Q. 4.41 - Intermediate Algebra

Question

Translate to a system of equations and then solve:

Mitchell left Detroit on the interstate driving south towards Orlando at a speed of $60$miles per hour. Clark left Detroit$1$hour later traveling at a speed of $75$miles per hour, following the same route as Mitchell. How long will it take Clark to catch Mitchell?

Step-by-Step Solution

Answer

Clark will catch up Mitchell in $4$ hours.

Step 1. Given

Mitchell, Detroit, and Clark are travelling towards south on the same route.
Mitchell left Detroit at a speed of $60$mph. One hour later, Clark left Detroit at a speed of $75$ mph on the same route.

Step 2. Finding the distance.

Assuming the time of Mitchell and Clark as $m\&c$, we will find the distance for them.
As we know that $D=R\times T$,we will use the same formula.

1. Mitchell's rate is $60 mph$and the time is $m$, so the distance comes out to be $60m$.
2. Clark's rate is $75 mph$ and the time is $c$, so the distance comes out to be $75c$.

And as we know that Clark left Detroit one hour later than Mitchell, so Clark's time will be an hour less than that of Mitchell. This can be written as $c=m-1$.

Step 3. Substituting the values.

To get a system of equations, we must recognize that both will drive the same distance. So this gives us $60m=75c$.

We have two equations, i.e.
$$\begin{gathered} 60m=75c \\ c=m-1 \end{gathered}$$

Substituting the value of $c$ in first equation gives us $60m=75(m-1)$.

Step 4. Finding the values.

We have $60m=75(m-1)$.

Solving the equation,

$$\begin{gathered} 60m=75m-75 \\ 15m=75 \\ m=5 \end{gathered}$$

Substituting this value in $c=m-1$ give us $c=4$.

This means that Clark will take $4$ hours to catch Mitchell and Mitchell would have travelled $5$ hours.

Step 5. Check the solution

We need to check the distance travelled by both Mitchell and Clark. If the distance comes out to be the same, our answer is right.

So,

1. Mitchell has travelled $5$ hours with a speed of $60 mph$, so distance$=5\times 60$ miles
   $=300$ miles.
2. Clark has travelled $4$ hours with a speed of $75 mph$, so distance$=4\times 75$ miles
   $=300$ miles.

The distance covered is same, i.e., our answers are right.