Tag Archives: fractions

Guidelines for Writing Math Solutions

Writing a math answer for a Problem of the Week is very different from writing an essay in English class or a term paper in History class, so we would like to give you some guidelines. You write only one document, but we receive sometimes as many as 300-400 (or more) answers per week to read and analyze; when your presentation style is at its best, much time can be saved, a more efficient service can be provided, and everybody will be happier.


Readability

The first thing that would speed up the evaluation process can be called readability. Sometimes an individual sends us an answer in one long continuous paragraph, with equations embedded in it. Such paragraphs are very hard to read.

The solution is simple: just break the long paragraph up into several short ones, each one with its own concept, and leave a blank line between paragraphs.

Another matter regarding readability concerns polynomial expressions and equations. Notice the difference between these items:

    EASY TO READ                      HARDER TO READ
    
    x^2 + 2x + 1                      x^2+2x+1
    
    x^3 + 4x^2 - 6x + 10              x^3+4x^2—6x+10
    
    (3a + 4b)(3a - 4b)                (3a+4b)(3a-4b)

See how a space on either side of a plus or a minus sign makes the reading easier? (This is what good textbooks do.)

Similarly, when you show the steps in solving equations, add spaces and align the equals signs, like this (when you align text, never use tabs!):

    EASY TO READ                      HARDER TO READ
    
    2x + 48 = 58                      2x+48=58
                                      2x=10
         2x = 10                      x=5
    
          x =  5

Getting Off To a Good Start

After carefully reading a problem, it is essential to determine just what you need to find to answer the question posed. You should now select a letter, or letters, that will represent your unknown quantity, or quantities. This is the famous “Let” statement. Then, and only then, are you ready to begin forming your expressions or equations.

Be careful here, however. Many times “Let” statements aren’t clear. Examples:

GOOD ==> Let x = the number of apples in the basket

BAD ==> Let x = apples

In the latter case it’s not clear if we’re counting apples, or weighing pounds of apples. So be as precise as possible. It will save troubles later in your solving process.


Use of Guess-and-Check Procedure

In general, the method of “guess-and-check” is not allowed in AlgPoW as your primary strategy to solve the problem. This is not saying guess-and-check is not a good way to solve problems. In fact, it is often a good way to start to understand a problem, and therefore recommended for that. But for most of our problems, you must define variables or unknowns, then form equations to solve by logical steps.

Historically, it was the main way that problems were solved. But as advances were made in symbolic notation, mathematicians moved away from it and toward the more efficient and time-saving methods of step-by-step manipulations on equations.

One of our mentors advises students in the following way:

Guess and check is a valid problem-solving approach. However, it also one of the most difficult to explain. If you are going to use guess and check, you must list every guess, along with the reason that you know the answer is incorrect. You also must explain why you know your final answer is the only possible answer. In all, a pretty long process; however, since this is the Algebra Problem of the Week, you might want to try algebra. Please read the “Guidelines for Writing POW Answers.” The link is at the bottom of the problem.
So, unless otherwise indicated, please do not use guess-and-check as your principal solving procedure.


Writing a Complete Answer

The Problem of the Week (PoW) project here at the Math Forum, as you probably know, is a very unique one. Unlike other math tests in school or competitions (such as SAT), here we are not only interested in the right answer, but also how you arrived at it. This means, you must show your procedure and steps and thinking along with the final answer.

Even more so, your presentation must be explained well as you go from start to finish. Just imagine if you were to show your solution to a friend who was unfamiliar with the problem. Would that friend be able to read it and understand what you were saying?

ElemPoW has its own Guidelines document such as this one. Here is what is said there:
One good way to make sure you include enough information in your solution is to pretend you are explaining the problem to a friend who does not know anything about it. Imagine yourself leading your friend on a tour of your thinking as you solved the problem. How did you start? Where did you find the information you used? What were your calculations? How did you check your solution?

Math steps without a math explanation in words is much like watching a talented magician on stage. You see all the moves go by rapidly and you are “amazed”, but still you are left with the question, “How’d he do that?” Problem solving in PoW is not magic. Our goal is for everybody to understand as much as possible, according to his/her capacity.

Again, a thought from the ElemPoW service:

Our focus here at the Math Forum is not only on getting the correct answer, but also on communicating the steps involved in finding the correct answer.

To see the entire ElemPoW document about writing good answers, consult this page: How do you write a good math solution? There is much good advice to be found there.


E-Mail Notation

Sometimes we cannot write certain symbols (like exponents or square roots) in e-mail as we do using paper and pencil. Here are some examples:

Exponents

    It is standard now in e-mail to use the ^ (caret sign) found above the 6 on the keyboard for exponents. If we wish to say ‘four squared’, we write 4^2. For higher powers we do the same: ‘The volume of a cube is e-cubed’ would be V = e^3.

Square roots

    •                __                 _____
                    V64       or      \/a + b
  • Some people use the notation popular in spreadsheet applications, e.g. sqrt(16), to mean ‘the square root of 16′. This even applies in formulas; for the Pythagorean theorem, we can write:

     

    c = sqrt(a^2 + b^2).

    Other students ‘draw’ a square root symbol this way:

    [A few people try decimal or fractional exponents: 64^0.5 or 64^(1/2),
    but depending on the font this method can be difficult to read, so it is not recommended. However, there are occasions in which such exponents are better.]

Fractions

    •  1               15               3a  +  4b
      ---             -----            ------------
       2               25               5c  -  6d
      Two-fourths  2/4     five-sixths  5/6     etc.
      (3a + 4b)/(5c - 6d)
                 2
      Vertical: --- x      Horizontal: (2/3)x
                 3
                 2
      Vertical: ----       Horizontal: 2/(3x)
                 3x
  • Writing fractions is more complicated. There are two basic styles: vertical (sometimes called ‘stacked’) fractions, and horizontal fractions. Vertical fractions are what we are used to writing with pencil and paper, and are what you see in books. We can make them in e-mail as well; it just takes more effort and more keystrokes. But they are more readable when we need to write algebraic fractions.

    Horizontal fractions consisting only of numerals are easy to write, as these examples show:

    Even fractions that contain binomials, as shown above, can be written horizontally, if you employ parentheses. Observe:

    The difficulty arises when you need to express something like ‘two-thirds of x’. If you write this as 2/3x, it could be misinterpreted as 2 over 3x. Luckily we have ways of clarifying our meaning:

    Now if your intention really was 2 over 3x, you still have two options:

Subscripts

    • a1, a2, a3, a4, …
           y1 - y2
      m = ---------   instead of  m = (y_1 - y_2)/(x_1 - x_2)
           x1 - x2
      m = (y1 - y2)/(x1 - x2)
  • Unlike exponents, which go above the line (that’s why they’re sometimes called ‘superscripts’), subscripts go below the line. Unfortunately, the standard keyboard doesn’t have a true subscript key. Some people write a_1 for ‘a-sub-one’, but since many cases that need subscripts occur in sequences, we could write the following:

    to stand for a sequence of terms (a-sub-one, a-sub-two, …). In this context there is no real confusion with multiplying ‘a’ by 4. We universally write that as 4a.

    Notice how nice the slope formula can look using vertical fractions with this subscript style:

    The vertical equation looks almost like a line from a textbook, but even a horizontal equation like this one would be preferable:

Quadratic Formula

The quadratic formula is often needed in algebra problems. Here are two good ways to write it in email answers:

x = (-b +/- sqrt(b^2 - 4ac))/(2a)
                            -b +/- sqrt(b^2 - 4ac)
                       x = -----------------------
                                      2a

Determinants

    When you are using determinants to solve a system of equation by Cramer’s
    Rule, they may be nicely formed as shown here:
        |   3      5  |
    D = |             | = (3)(6) - (-1)(5) = 18 - (-5) = 18 + 5 = 23
        |  -1      6  |
    For a 3-by-3 case, the same idea applies:
        | a    b    c |
        |             |
    D = | d    e    f | = aei + dhc + gbf - gec - dbi - ahf
        |             |
        | g    h    i |

 

The method you use will often depend on the needs of the specific problem you are working; these comments should be understood as suggestions and general guidelines only.

 


 

Distinct Digit Fraction Sums

Observe the following fraction addition carefully:

                  1     6
                 --- + ---  =  1
                  4     8

On the left side of the equation there are four distinct digits — 1, 4, 6, and 8. While that may not look like earth-shattering news to some people, I think it looks nice.
Can you make up a similar example?
This means, can you find another equation of the form

                  a     c
                 --- + ---  =  1
                  b     d

where a, b, c, and d are distinct digits?
You know, it may not be as easy as it looks. This shall be called a Type I expression.


Now, how about another variation on that theme? Observe this structure…

                  a     c       e
                 --- + ---  =  ---
                  b     d       f

where a, b, c, d, e, and f are distinct digits, and e/f < 1.

Can you find a solution to that?

This shall be called Type II.


Want to go for more? Well, then look at this.

                  a      d
                 ---- + ---  =  1
                  bc     e

where “bc” represents a “two-digit number” (like 27 or 83), and not the algebraic multiplication of 2 values.

This shall be called Type III.


Hey, I’m not done yet. Try your luck, er skill, on this one.

                  ab     e
                 ---- + ---  =  1
                  cd     f

where again “ab” and “cd” represent “two-digit numbers” (like 14 or 65), and not the algebraic multiplication of 2 values.

This shall be called Type IV.


If you can show me an answer to any of these questions, send it to me by email and I will present it here on this page in the charts below.

Please note: that in order for your solution to even be considered for posting, you must write “DDFS” in the subject line of your email; otherwise I will merely ignore it and delete it. Thank you.

trottermath@gmail.com or ttrotter3@yahoo.com

For an important UPDATE, see below Chart IV…


Type I

# Solution Name Date
1 1/4 + 6/8 Daniel Lu 4/30/01
2 1/2 + 3/6 Konstantin Knop 8/9/01
3 1/3 + 4/6 Jacqueline Hu 10/24/01
4 3/4 + 2/8 Jacqueline Hu 10/24/01
5 1/2 + 4/8 Leonard Lee 11/4/01
6 2/4 + 3/6 Leonard Lee 11/4/01
7
8

Type II

# Solution Name Date
1 1/4 + 2/8 = 3/6 Konstantin Knop 8/9/01
2 3/9 + 1/6 = 2/4 Leonard Lee 11/4/01
3
4
5
6

Type III

# Solution Name Date
1 2/10 + 4/5 Konstantin Knop 8/9/01
2 2/16 + 7/8 Jacqueline Hu 10/24/01
3 2/14 + 6/7 Jacqueline Hu 10/24/01
4 8/14 + 3/7 Jacqueline Hu 10/24/01
5 5/10 + 4/8 Leonard Lee 11/4/01
6 4/12 + 6/9 Leonard Lee 11/4/01
7 5/20 + 6/8 Leonard Lee 11/4/01
8 7/21 + 6/9 Leonard Lee 11/4/01
9
10
11
12

Type IV

# Solution Name Date
1 13/26 + 4/8 Konstantin Knop 8/9/01
2 15/30 + 2/4 Leonard Lee 11/4/01
3 15/30 + 4/8 Leonard Lee 11/4/01
4 15/60 + 3/4 Leonard Lee 11/4/01
5 19/38 + 2/4 Leonard Lee 11/4/01
6 16/48 + 2/3 Leonard Lee 11/4/01
7
8
9
10

On August 9 and 10, 2001, Konstantin Knop, from St. Petersburg, Russia, sent in some solutions to our problems posed above. But he extended the concept to include more types. And he provided solutions as well.
So we now present his extension ideas with two samples of solutions for each one. Wouldn’t you like to join him and send in a solution or two of your own?

Here is Type V.

                  ab      e
                 ----- + ---  =  1
                  cde     f

Type V

# Solution Name Date
1 34/102 + 6/9 Konstantin Knop 8/9/01
2 26/130 + 4/5 Konstantin Knop 8/9/01
3 35/140 + 6/8 Leonard Lee 11/4/01
4 72/108 + 3/9 Leonard Lee 11/4/01
5 53/106 + 2/4 Leonard Lee 11/4/01
6 78/156 + 2/4 Leonard Lee 11/4/01
7
8
9
10

Next is Type VI.

                  ab      f
                 ----- + ----  =  1
                  cde     gh

Type VI

# Solution Name Date
1 64/208 + 9/13 Konstantin Knop 8/9/01
2 85/136 + 9/24 Konstantin Knop 8/9/01
3
4
5
6

And now Type VII.

                  ab      fg
                 ----- + ----  =  1
                  cde     hi

Type VII

# Solution Name Date
1 24/136 + 70/85 Konstantin Knop 8/9/01
2 96/324 + 57/81 Konstantin Knop 8/9/01
3 45/180 + 27/36 Leonard Lee 11/4/01
4
5
6

This is Type VIII.

                  abc     gh
                 ----- + -----  =  1
                  def     ij

Type VIII

# Solution Name Date
1 148/296 + 35/70 Konstantin Knop 8/9/01
2 204/867 + 39/51 Konstantin Knop 8/9/01
3
4
5
6

This is Type IX.

                  ab      fg
                 ----- + -----  =  1
                  cde     hij

Type IX

# Solution Name Date
1 57/204 + 98/136 Konstantin Knop 8/9/01
2 59/236 + 78/104 Konstantin Knop 8/9/01
3
4
5
6

We like Type X.

                  abcd     i
                 ------ + ---  =  1
                  efgh     j

Type X

# Solution Name Date
1 1278/6390 + 4/5 Konstantin Knop 8/10/01
2 1485/2970 + 3/6 Konstantin Knop 8/10/01
3
4
5
6

August 12, 2001…

Let’s continue our patterns. Here’s another variation on Type II.

                  a     c     e
                 --- + --- + ---  =  1
                  b     d     f

Type XI

# Solution Name Date
1 1/4 + 2/8 + 3/6 Leonard Lee 11/4/01
2 3/9 + 1/6 + 2/4 Leonard Lee 11/4/01
3
4
5
6

Adding Fractions

A Case Approach

Everyone, or nearly everyone it seems, hates fractions, especially their addition and the other operations. Of particular note is the fact that the rules for addition and multiplication are often confused and interchanged, and the rule for division is just plain forgotten.

This article attempts to present WTM’s own approach to one aspect of this situation, the addition of just two fractions. The thesis is: that if this is thoroughly understood, then the other operations will cause less misunderstanding and frustration; and later the student will operate with “algebraic” fractions in an Algebra I or II course with
greater comprehension of the basic structures involved.


This method, here called a “case approach”, takes the a priori assumption that the student already knows (or can acquire soon) the simple fundamentals of elementary number theory, e.g. such terms as least common multiple (LCM), greatest comon factor (GCF), and relatively prime numbers. Also it is assumed that the basic concepts and terminology of fractions (numerator, denominator, reduce, equivalent fractions) are likewise understood.

The key idea in this approach is to focus our attention first on the denominators, according to the following outline:

Case 1. Same denominators
Case 2. Different denominators
a) Relatively prime denominators
b) Not relatively prime denominators
c) Special case


Case 1 is practically self-explanatory. Both denominators are the same number, hence the addition comes naturally. Few individuals have much trouble here. Examples:

                                    2       5       7
                without reduction: ---  +  ---  =  ---
                                    9       9       9

                                  1        7        8       2
                with reduction: ----  +  ----  =  ----  =  ---
                                 12       12       12       3

However, Case 2 is the situation where the most difficulties arise. Let’s examine it more carefully now.


It is a fact in number theory that when two numbers are relatively prime (i.e. their GCF is 1), that their LCM is the product of the given numbers. Example:

The LCM of 5 and 8 is 5 × 8, or 40.

Therefore, when the two numbers are then used as denominators of fractions, the lowest common denominator (LCD) is, in effect, the aforementioned LCM. So, for Case 2a, the LCD is fairly automatic. Note:

                         2       3
                        ---  +  ---  =  ----  +  ----
                         5       8       40       40

In this problem the numerators are 2×8, or 16, and 3×5, or 15, respectively, yielding

                         2       3       16       15
                        ---  +  ---  =  ----  +  ----
                         5       8       40       40

The problem has now been converted to Case-1 style, hence the answer, 31/40 is now obvious. It is interesting to point out that if in a Case-2a problem the original fractions are given in “reduced” form to start with, then the sum itself will also be in “reduced” form, or as is often said, “in lowest terms”.

The foregoing example could be written in a condensed form as follows:

                         2       3       16 + 15      31
                        ---  +  ---  =  --------- =  ----
                         5       8          40        40

This mode of presentation suggests the formula for the addition of two fractions in Case-2a:

                         a       c       ad + bc
                        ---  +  ---  =  --------- 
                         b       d         bd

[As will soon be pointed out, this formula can technically be used for ANY two fractions, though this may not necessarily be the most desirable or practical thing to do in specific instances.]


Now let’s examine the most intriguing case: Case 2b. Here the denominators are NOT relatively prime, that is, their GCF is greater than 1. This means that their LCD (alias LCM) is some value less than their product. An example will clarify this:

			 3	 5
			---  +  ---  =   ?
			 4	 6

Here the LCD is not 24, but rather 12. So, we have (step by step):

		   3	   5
	Step 1:   ---  +  ---  =  ----  +  ----
		   4	   6	   12	    12

		   3	   5	    9	    10
	Step 2:   ---  +  ---  =  ----  +  ----
		   4	   6	   12	    12

		   3	   5	    9	    10	     19
	Step 3:   ---  +  ---  =  ----  +  ----  =  ----
		   4	   6	   12	    12	     12

Although the example above usually causes no great difficulty for most individuals (due to the smallness of the denominators), in other problems the search for the appropriate LCD can produce a certain degree of anxiety. [For example, try this problem and see how much time you spend finding the LCD, then using it:

			  7        8
			----  +  ----.]
			 26	  39

This then raises the question: What about the formula? Can it still be used? As was suggested earlier, the answer is “yes.” Note:

		   3	   5	   18 + 20	 38	  19
	          ---  +  ---  =  ---------  =  ----  =  ----
		   4	   6	    4 × 6	 24	  12

The only difference now is that a reduction step will always be necessary in order to give the answer in lowest terms, as is normally required.

Now it can be stated: Case 2b can be done in either of two ways: 1) using the LCD, or 2) using the formula. There are advantages and disadvantages for both methods.

A few words are in order for the formula’s disadvantages. In this modern era, the easy access to electronic calculators make the large number aspect no problem at all. The cross products are quickly done and (if you are using the simple 4-function models with memory keys) stored in the memory, where they are even added for you as well.
Note our big-number example earlier:

		  7        8      273 + 208	   481
		----  +  ----  = -----------  =  ------
		 26	  39       26 × 39        1014

The reduction step is also not as difficult as it often used to be before calculators either. This is because the GCF of the two original denominators (13 in the example above) is usually easier to find, or at least easier (and smaller) than the LCD. Dividing both parts of the fraction 481/1014 by 13 gives its reduced form: 37/78.

So when should one use a certain method? I suggest that it’s largely a matter of personal preference. We need to show students that there are times when a problem can be done in more than one way. This should tend to relieve a lot of the math anxiety prevalent in people’s minds.


Lastly, we turn our attention to Case 2c, the Special Case. This arises when one of the given denominators is already a multiple of the other, or equivalently the former is a factor of the latter.
Example:

		 3	 7	 6	 7	 13
		---  +  ---  =  ---  +  ---  =  ----
		 4	 8	 8	 8	  8

Note that here only one fraction had to be changed to an equivalent one, thereby keeping the numbers as small as possible. This case
should be stressed more than it normally is because it’s the one that probably occurs more frequently in “real life” situations.


In this brief article, I have not pretended to be exhaustive in my presentation of all the ramifications normally encountered while doing the variety of textbook problems that students are expected to master. Nor has mention been made of a) improper-vs-mixed-number answers, or b) that with the calculator* all the fractions could be converted to decimal form and added in the memory even more easily still. Rather it has been to offer a basic structure than can be readily used to organize one’s thoughts along the lines of number theory and that the types of denominators govern what strategy must or might be taken in a particular situation.


*Footnote: The original text of this article I wrote at a time when most young students might have access only to simple 4-function models with memory keys. Now scientific models are very common. And lately there have even appeared models that “do fractions” in pure fraction form! My, how the world is changing!


Appendix

An interesting fact from number theory not well known by the average student regarding LCM’s and GCF’s greatly aids the search for the LCD (LCM) in fraction work. It is that the product of the LCM and GCF of two numbers is equal to the product of the two numbers themselves. Symbolically this means:

LCM(a, b) × GCF(a, b) = a × b

This permits us to state a formula for LCD’s (LCM’s):

			                  a × b
			LCM (a, b)  =  -----------
			                GCF(a, b)

For our “big number” problem above, we can write:

                                   26  ×  39       26 × 39
                LCM(26, 39)  =  --------------  =  -------
                                 GCF (26, 39)         13

                             =  2 × 39  (or  26 × 3)

                             =  78