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Tutorials
LABELS
"Some things
which can be neither counted nor measured can still be savored."
Edward MacNeal,
"Mathsemantics," (Viking, 1994), page 39
You have probably had math teachers who told you to "label your answers." We're even more of a nuisance, because we are going to tell you to carry labels throughout each problem that you solve. Most math isn't done with plain numbers; it's done in the context of practical problems involving quantities of things. Labeling each quantity allows you to remember what the numbers refer to and to better organize your problem solving.
Consistency is a must in labeling, but consistency doesn't necessarily mean that the labels must all be the same. The surprising thing is that labels can allow us to do calculations with apparently dissimilar things. Of course, the things must have some connection in the context of the problem being solved. So, we can add apples and oranges to get pieces of fruit. If we want to add hours and minutes, we need to make clear, at each stage, which unit (hours or minutes) the numbers refer to.
Remember that we can't calculate with a mixture of labeled numbers and pure numbers. 12 people cannot be divided by the number 3. 12 people can be divided by 3 teams. Likewise, the number 12 can be divided by the number 3.
examples:
1. The passenger
list for flight 508 consists of twentytwo men and thirteen women. How
many passengers are on flight 508?
Explanation
Ill bet you think this is easy, and it is  just add 22 + 13 = 35. Notice, however, that you are adding "apples and
oranges" or, in this case, men and women. It's O.K., because they have something in common within the context of the problem: all are passengers. So, our problem is really 22 men + 13 women = 35 passengers. Can you think of a situation in which you literally add apples and oranges?
2. 15 dogs
 3 beagles = ?
Explanation
Since beagles are dogs, we can subtract: 15 beagles  3 dogs = 12 dogs. Notice that we don't know anything about the 12 except that they are dogs.
3. Last week,
JoAnne worked 3 hours and 47 minutes on Tuesday, 4 hours and 19 minutes
on Wednesday, and 53 minutes on Friday. How long did she work last week?
Explanation
You can approach this problem in at least two different ways. Hours and minutes are units of measurement. In order to add them, we must only add units which are measured in the same way, i.e. hours to hours or minutes to minutes. The same is true for any unit of measurement, for instance feet and inches.First method: 3 hours 47 minutes Add the hours; then add the minutes separately.
4 hours 19 minutes
+ 53 minutes

7 hours 119 minutes
To convert 119 minutes into hours and minutes, you need to know that there are 60 minutes in an hour. So 1 hour = 60 minutes, 2 hours = 120 minutes (2 * 60) which is 1 minute more than the 119 minutes in our problem. That means that 119 minutes = 1 hour 59 minutes.7 hours8 hours 59 minutes ans. Total amount JoAnne worked last week.
+1 hour 59 minutes
Second method: Convert everything to minutes.
3 hr. 47 min. = (3 hr. * 60 min. per hr) + 47 min. = 180 min. + 47 min. = 227 min.In this case, we would normally convert back to a mixture or hours and minutes. To find the number of hours, do: 539 min. / 60 min. per hr. = 8 hr. (rounded to the nearest hour)
4 hr. 19 min. = (4 hr. * 60 min. per hr) + 19 min. = 240 min. + 19 min. = 259 min.
+ 53 min.

539 min.
8 hr. = 8 hr. * 60 min. per hr = 480 min..
539 total min.  480 min. in 8 hr. = 59 min..
So, our answer is the same: 8 hr. 59 min..
4.
8 apples * 2 = ?
Explanation
Strictly speaking, we cannot do this problem, because we do not know what 2 represents. For instance, 8 apples * 2 oranges doesn't make sense. However, 8 apples * 2 pies could represent doubling the recipe for apple pie. In general, it's best not to mix unlabeled and labeled numbers.
Remember that we usually do math to solve a problem. Part of an effective solution is organizing our information. Another part is communicating our solution to others. Clear, consistent labels help with both aspect of problem solving.
You can test your skills in the Labels portion of the Basic Math SelfAssessment.
ORDER OF OPERATIONS
Unfortunately, not all calculators follow the mathematical order of operations. If you used a calculator for this section of the assessment and still had wrong answers, you may have a calculator which does each calculation as it is keyed in from left to right. Very simple calculators often do this. To correct for this problem, you need to learn the correct order of operations described below and key in the calculations in the right order.
Calculators which store the whole string of calculations and do them all according to the order of operations when you press = can be purchased for $10 or less. Even so, you will need to know the order of operations for more complex calculations and equations.
Rules of the Order of Operations:
You may want to try each of the examples without a calculator first. When you understand the order, try it using your calculator. Can you reach the same solution?
1. Do everything inside brackets [ ] and parentheses ( ) first, working outward from the innermost set.
example: [ 3 + (8  (4 / 2)) 1] = 8Within a given set of parentheses or brackets ( or if there are none), do calculations in the following order:
do: (4 / 2) = 2
(8  2 ) = 6
[3 + 6  1] = 8
2. Do exponents and roots first.
example: (3 + 2^2) + (4 + 1)^23. Do division and multiplication next, before addition and subtraction.
do : 2^2 = 4 within first set of ( ) do the exponent first
(3 + 4) = 7 do addition within first set of ( )
(4 + 1) = 5 do the addition within the second set of ( )
5 ^2 = 25 the exponent is outside the ( ), so do it next
7 + 25 = 32 ans. add the results from the two sets of ( )
example: 3 + (8  4 / 2)  2 * 24. Do addition and subtraction last.
do: 4 / 2 = 2 do the division within ( ) first
(8  2) = 6 finish the subtraction within ( )
since the ( ) are gone, the problem is now 3 + 6  2 * 2
2 * 2 = 4 do the multiplication before addition & subtraction
3 + 6  4 = 5 do addition and subtraction from left to right
example: Alison had a starting balance of $1567 wrote checks for the following transactions this month: $53 for phone; two loan payments of $127; the minimum payment of $25 plus 10% of the $2734 balance on her credit card. a) Can you represent what happened in her checking account arithmetically, considering the order of operations? b) What is Alisons balance after paying her bills?
do:
a) $1567  $53  $127 * 2  ($25 + $2734 * .10) = ? This is one of the correct answers.
b) ($25 + $2734 * 10) = $298.40 This is done first; it is in ( ). Multiply before adding.
$ 1567  $53  $127 * 2  $298.40 = ? The equation now looks like this.
$127 * 2 = $254 There are no more ( ), so do multiplication next.
$1567  $53  $254  $298.40 = $961.60 is Alisons balance.
FRACTIONS
Fractions
and decimals get bumped around quite a bit. There's little difficulty when
asked how to share a pizza fairly, but many cringe at seeing something
like the following in print:
1/4
+ 1/8 = ?
There are many ways to solve problems with fractions. For some, using a visual clue, such as a pizza pie or a square can do the trick:




Here is a square
divided into four parts.
Each part
is 1/4 of the square.







