Working with Formulas
Chapter 5: Working with Formulas
3. Add and subtract
Try your hand at the order of operations with the example problems in the following sections.
Example 1
Simplify the following expression using the order of operations:
6 2 4 3
2 18 121 + 3-
$
+Because the expression has no grouping symbols, you know that you first need to deal with the power and root. So raise 2 to the third power and find the square root of 121. Be sure to put a dot or parenthesis to indicate multi- plication between the fraction and the answer to the root of 121. After the radical is dropped, you lose the grouping symbol that indicates multiplica- tion. Here’s what your new expression looks like:
6 8 4 3
2 18 11 + -
$
+$
Now, moving left to right, multiply the 4 and 3, divide the 18 by 2, and then take the result of the division and multiply it by 11. After all these calcula- tions, you get the following:
6+8-12+9 11
$
=6+8-12+99Now, add the 6 and 8 to get 14. Subtract the 12 to get 2, and add the 2 to 99.
The final answer is 101.
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Chapter 5: Working with Formulas
Example 2
Evaluate numbers in the following formula for compound interest (see Chapter 9):
, .
A P 1 nr 10 000 1
4 0 06 ( )
n t
4 25
= +
= +
c c
m m
Note:In this version of the formula, the rate, r, is replaced with 6% (or 0.06), the nis replaced with 4 to represent quarterly compounding, and the tis replaced with 25 for that many years.
Start by simplifying the fraction in the parentheses and adding it to 1. Then raise the result to the appropriate power and multiply the result by 10,000.
Here’s what your math should look like:
, .
, .
, .
, . , .
A 10 000 1 0 015 10 000 1 015 10 000 4 43204565 44 320 4565 44 320 46
( ) 4 25
100
.
= +
=
=
=
^
^
^
h h
h
Example 3
Simplify the numbers in the following formula for the payment amount of an amortized loan (see Chapter 12 for more on amortized loans) where $120,000 is borrowed at 0.08 interest for 60 months:
.
, .
R 1 1 0 008 120 000 0 008
= 60
-^ + -
^ h
h
You start by first working on the denominator. Add the numbers in the paren- theses and then raise the result to the appropriate power. Subtract that result from 1. Multiply the two numbers in the numerator together, and then divide the sum by what’s in the denominator. Your math should look like this:
.
, .
.
, .
.
, .
. , .
R 1 1 008
120 000 0 008
1 0 61996629 120 000 0 008
0 38003371 120 000 0 008
0 38003371
960 2 526 09
60
.
= -
= -
=
=
^ -
^
^
^ h
h h h
Making sure what you have makes sense
When simplifying expressions involving lots of numbers and operations, it’s easy to make a simple arithmetic error. You may have a mental meltdown, or you may enter the numbers into the calculator incorrectly. For these and sev- eral other reasons, it’s always a good idea to have an approximate answer in mind — or, at least, a general range of possible answers.
For example, say you want to find the interest earned on a deposit of $10,000 for 10 years. If you come up with an answer of $100,000 in interest, hopefully you’ll realize that the answer isn’t reasonable. After all, that’s more interest than the initial deposit! Unheard of! (Or, if it’s correct, tell me what bank you’re using because I’m going to sign up!)
Here’s another example: If you’re determining the monthly payments on a
$4 million house and come up with $500 per month, you should be a bit sus- picious of your answer (and ever hopeful). Why? Well, if you think about it,
$500 per month is awfully low for such a large home — and it would take almost 700 years to pay back just the principal at this rate.
True, you’ll come across some situations where you have no clue what the answer should be. I’m faced with them all the time. You just work as carefully as you can, and perhaps check with someone else, too. In the big picture of mathematics, though, using common sense goes a long way toward accuracy.
Computing with Technology
Gone are the days of the abacus, the rolled-up sleeves, the green visor, and the quill pen. Okay, I’ve really mixed up some computing centuries, but you probably get my point. Technology is here to stay. And that technology can make your life much easier. However, you also can get into trouble much more quickly when using technology; in a complex spreadsheet, a simple error in only one cell can create calculating havoc.
In this section, I don’t try to sell you any particular brand of calculator or any particular spreadsheet product. I’m just going to whet your appetite — tease you with some neat features that technology brings to your business math table. The directions and suggestions are as general as possible. So it’s up to you to check out your own personal calculator or computer spreadsheet to find the specific directions and processes necessary.
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Chapter 5: Working with Formulas
Calculators: Holding the answer in the palm of your hand
No matter how much you love to crunch numbers in your head, you eventu- ally come across calculations that are too big or too difficult. In that case, you look to a calculator for help. You can work most of these computations with a simple scientific calculator. A scientific calculator raises numbers to powers and finds roots and the values of logarithms.
You can go fancy-schmantzy and invest in a graphing calculator (or even in one that does calculus), but be aware that the more features you have in your cal- culator, the more opportunities you have of going astray. Besides, if you don’t need to do fancy calculations, why spend more money on a fancy calculator?
The main challenge of using a calculator is to direct the calculator to do what you mean.The calculator computes with a certain set of rules — mostly based on the order of operations (see the earlier section, “Operating according to the order of operations,” for more). So you need to push the right button, use enough parentheses, and interpret the resulting answer. That way your calcu- lator computes the right information and you answer your question correctly.
The four basic operations each have their own button on a calculator. When you hit the + or - button, you see the + or – right on the screen. When you hit #, most calculators show the multiplication as an asterisk (*). The ' button usually conjures a slash (/).
Wielding the power of exponents
Many financial and geometrical formulas involve exponents (powers) of the values in the expression. Some examples of these types of formulas include the following:
Area of a circle:A= πr2
Accumulated money from compound interest: A P 1 nr
n t
= c + m Regular payment amount in a sinking fund: R
i Ai
1 n 1
=^ + h -
You access the power (exponent) button on a scientific calculator in one of several ways. If you want to square a value (raise it to the second power), you hit the button that looks like this:
x2
The square, or second power, is usually the only power to get its own button on a calculator. Too many other powers are used in computing, so the rest are taken care of with a general power button, which looks like one of the following:
, xy, yx /
The two buttons with xraised to the yor yraised to the xare sort of scripted.
You have to enter the numbers exactly in the correct order for the calculator to compute what you mean. To use this button, you put in the xvalue first, you hit the button, and then finally you put in the yvalue.
I’m sure you’re dying to try out these new tricks on your calculator. As prac- tice, determine how to enter the following expressions in a calculator:
72+ 1 35– 41/3 2(58)