Unit Rates

Written by tutor Steve C.

We’d like to simplify TWO issues with ONE idea.

In the top rated TV show, NCIS,
what do they do when they have no evidence?
They FOLLOW THE MONEY.

Basically, if Fred dies & Sue makes lots of money,
then Sue is gonna need a good alibi.

So, let’s tweak FOLLOW THE MONEY
And turn it into FOLLOW THE UNITS.

This will be our new rule

Let’s consider the frequently dreaded
Rate – Distance – Time problems (or RDT problems).

Some teachers try to have students memorize codes.
A common one is “DIRT” where D = RT.
The problem with memorizing is … sometimes, you can forget.
They might recall DR = T or D = R/T, both of which are incorrect.

So, let’s get down to basics and consider the new rule:
FOLLOW THE UNITS

When you think about Rate, what is a Rate?
Somebody says that they’re going 65 down the highway.
The question is, 65 what?

Well, the answer is 65 miles per hour, or 65 mph.
Every 7 yr old who watches TV or rides with their folks knows that.

So … Consider Rate – Distance – Time problems.

If 65 miles per hour is the Rate…
Then, what is miles? The Distance.
Then, what is hour? The Time.

Therefore, by definition, since Rate = miles / hours
Then Rate = Distance / Time

This was not from Memory. What’d we do to get there?
FOLLOW THE UNITS

So, if R = ^D/_T
We can solve simple RDT problems, like

What the rate if you go 200 miles in 4 hours?

R = ^D/_T
R = 200/4
R = 50 mph

Likewise, if you invoke the Downstairs Swap, you can turn
R = ^Dist/_Time       into       T = ^Dist/_Rate

What’s the time if you go 200 miles at 50 mph?

T = ^D/_R
T = 200/50
T = 4 hours

This was not from a memorized equation. This results when you
FOLLOW THE UNITS

You can confirm this with the well known “DIRT” equation
(if you really want to use a memorized equation.)

What’s the distance if you go 50 mph for 4 hours?
D = RT
D = (50)4
D = 200 miles

This was not from a memorized equation. This was from basics:
the definition of a Rate, which we all learned when we were 7 yrs old.

This set of equations are the result when you

FOLLOW THE UNITS

So, that’s the first issue.

Now, for the second issue:

What happens if you need to convert a value, but you don’t have an internet converter handy? Something like mph into ft/sec; How do you do it?

FOLLOW THE UNITS

Let’s review two basic foundation “1” facts in Math: theory

Anything times 1 is that thing.
(A)(1) = A

Anything over itself = 1
A / A = 1
Therefore… If 12in = 1 ft, then

1 = ^{1 ft}/_{12 in}       and       1 = ^{12 in}/_{1 ft}

How to know which fraction to use in any conversion?

FOLLOW THE UNITS

If we multiply    ^mi/_hr(^ft/_mi), we get ^ft/_hr

If we multiply    ^mi/_hr(^hr/_s), we get ^mi/_s

So what is 60 mph in ^ft/_s?

60 ^mi/_hr(5280 ^ft/_mi)(^{1 hr}/_{3600 s}
60 ^mi/_hr(5280/3600)
60 ^mi/_hr = 88 ^ft/_s

This will work with any complex unit conversion with basic “1” facts. Just another examplpe of using the definitions and the new rule.

FOLLOW THE UNITS.