutes in one hour.
Taking the second detail, we divide 75 kilometers by 10 hours. This gives us an answer of 7.5 kilometers per hour.
Kenny and Marie, a married couple, work in the same building.
One morning, both left at 9:00, but in different cars. Kenny arrived at 10:10; Marie arrived 10 minutes later. If Kenny's average speed was 6 miles per hour faster than Marie's, how far is their work place from their home (nearest whole mile)?
Kenny and Marie, a married couple, work in the same building.
One morning, both left at 9:00, but in different cars. Kenny arrived at 10:10; Marie arrived 10 minutes later. If Kenny's average speed was 6 miles per hour faster than Marie's, how far is their work place from their home (nearest whole mile)?
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Let 
be the rate at which Kenny drove. Then Marie drove at a rate of 
miles per hour. The two drove the same distance, so, since Kenny drove 70 minutes, or 
hours, and Marie drove for 80 minutes, or 
hours, we can use the formula 
to set up the equation:
Since Kenny traveled at 48 miles per hour for 
hours, the distance each drove is
miles.
Let
Since Kenny traveled at 48 miles per hour for
Jerry took a car trip of 320 miles. The trip took a total of six hours and forty minutes; for the first four hours, his average speed was 60 miles per hour. What was his average speed for the remaining time?
Jerry took a car trip of 320 miles. The trip took a total of six hours and forty minutes; for the first four hours, his average speed was 60 miles per hour. What was his average speed for the remaining time?
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Jerry drove 60 miles per hour for 4 hours - that is, 
miles.
He drove the remainder of the distance, or 
miles over a period of 
hours, so his average speed was
miles per hour.
Jerry drove 60 miles per hour for 4 hours - that is,
He drove the remainder of the distance, or
Andy and his wife Donna both work at the same building.
One morning, Andy left home at 8:00; Donna left 5 minutes later. Each arrived at their common destination at 8:50. Andy drove at an average speed of 45 miles per hour; what was Donna's average speed, to the nearest mile per hour?
Andy and his wife Donna both work at the same building.
One morning, Andy left home at 8:00; Donna left 5 minutes later. Each arrived at their common destination at 8:50. Andy drove at an average speed of 45 miles per hour; what was Donna's average speed, to the nearest mile per hour?
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Andy traveled for 50 minutes, or 
of one hour, at 45 miles per hour, so the office building is 
miles from Andy and Donna's house. Donna traveled this distance for 45 minutes, or 
of one hour, so her speed was:
miles per hour.
Andy traveled for 50 minutes, or
In order to qualify for the next heat, the race-car driver needs to average 60 miles per hour for two laps of a one mile race-track. The driver only averages 40 miles per hour on the first lap. What must be the driver's average speed for the second lap in order to average 60 miles per hour for both laps?
In order to qualify for the next heat, the race-car driver needs to average 60 miles per hour for two laps of a one mile race-track. The driver only averages 40 miles per hour on the first lap. What must be the driver's average speed for the second lap in order to average 60 miles per hour for both laps?
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If the driver needs to drive two laps, each one mile long, at an average rate of 60 miles per hour. To find the average speed, we need to add the speed for each lap together then divide by the number of laps. The equation would be as follows:
In our case we know lap one was driven at 
miles per hour. We substitute this value in for 
and solve for 
.
Thus to average 
miles per hour for two laps with lap one being 
miles per hour, lap two would have to have a rate of 
miles per hour.
If the driver needs to drive two laps, each one mile long, at an average rate of 60 miles per hour. To find the average speed, we need to add the speed for each lap together then divide by the number of laps. The equation would be as follows:
In our case we know lap one was driven at
Thus to average
Ray travels 
in three hours. At this rate, how long (in hours) will it take him to travel 
?
Ray travels
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If Ray covers 
in three hours, that means he covers 
in one hour:
Perform the following calculation to find how long it takes to cover 
.
If Ray covers
Perform the following calculation to find how long it takes to cover
If a plane flies the 3000 miles from San Francisco to New York at an average speed of 600 mph, and then, buffetted by a hefty headwind, makes the return trip at an average speed of 300 mph. What was its average speed over the entire round trip?
If a plane flies the 3000 miles from San Francisco to New York at an average speed of 600 mph, and then, buffetted by a hefty headwind, makes the return trip at an average speed of 300 mph. What was its average speed over the entire round trip?
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In combined rate problems such as this, we must first find units of the desired answer: 
and then find the totals of each piece of those units. Total miles is easy as we can just add together the two legs of the trip:
To find total hours, we just have to use each leg's speed:
The trip therefore took 15 total hours.
Now we simply divide the totals to find the average speed:
In combined rate problems such as this, we must first find units of the desired answer:
To find total hours, we just have to use each leg's speed:
The trip therefore took 15 total hours.
Now we simply divide the totals to find the average speed:
Frank can eat a huckleberry pie in 15 minutes. His formidable sister, Francine, can eat it in 10 minutes. How long does it take them to eat a pie together?
Frank can eat a huckleberry pie in 15 minutes. His formidable sister, Francine, can eat it in 10 minutes. How long does it take them to eat a pie together?
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To solve this combined rate problem, we must use the equation: 
Where A and B are the times it takes Frank and Francine, respectively, to eat a pie.
Therefore, it takes Frank and Francine
to eat the pie.
To solve this combined rate problem, we must use the equation:
Where A and B are the times it takes Frank and Francine, respectively, to eat a pie.
Therefore, it takes Frank and Francine
to eat the pie.
If it takes Sally 3 hours to drive q miles, how many hours will it take her to drive r miles at the same rate?
If it takes Sally 3 hours to drive q miles, how many hours will it take her to drive r miles at the same rate?
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If Sally drives q miles in 3 hours, her rate is 3/q miles per hour. Plug this rate into the distance equation and solve for the time:
Distance = ratetimes time
r=\frac{q}{3}$times t
t=\frac{3r}{q}$
If Sally drives q miles in 3 hours, her rate is 3/q miles per hour. Plug this rate into the distance equation and solve for the time:
Distance = ratetimes time
r=\frac{q}{3}$times t
t=\frac{3r}{q}$
A cat runs at a rate of 12 miles per hour. How far does he run in 10 minutes?
A cat runs at a rate of 12 miles per hour. How far does he run in 10 minutes?
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We need to convert hours into minutes and multiply this by the 10 minute time interval:
$\frac{12 miles}{1 hour}$x$\frac{1 hour}{60 min}$x$\frac{10 min}{1}$=\frac{120 miles}{60}$=2 miles
We need to convert hours into minutes and multiply this by the 10 minute time interval:
$\frac{12 miles}{1 hour}$x$\frac{1 hour}{60 min}$x$\frac{10 min}{1}$=\frac{120 miles}{60}$=2 miles
A group of students are making posters to advertise for a bake sale. 12 large signs and 60 small signs are needed. It takes 10 minutes to paint a small sign and 30 minutes to paint a large sign. How many students will be needed to paint all of the signs in 2 hours or less?
A group of students are making posters to advertise for a bake sale. 12 large signs and 60 small signs are needed. It takes 10 minutes to paint a small sign and 30 minutes to paint a large sign. How many students will be needed to paint all of the signs in 2 hours or less?
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In 2 hours, 1 student can paint 4 large signs or 12 small signs. Therefore, 3 students are required to paint the large signs () and 5 students are required to paint the small signs (). In total, 8 students are required.
In 2 hours, 1 student can paint 4 large signs or 12 small signs. Therefore, 3 students are required to paint the large signs (
Jason is driving across the country. For the first 3 hours, he travels 60 mph. For the next 2 hours he travels 72 mph. Assuming that he has not stopped, what is his average traveling speed in miles per hour?
Jason is driving across the country. For the first 3 hours, he travels 60 mph. For the next 2 hours he travels 72 mph. Assuming that he has not stopped, what is his average traveling speed in miles per hour?
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In the first three hours, he travels 180 miles.
In the next two hours, he travels 144 miles.
for a total of 324 miles.
Divide by the total number of hours to obtain the average traveling speed.
In the first three hours, he travels 180 miles.
In the next two hours, he travels 144 miles.
for a total of 324 miles.
Divide by the total number of hours to obtain the average traveling speed.
Tom runs a 100m race in a certain amount of time. If John runs the same race, he takes 2 seconds longer. If John ran at 8m/s, approximately how fast did Tom run?
Tom runs a 100m race in a certain amount of time. If John runs the same race, he takes 2 seconds longer. If John ran at 8m/s, approximately how fast did Tom run?
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Tom runs a 100m race in a certain amount of time. If John runs the same race, he takes 2 seconds longer. If John ran at 8m/s, how fast did Tom run?
Let denote the amount of time that it took Tom to run the race. Then it took John seconds to run the same race going 8m/s. At 8m/s, it takes 12.5 seconds to finish a 100m race. This means it took Tom 10.5 seconds to finish. Running 100m in 10.5 seconds is the same as
Tom runs a 100m race in a certain amount of time. If John runs the same race, he takes 2 seconds longer. If John ran at 8m/s, how fast did Tom run?
Let
Jim and Julia, a married couple, work in the same building.
One morning, both left at 9:00, but in different cars. Jim arrived at 10:10; Julia arrived 10 minutes later. If Jim's average speed was 54 miles per hour, what was Julia's average speed (nearest whole number)?
Jim and Julia, a married couple, work in the same building.
One morning, both left at 9:00, but in different cars. Jim arrived at 10:10; Julia arrived 10 minutes later. If Jim's average speed was 54 miles per hour, what was Julia's average speed (nearest whole number)?
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Jim arrived at the common destination in 70 minutes, or hours. His average speed was 54 miles per hour, so their workplace is
miles away from Jim and Julia's home.
Julia traveled those 63 miles in 80 minutes, or hours, so her average speed was
,
or, rounded, 47 miles per hour.
Jim arrived at the common destination in 70 minutes, or
Julia traveled those 63 miles in 80 minutes, or
or, rounded, 47 miles per hour.