# Find Total Distance

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##### Full Member
John took a drive to town at an average rate of 40 mph. In the evening, he drove back at 30 mph. If he spent a total of 7 hours traveling, what is the distance traveled by John?

Solution:

Round trip distance = D

Going rate = 40 mph
Returning rate = 30 mph

Going time = x
Returning time = 7 - x

40x = 30(7 - x)

My problem here is the TIME set up. I do not understand why x is the going time and (7 - x) the returning time. Is the solution provided dividing 7 into two parts? If so, what WORDS in the problem reveal this information?

#### lookagain

##### Senior Member
My problem here is the TIME set up. I do not understand why x is the going time and (7 - x) the returning time. Is the solution provided dividing 7 into two parts? If so, what WORDS in the problem reveal this information?
No, that is immaterial. It's fine if x is the return time and (7 - x) is the start time.
Yes, it divides seven hours into two parts. The words that stand out to me are
"he spent a total of 7 hours traveling."

That is similar to the discussion I had on another thread with you about two numbers that added to 90, where one part could be x, and the other part could
then be (90 - x).

##### Full Member
No, that is immaterial. It's fine if x is the return time and (7 - x) is the start time.
Yes, it divides seven hours into two parts. The words that stand out to me are
"he spent a total of 7 hours traveling."

That is similar to the discussion I had on another thread with you about two numbers that added to 90, where one part could be x, and the other part could
then be (90 - x).
Nicely explained.

##### Full Member
40x = 30(7 - x)

40x = 210 - 30x

40x + 30x = 210

70x = 210

x = 210/70

x = 3

John traveled 3 total miles.

#### MarkFL

##### Super Moderator
Staff member
40x = 30(7 - x)

40x = 210 - 30x

40x + 30x = 210

70x = 210

x = 210/70

x = 3

John traveled 3 total miles.
You are using $$x$$ to represent time, not distance. You have found that John took 3 hours to travel one way at 40 mph.

##### Full Member
You are using $$x$$ to represent time, not distance. You have found that John took 3 hours to travel one way at 40 mph.

Round trip distance = D

Going rate = 40 mph
Returning rate = 30 mph

Going time = x
Returning time = 7 - x

40x = 30(7 - x)

x = 3

40(3) = 120

Or

30(7 - 3)

30(4) = 120

D = 120 miles

#### MarkFL

##### Super Moderator
Staff member
You found the one way distance.

$$\displaystyle D=40t+30(7-t)=10t+210$$

Now, given that:

$$\displaystyle 40t=30(7-t)\implies t=3$$

We find:

$$\displaystyle D=10(3)+210=240$$

##### Full Member
You found the one way distance.

$$\displaystyle D=40t+30(7-t)=10t+210$$

Now, given that:

$$\displaystyle 40t=30(7-t)\implies t=3$$

We find:

$$\displaystyle D=10(3)+210=240$$
Wow! I got it wrong. Amazing. Oh boy!!

#### Otis

##### Senior Member
... 40(3) = 120

Or

... 30(4) = 120 ...
Use AND, not OR

That is, John is going AND returning, not going OR returning.

D = 120 + 120

#### MarkFL

##### Super Moderator
Staff member
Wow! I got it wrong. Amazing. Oh boy!!

##### Full Member
Minor to me now that I am not a student. Crucial in a classroom.

##### Full Member
Use AND, not OR

That is, John is going AND returning, not going OR returning.

D = 120 + 120

And not or, what? English lessons now???

#### Otis

##### Senior Member
And not or, what? English lessons now???
No, it's a math lesson.

In math, AND and OR are logical operators. In this exercise, there are two trips (one going; one returning).

AND means both.

OR means just one.

You were thinking OR (one trip's distance or the other), when you ought to have thought AND (both trips together).

##### Full Member
No, it's a math lesson.

In math, AND and OR are logical operators. In this exercise, there are two trips (one going; one returning).

AND means both.

OR means just one.

You were thinking OR (one trip's distance or the other), when you ought to have thought AND (both trips together).

Thank you for clearing that up for me.

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