Distance - Rate - Time – Same Direction

Distance - Rate - Time – Same Direction exercise

• Identify all forms of the equation used to solve the distance-rate-time problem.

  Hints

  Look at this triangle:

  • $d$ is above $t$ and $r$
  • $r$ and $t$ are on the same level

  If you know one version of the distance-rate-time equation, manipulate it using inverse operations so that you isolate each variable on the left side of the equation.

  Solution

  You can use the triangle on the right to help you remember the equations concerning the distance-rate-time problem.

  • Rate is distance divided by time, $r=\frac dt$ because you have to divide distance by time to get the rate. Also, notice that the $d$ is above the $t$ in the provided figure.
  • You can transform $r=\frac dt$ by multiplying both sides of the equation by $t$ and then dividing both sides by $r$ to isolate the $t$, giving us $t=\frac dr$. Notice that the $d$ is above the $r$ in the triangle.
  • Multiplying $t=\frac dr$ by $r$ gives us our final equation: $d=r\times t$. $r$ and $t$ are on the same level in the triangle.

  Remember the units for time need to be consistent in your calculations. For example, if you are calculating miles per hour, you should not use minutes without converting minutes to hours.

• Set up an equation to see if Tom arrives earlier than his mom.

  Hints

  You can transform the equation to fit your needs by manipulating the equation shown here using multiplication and division.

  Calculate the time the school bus takes to bring Tom home and add this to Tom's starting time.

  Solve the problem in a similar manner for Tom's mom.

  There exist three variations of distance-rate-time equation:

  • $t=\frac dr$
  • $r=\frac dt$
  • $d=t\times r$

  Think about what information you have and what information you're looking for, then isolate the variable you want to find.

  Remember, the units for time need to be consistent in your calculations. For example, if you are calculating miles per hour, you should not use minutes without converting minutes to hours.

  Solution

  Tom played a prank on his teacher and needs to intercept the letter his teacher sent home before his mom gets a chance to read it.

  What we know:

  • Tom's starting time, $2:50$ pm, and Tom's mom's starting time, $3:00$ pm
  • The distance between Tom's school and home and the distance between Tom's mom's work to home is the same: $12.5$ miles
  • The rate of Tom's school bus, $25$ miles per hour, and the rate of Tom's mom's driving, $50$ miles per hour

  This is a distance-rate-time-problem. Because we're looking for the time, we need the following equation $t=\frac dr$.

  • First, we want to calculate the time it takes Tom to get home: $t=\frac{12.5}{25}=0.5$ hours. Since $1$ hour is $60$ minutes, we have to add $0.5 \times 60$ minutes to Tom's starting time to get his arrival time: $3:20$ pm.
  • Then, we evaluate the time it takes Tom's mom to get home: $t=\frac{12.5}{50}=15$ minutes. We also have to add this time to Tom's mom's starting time to get her arrival time: $3:15$ pm.

  Now we know that Tom's mom arrives $5$ minutes before Tom...poor Tom.

• Calculate the time Tom's mom takes to drive home if she drives $37.5$ miles per hour.

  Hints

  Use the equation $\large{t=\frac dr}$ to compute the time it takes for Tom's mom to drive home.

  The result is a fraction. Convert your answer into minutes.

  For example:

  • $\frac12$ hours is the same as $\frac 12\times 60=30$ minutes
  • $\frac 15$ hours is the same as $\frac15\times 60=12$ minutes

  Add your answer to the starting time to get the arrival time.

  Solution

  We have the following information:

  • The distance from Tom's mom's work to home: $12.5$ miles
  • Tom's mom drives at a rate of $37.5$ miles per hour
  • Tom's mom's starting time: $3:00$ pm

  Using the equation for distance-rate-time, we can solve the problem as follows:

  • We're looking for the time it takes Tom's mom to drive home using the equation: $t=\frac dr$.
  • Plugging in the values gives us $t=\frac{12.5}{37.5}=\frac13$ hours.
  • To convert $\frac13$ hours to minutes, we have to multiply $\frac 13$ by $60$, giving us $20$ minutes.
  • We get Tom's mom's arrival time by adding $20$ minutes to her starting time: $3:00 + 0:20 = 3:20$ pm.

• Find the distance each friend has to travel to arrive at the movies.

  Hints

  Think about which of the following forms of the distance-rate-time equation you need to use:

  • $r=\frac dt$
  • $t=\frac dr$
  • $d=t\times r$

  Think about the information you already have:

  • The time it takes each friend to get to the theater
  • The rate of travel for each of the friends

  You're looking for the distance each friend must travel.

  Convert the time in minutes to hours.

  For example: $15$ minutes corresponds to $\frac{15}{60}=\frac14$ hours.

  Solution

  We have to calculate the distances Tom and each of his friends need to travel to get to the theater. For every friend, including Tom, we have two pieces of information:

  • the time it takes each friend to get to the theater
  • the rate of travel for each friend

  Since we're looking for the distance, we should use the equation $d=t\times r$.

  Because the unit of measure for rate is miles per hour, we first have to convert minutes into hours.

  • Tom walks $20$ minutes at a rate of $3$ miles per hour.
  • Anne takes $0.5$ hours by bus at a rate of $30$ miles per hour.
  • Paul rides his bike $10$ minutes at a rate of $12$ miles per hour.
  • George also rides his bike $20$ minutes at a rate of $15$ miles per hour.

  • Tom's distance is $d=\frac13\times 3=1$ mile because $20$ minutes corresponds to $\frac{20}{60}=\frac 13$ hour.
  • Anne travels by bus at $30$ miles per hour for $0.5$ hours. We have to evaluate $d=\frac12\times 30$, which means Anne traveled $15$ miles.
  • Paul rides his bike for $10$ minutes, or $\frac{10}{60}=\frac 16$ hours. So Paul's distance is given by $d=\frac16\times 12$, which means Paul rode his bike for $2$ miles.
  • George rode his bike a bit faster than Paul at $15$ miles per hour. He also rode a little longer than Paul did. If we convert minutes to hours, we get $\frac{20}{60}=\frac 13$ hour. That means Paul rode for $d=\frac13\times 15$ miles, or $5$ miles.

• Describe the correlation between distance, rate, and time.

  Hints

  Convert minutes to hours:

  • $15$ minutes is $0.25$ hours
  • $30$ minutes is $0.5$ hours

  Pay attention to the units of measure:

  • Time is measured in minutes or hours
  • Distance is measured in miles
  • Rate is measured in miles per hour, or mph

  Solution

  Looking at the triangle on the right, we can establish the following equations:

  • $t=\frac dr$
  • $r=\frac dt$
  • $d=t\times r$

  Which equation we need depends on the unknown from the questions.

  If we're looking for the rate of Tom's mom, since she drives $12.5$ miles in $15$ minutes, we have to use the second equation. First, we have convert $15$ minutes to $0.25$ hours. This gives us $r=\frac{12.5}{0.25}=50$ mph.

  If Tom's school bus drives $25$ miles per hour for $30$ minutes, he will cover a distance of $12.5$ miles. For this question we need the third version of this equation. First, we have to convert $30$ minutes to $0.5$ hours. This leaves us with $d=0.5\times 25=12.5$ miles.

• Determine the time Tom needs to leave home in order to get to school on time.

  Hints

  The information we're given is:

  • The rate Tom rides his bike: $12$ miles per hour
  • The distance to Tom's school: $8$ miles

  We are interested in how much time it takes Tom to get to school. Therefore, we can use $\large{t=\frac dr}$.

  Subtract the time it takes Tom to get to school from Tom's school's starting time, $8$ am.

  To convert hours into minutes multiply the fraction of an hour by $60$ minutes.

  Solution

  Tom's school starts at $8$ am. When must he leave his home to arrive at school on time?

  We know that Tom bikes at an average rate of $12$ miles per hour. The distance to his school is $8$ miles. We need to know how long it takes Tom to get from home to his school.

  First of all, we have to decide which equation we need. We have to use $t=\frac dr$ because we know Tom's distance as well as his rate. We are looking for time.

  When we organize our given information we get:

  • $r=12$ miles per hour
  • $d=8$ miles

  Now we can plug in the known values into a version of the distance-rate-time equation: $d=\frac 8{12}=\frac 23$. This gives us time in hours. To convert this into minutes, we do the following: $\frac 23\times 60=40$ minutes.

  In order to calculate Tom's starting time, we have to subtract $40$ minutes from $8:00$ am. Since there are $60$ minutes in an hour, Tom must leave at $7:20$ am if he doesn't want to be late for school.