How to do distance problems - HSPT Math
Card 1 of 88
Sophie travels f miles in g hours. She must drive another 30 miles at the same rate. Find the total number of hours, in terms of f and g, that the trip will take.
Sophie travels f miles in g hours. She must drive another 30 miles at the same rate. Find the total number of hours, in terms of f and g, that the trip will take.
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Using d = rt, we know that first part of the trip can be represented by f = rg. The second part of the trip can be represented by 30 = rx, where x is some unknown number of hours. Note that the rate r is in both equations because Sophie is traveling at the same rate as mentioned in the problem.
Solve each equation for the time (g in equation 1, x in equation 2).
g = f/r
x = 30/r
The total time is the sum of these two times
Note that, from equation 1, r = f/g, so
=
Using d = rt, we know that first part of the trip can be represented by f = rg. The second part of the trip can be represented by 30 = rx, where x is some unknown number of hours. Note that the rate r is in both equations because Sophie is traveling at the same rate as mentioned in the problem.
Solve each equation for the time (g in equation 1, x in equation 2).
g = f/r
x = 30/r
The total time is the sum of these two times
Note that, from equation 1, r = f/g, so
=
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Gary is the getaway driver in a bank robbery. When Gary leaves the bank at 3 PM, he is going 60 mph, but the police officers are 10 miles behind traveling at 80 mph. When will the officers catch up to Gary?
Gary is the getaway driver in a bank robbery. When Gary leaves the bank at 3 PM, he is going 60 mph, but the police officers are 10 miles behind traveling at 80 mph. When will the officers catch up to Gary?
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Traveling 20 mph faster than Gary, it will take the officers 30 minutes to catch up to Gary. The answer is 3:30 PM.
Traveling 20 mph faster than Gary, it will take the officers 30 minutes to catch up to Gary. The answer is 3:30 PM.
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Trevor took a road trip in his new VW Beetle. His car averages 32 miles per gallon. Gas costs $4.19 per gallon on average for the whole trip. How much would it coust to drive 3,152 miles?
Trevor took a road trip in his new VW Beetle. His car averages 32 miles per gallon. Gas costs $4.19 per gallon on average for the whole trip. How much would it coust to drive 3,152 miles?
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To find this answer just do total miles divided by miles per gallon in order to find how many gallons of gas it will take to get from point A to Point B. Then multiply that answer by the cost of gasoline per gallon to find total amount spent on gasoline.
To find this answer just do total miles divided by miles per gallon in order to find how many gallons of gas it will take to get from point A to Point B. Then multiply that answer by the cost of gasoline per gallon to find total amount spent on gasoline.
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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.
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1 mile = 5280 feet
If Greg's house is 5.3 miles away, how far is it in feet?
1 mile = 5280 feet
If Greg's house is 5.3 miles away, how far is it in feet?
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Using the conversion formula, you would multiply 5.3 miles by 5280 feet and you will get 27,984 feet.
Using the conversion formula, you would multiply 5.3 miles by 5280 feet and you will get 27,984 feet.
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Joe drove an average of 45 miles per hour along a 60-mile stretch of highway, then an average of 60 miles per hour along a 30-mile stretch of highway. What was his average speed, to the nearest mile per hour?
Joe drove an average of 45 miles per hour along a 60-mile stretch of highway, then an average of 60 miles per hour along a 30-mile stretch of highway. What was his average speed, to the nearest mile per hour?
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At 45 mph, Joe drove 60 miles in 
hours.
At 60 mph, he drove 30 miles in 
hours.
He made the 90-mile trip in 
hours, so divide 90 by 
to get the average speed in mph:
At 45 mph, Joe drove 60 miles in
At 60 mph, he drove 30 miles in
He made the 90-mile trip in
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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, 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
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Kate and Bella were both travelling at the same speed. Kate went 300 miles in 5 hours. Bella travelled 450 miles. How many hours did it take for Bella to reach her destination?
Kate and Bella were both travelling at the same speed. Kate went 300 miles in 5 hours. Bella travelled 450 miles. How many hours did it take for Bella to reach her destination?
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The distance formula is essential in this problem.
First, use Kate's info to figure out the rate for both girls since they're travelling at the same speed, which is 
. Then, plug in that rate to the formula with Bella's information, which gives her a time of 
hours.
The distance formula is essential in this problem.
First, use Kate's info to figure out the rate for both girls since they're travelling at the same speed, which is
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Find the distance from point 
to point 
.
Find the distance from point
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Write the distance formula.
Substitute the values of the points into the formula.
The square root of 
can be reduced because 
, a factor of 
, is a perfect square. 
.
Now we have 
Write the distance formula.
Substitute the values of the points into the formula.
The square root of
Now we have
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Find the distance between the points 
and 
.
Find the distance between the points
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Write the distance formula.
Plug in the points.
The distance is: 
Write the distance formula.
Plug in the points.
The distance is:
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Suppose a student ran a pace of eight minutes per mile at consistent pace. He arrived at the school in thirty minutes. How far is the school in miles?
Suppose a student ran a pace of eight minutes per mile at consistent pace. He arrived at the school in thirty minutes. How far is the school in miles?
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The consistent pacing tells us that this is a linear relationship between distance and the student's speed and time.
Write the equation for the distance travelled.
The speed can be rewritten as: 
Substitute the speed and time.
The consistent pacing tells us that this is a linear relationship between distance and the student's speed and time.
Write the equation for the distance travelled.
The speed can be rewritten as:
Substitute the speed and time.
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On a map, one half of an inch represents thirty miles of real distance.
The towns of Waterbury and Nashua are three and one half inches apart on this map. How long, in hours, would it take for someone to drive from Waterbury to Nashua if his speed averaged 
miles per hour?
On a map, one half of an inch represents thirty miles of real distance.
The towns of Waterbury and Nashua are three and one half inches apart on this map. How long, in hours, would it take for someone to drive from Waterbury to Nashua if his speed averaged
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On a map, one half of an inch represents thirty miles of real distance, so one inch represents twice this, or sixty miles. The actual distance from Waterbury to Nashua, which is three and a half inches on the map, is
Therefore, the two cities are 210 miles apart. Divide the distance by the rate to get the time:
, or 
.
On a map, one half of an inch represents thirty miles of real distance, so one inch represents twice this, or sixty miles. The actual distance from Waterbury to Nashua, which is three and a half inches on the map, is
Therefore, the two cities are 210 miles apart. Divide the distance by the rate to get the time:
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On a map, one half of an inch represents forty miles of real distance.
It takes John 90 minutes to get from Kingsbury to Willoughby driving an average of 
miles per hour. How many inches apart, in terms of 
, are the two cities on the map?
On a map, one half of an inch represents forty miles of real distance.
It takes John 90 minutes to get from Kingsbury to Willoughby driving an average of
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The distance in real miles between Kingsbury in Willoughby can be found by multiplying rate 
miles per hour by time 90 minutes, or one and a half hours:
Let 
represent map distance between the cities, One half of an inch represents forty miles of real distance, so one inch represents twice this, ir eighty miles. The ratio that compares map distance and real distance is
The distance in real miles between Kingsbury in Willoughby can be found by multiplying rate
Let
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Mr. Thomas's car holds exactly 14 gallons of gasoline, and gets 24 miles per gallon. He gets into his car, which has a full gas tank, and drives 
miles. He then refills the car until the gas gauge reads "full" again. In terms of 
, how many gallons of gasoline did he put in his car?
Mr. Thomas's car holds exactly 14 gallons of gasoline, and gets 24 miles per gallon. He gets into his car, which has a full gas tank, and drives
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Mr. Thomas gets 24 miles per gallon, and has driven 
miles; divide distance by gallons used, and he has used 
gallons. Since he is refilling his car, he is putting 
gallons in his car.
Note that the amount of gasoline that the tank will hold is irrelevant to the problem.
Mr. Thomas gets 24 miles per gallon, and has driven
Note that the amount of gasoline that the tank will hold is irrelevant to the problem.
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Mrs. Williams' car gets 
miles to the gallon; its tank holds 
gallons. Mrs. Williams gets into her car, which has a full tank; she drives 
miles, and then refills the tank completely with gasoline that costs 
dollars per gallon.
In order to determine the amount Mrs. Williams paid for the gasoline, you need to know the value of each of the following varables except:
Mrs. Williams' car gets
In order to determine the amount Mrs. Williams paid for the gasoline, you need to know the value of each of the following varables except:
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The price of the gasoline Mrs. Williams purchased is the number of gallons multiplied by the price per gallon. 
is the latter quantity, so it is needed to answer the question.
To find the amount of gasoline purchased - which is the amount she used - it is necessary to divide the number of miles she drove by the gas mileage in miles per gallon. These are 
and 
, so both are needed to answer the question.
is not relevant to the problem, and is the correct choice.
The price of the gasoline Mrs. Williams purchased is the number of gallons multiplied by the price per gallon.
To find the amount of gasoline purchased - which is the amount she used - it is necessary to divide the number of miles she drove by the gas mileage in miles per gallon. These are
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Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, which one called in sick at 7:30 AM and stayed in bed?
Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, which one called in sick at 7:30 AM and stayed in bed?
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Someone who called in sick would remain at constant zero distance from his home during the entire time. This would be represented by a horizontal line along the zero axis; this describes the graph for Mr. Jones.
Someone who called in sick would remain at constant zero distance from his home during the entire time. This would be represented by a horizontal line along the zero axis; this describes the graph for Mr. Jones.
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Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, who stopped for some breakfast on the way to work?
Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, who stopped for some breakfast on the way to work?
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We are looking for someone whose distance from home increased steadily for a while, then became constant (since the person had to have not been moving), then increased steadily again. This describes the graph for Mr. Chapman.
We are looking for someone whose distance from home increased steadily for a while, then became constant (since the person had to have not been moving), then increased steadily again. This describes the graph for Mr. Chapman.
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Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, which one started out, realized he forgot his briefcase, went back for it, and went to work?
Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, which one started out, realized he forgot his briefcase, went back for it, and went to work?
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The distance this person was from his home increased steadily as he went to work, then decreased as he went back home for his briefcase, then increased again as he went on to work. This describes the graph for Mr. Idle.
The distance this person was from his home increased steadily as he went to work, then decreased as he went back home for his briefcase, then increased again as he went on to work. This describes the graph for Mr. Idle.
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Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, which one works the night shift, and therefore went from work to home between 8 AM and 9 AM?
Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, which one works the night shift, and therefore went from work to home between 8 AM and 9 AM?
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Since the person in question went home, his distance from home decreased rather than increased as time went on. The graph will be a falling line; this graph represents Mr. Palin.
Since the person in question went home, his distance from home decreased rather than increased as time went on. The graph will be a falling line; this graph represents Mr. Palin.
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Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, who got halfway there, got sick, and returned home?
Six friends work for a company as maintenance staff.
Above are six graphs. Each graph shows the distance that one of the six is from his home from 8 AM to 9 AM on a particular day, relative to the time. The time of day is represented by the horizontal axis, and the distance from home is represented by the vertical axis. The name of the person represented by each graph is under the graph.
Of the six, who got halfway there, got sick, and returned home?
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The person's distance increased as he drove to work, but decreased as he returned home. This describes the graph for Mr. Gilliam.
The person's distance increased as he drove to work, but decreased as he returned home. This describes the graph for Mr. Gilliam.
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