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### Isolating Groceries from Total Expenses, and Problems from Paragraphs

```Date: 09/03/2011 at 20:18:01
From: Wendi
Subject: R(t)=T(t)-G(t)

Average monthly expenditures for the Mays family's groceries and rent has
increased year after year. The total amount spent on groceries and rent,
combined, can be estimated using

T(t) = 0.1x^2 + 49x + 1125

Here, T(t) is the average monthly amount in dollars, and t is the number
of years since 1995.

The amount spent on groceries alone can be estimated using

G(t) = 1.1x^2 + 2x + 465

Here, G(t) is the average number of dollars spent per month for groceries,
and t is again the number of years since 1995.

Write an equation for the average monthly rent expense R(t) = T(t) = G(t).

A) R(t) = x^2 - 47x - 660
B) R(t) = x^2 - 47x_660
C) R(t) = 1.1x^2 + 47x + 660
D) R(t) = x^2_47x + 660

This whole thing is confusing to me. I am now taking college classes after
26 years out of school.

```

```
Date: 09/03/2011 at 23:20:26
From: Doctor Peterson
Subject: Re: R(t)=T(t)-G(t)

Hi, Wendi.

I've worked with lots of students in your position, and believe me, I
appreciate the effort it takes!

This is the sort of problem that uses a long story to disguise what is
really a simple problem. Sometimes what you have to do, at least in your
mind, is paraphrase the problem with all the extraneous details removed.

Replacing x with t (as the original problem stated, I'm sure), and
changing the last "=" to "-" (a typo):

T(t) = 0.1t^2 + 49t + 1125

Here, T(t) is the average monthly amount in dollars, and t is the
number of years since 1995.

G(t) = 1.1t^2 + 2t + 465

Here, G(t) is the average number of dollars spent per month for
groceries, and t is again the number of years since 1995.

Write an equation for the average monthly rent expense,

R(t) = T(t) - G(t)

In fact, we can do without the definitions of the variables and the
meaning of the functions:

T(t) = 0.1t^2 + 49t + 1125
G(t) = 1.1t^2 + 2t + 465
R(t) = T(t) - G(t)

They've just defined two functions, and asked you to subtract one from the
other. You have to subtract because of the meaning: the total is groceries
plus rent, so the rent is the total minus groceries.

The main purpose of this problem is to get you used to function notation.
Subtracting two functions just means subtracting the expressions they
represent, so what they're asking for is something you've probably done
many times:

Simplify the expression

(0.1t^2 + 49t + 1125) - (1.1t^2 + 2t + 465)

(By the way, the correct answer is not in your list, but you've got more
typos there anyway.)

Again, although now you should have no trouble completing the problem,
you'll want to focus on how to extract the real problem from all the
"noise" to get to this point. One way is to read backward (after reading
the problem through the usual way, of course). That is, look for what they
are asking (here, an expression for the function R); then for how R is
defined (as the difference of two functions); and in turn, for how those
functions are defined (by expressions). That allows you to focus on what
is really needed, and bypass the trivia. In a "real" problem, of course,
they wouldn't have given you the expressions; you'd have had to work them
out from the meanings of all the words. But in this problem, there's not
enough information to have done that, anyway.

Finally -- unless the book did this, which would be unforgivable -- your
typos suggest you may need help on one more point. You changed many of the
t's of the expressions to x's. Possibly you are not yet quite comfortable
with an essential idea in function notation: the variable in the
expression for a function is not always x, but is whatever is inside the
parentheses. Writing T(t) tells you that the variable they are using is t,
so that when you evaluate T(1), say, you'd replace t with 1. This t is a
placeholder; any letter could have been used just as well.

You may fully understand this, and have typed the x's purely out of
finger-habit, but I thought I'd mention it in case it's more than your
fingers that need reminding!

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Equations
Middle School Word Problems

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