Fractions in a Nutshell

Instructors: The example below involve boxes or tables. Further example could be generated using pies, one or many, and  rulers or divisions of lengths along line segments. The material below could be expanded into several lesson in which you cover the underlying concept carefully and less briefly. 

Students: The explanation below of fractions and operations on them may be too brief or cryptic. See what you can follow now, but you may want to read after the algebra lessons at this site. A later version of this lesson (with more numbers and examples) may be easier to read. 

Would you like to show yourself or others how to be  algebra power users?.

1. Meaning of Fractions

• When a object can be divided into 2 parts of equal value, we say that each part equals a half (½ )
• When a object can be divided into 4  parts of equal value, we say that each part equals a quarter (¼ )
• When a object can be divided into 4 parts of equal value, two of those parts provide a half (½) of the object and three of the parts provide three quarters ( ¾) of the object.

The simple fraction 

In symbols or shorthand notation (with a * meaning times)

 Here 

is unit fraction. All fractions are multiples of simple fractions 

Think of a unit fraction like a third

1
3

as unit for counting not whole objects, but parts of them. With that viewpoint, can count 2 thirds,  4 thirds and  8 thirds of an object. We also have 

2 thirds + 5 thirds = 8 thirds.

or in fraction notation

Whole numbers like 20 will be regarded as 20*1. Below we have to add whole numbers to fractions.

2. Unit Fractions of Unit Fractions

First Example: When we divide each third into four equal parts, we get  3 x 4 = 12 equal parts. 

So a quarter of a third is a twelfth. In words and shorthand symbols

where 12=3*4. In shorthand symbols only (replace of by times sign *)

We also see that  4 twelfths is a third from above diagram. In symbols, we may write

Second Example: Dividing a half of a length into three equal parts turns gives 6 = 2*3 equal parts in all.  Each part is a sixth.

In words and shorthand symbols

and with shorthand symbols only

Here we see 

Algebraic Shorthand Description of a Unit Fraction of a Unit Fraction

In general for whole numbers M and N (take M = 5 and N =7 and M*N = 35 on first pass)

1 N

of

1 M

=

1   N*M

and  

3 Unit Fraction of a Simple Fraction

First Example:  Use the following diagram to find a fifth of three quarters:

Splitting each quarter into five equal parts gives 4*5 equal parts (third row). So each box in the third row is a 20th. Each box in the third row represents a fifth of a quarter. 

So a fifth of three quarters should be three times greater than a fifth of quarter. That is we should have

Count the silver boxes in the third row to confirm this. 

Second Example:  Use the following diagram to find a half of two thirds

Splitting each third into two equal parts gives 2*3  equal parts (third row). So each box in the third row is a sixth  Each box in the third row represents a half of third.. 

So a half of two thirds should be twice greater than a half a third.  That is we should have

Count the silver boxes in the third row to confirm this.  Observe that each third is two sixths. So the answer could also be written as a third.  The explanation of equivalent fractions below will go further into matter.

Algebraic Shorthand Description of  Unit Fraction of a Simple Fraction

In general for whole numbers M and N (take M = 5 and N =7 and M*N = 35 on first pass)

1 N

of

B M

=

B   N*M

or with a times symbol  * instead of the word of, we write.   

1 N

*

B M

=

B   N*M

In the first example above,  B = 3, M = 4 and N =5 while in the second example B = 2, M = 3 and N = 2 

4. Simple Fraction of a Simple Fraction

By definition or convention above,  4 fifths of an object is 4 times a fifth of the object.  If the object itself is a simple fraction of another object, we would have 4 fifths of the simple fraction would be 4 times a fifth of the simple fraction.  Examples follow.

So    7 quarters would be seven times the latter.

We may replace the word of by the times symbol * to get

In general, we may compute a simple fraction of a simple fraction as follows:

or with a times symbol  * instead of the word of, we write.   

In the resulting fraction, the the numerator (top) is a product of the numerators of the factors and  the denominator (bottom) is a product of the denominator of the factors.  

The foregoing describes the first method for multiplying fractions. After that, we would simplify the resulting fraction by canceling common factors in the products numerator and denominator. The rule here is multiply first and cancel second.  But this order can be changed. Cancellation first leads to smaller numbers and a quicker way (usually) to get the simplified form of the product. 

More examples of products of simple fractions

A)

B)

C) 

Here we are multiplying first and canceling common factors second following the general rule above. But a modification of it is more efficient. See below.

Do you see how the figure 

	3 4
2 3

suggests or confirms
 

4. Equivalent Fractions

Example:

Each quarter is two eighths.. Hence 3 quarters is 6 = 3*2 eighths. Hence 

The foregoing equation may be read forwards or backwards.  The fractions 

3    and   6
4             8

are equivalent. 

A general discussion, an Algebraic Shorthand Description of ideas,  follows.

• Instructors: Give numbers in place of letter below.  
• Learners:  Assume N = 4 and M = 5 and B = 3 on first reading below.

We may  use the property of divisible objects (fractions included)

Thus if we have a fraction  

 B 
M

of an object then 

In shorthand we see

or equivalently 

The latter in turn gives the common factor cancellation property 

The left and right hand side in foregoing equation are said to be equivalent fractions.  Replacing the left hand side by the right hand side in a calculation is called a simplification, a reduction, a cancellation or a lowering of terms.  On the other hand, replacing the right hand side by the left hand side is called raising terms.  Raising of terms is useful in the addition and multiplication of fractions. 

In the computation of fractions, we may also use 

5. Improper Fraction and Equivalent Mixed Numbers

A centimeter is one hundredth of a metre.  So 350 centimeters is 350 one hundredths of a meter. 

Now 

       350 cm  = 3* 100 cm  +50 cm

So 

We may like wise say

The left hand side fraction is equivalent to a mixed number 3+½ or 3½.

Remark 1 - A  hazard:. The notation for mixed numbers is an exception to the rule in algebra that two numbers written side by side indicates a multiplication. Another exception is given by multi-digit decimal notation. 

Remark 2 - A quicker way to simplify: Converting an improper fraction (numerator greater than denominator) to a mixed number gives a whole number part and a proper fraction part. The proper fraction has a smaller numerator. The latter may be easier to factor than the original numerator in the improper fraction.  So converting to a mixed number may speed the simplification process (lowering terms) for an improper fraction.

6. Comparison of Fractions

Justification of a cross-multiplication rule

Example 1. The question which is greater

is often answer by comparing 5*4 = 20 with 6*3 =18. Let use look at this in more detail.  The least common denominator is 12. The following diagram show both fractions in terms of twenty-fourths: Here 24 = 2* 12

Here by putting both fractions over the common denominator 4*6= 24, we see that 

Therefore 

By putting both fractions over the common denominator, the original comparison can be decided by comparing the over 24 = 4*6  numerators 

(i)   20 = 5*4 = (first numerator)*(second denominator) 

with

 (ii) 18 =6*3 = (first denominator)*(second denominator). These 

These over 6*4 = 24 numerators indicate how many (6*4)ths there are in the original fractions. 

Example 2:  The question which is greater

This can be answered by seeing how  (13*17)ths there are in each fraction. We see that

So the first fraction is greater. It provides more (13*17)ths than the second.

For those of you who insist on knowing, 13*17 =221, a number whose existence we need, but whose value is not required.

Algebraic Shorthand Description of Ideas- the General Case: The key to comparing fractions is to put them over a common denominator.

Read the following with literals  (a,b,c,d) = (8, 5, 7, 11) in the first instance. Then let the literals or letters a, b, c and d be any whole number you like.

To compare 

we put them over a common denominator b*d.   (Here we assume b and d are non-negative - why?)

Now we need to compare numerators a*d and b*c.

There are three possibilities:

The foregoing explains and justifies the so called cross-multiplication rule for the comparison of fractions with non-negative denominators. The product of the denominators b*d gives a common denominator, most likely not  the least common, but it will do. The use of the least common denominator is optional in the case of comparison.

Instructors: Students may also compare mixed numbers. The comparison there begins with comparison of the whole number parts. If those parts are equal, comparison then proceeds with the (proper)  fraction parts. The cross-multiplication or common denominator method then applies.  Note: there is no need to convert the mixed number into an improper fraction.

6. Addition (and Subtraction) with like Denominators

Instead of counting how much or how is present in terms of whole units we may count in terms of unit fractions. 

Example 1. Counting thirds

The foregoing says 2 thirds plus 4 thirds (of a unit of measure)  gives six thirds  (of the unit of measure) and the latter is equivalent to 2 (of the unit of measure).

Example 2. Counting Tenths

since 33 = 3*10+3.  The foregoing says 18 tenths plus 15 tenths (of a unit of measure)  gives 33 tenths (of the unit of measure) and the latter can be regrouped in to 3 units (as ten tenths equal 1) plus 3 tenths.

Example 3. Subtracting 12ths.

since 33 = 3*12+3.  The foregoing says 11 twelfths minus 5 twelfths is 6  twelfths (of a unit of measure). The latter gives a half. The following diagram attempts to illustrate the subtraction and the fact that 6 twelfths is a half. 

Algebraic Shorthand Pattern or Rule for Addition and Subtraction with like denominators

The general pattern or rule for addition  in terms of shorthand letters is as follows:

A N

+

B N

=

A+B  N

For addition of fractions with the same denominator,  add numerators and keep the denominator

The general pattern or rule for subtraction  in terms of shorthand letters is as follows:

A N

-

B N

=

A-B  N

For subtract of fractions with the same denominator,  subtract numerators and keep the denominator

7. Addition (and Subtraction) with unlike Denominators

What is 2 thirds plus 3 quarters of something. 

From  

we get 

Physically, we can do and illustrate the addition.

It is physically possible to take a line segment length of 2 thirds a unit and put it besides a line segment of length of 3 quarters a unit. The total length of the two combined segments will be 1 and 5 twelfths as

Remark: Efficient ways to add and multiply remain to be treated. Division of fractions remains too. 

8. Addition with unlike Denominators, efficiency matters

Two fractions may be added together using any common denominator. For example, the use of common denominator 12 = 2*6 = 3*4 leads to 

the use of common denominator 24 = 4*6 = 6*4 leads to 

and  use of common denominator 36 = 6*6 = 9*4 leads to 

For all three  choices of common denominators, the least and other, conversion to a like denominator, addition and simplification all lead to the result 4¼ . But the use of smaller common denominators involves smaller numbers in the computation and hence less simplification work in the end.  The use of the least common denominators usually gives the most efficient way to add and subtract fractions with unlike denominators. So try to use the least common denominator. 

There is one exception that comes to mind, that occurs when the product of the original denominators in the  addends (the fractions being added) gives a power of ten, for example 10, 100, 1000, 10000, and so on. In the latter case, divisibility rules for division by 2, 5 and 10 may lead to easier simplification despite the presence of larger numbers. 

9. Efficient ways to Multiply Fractions

In general, we may multiply  fractions  as follows:.   

In the resulting fraction, the the numerator (top) is a product of the numerators of the factors and  the denominator (bottom) is a product of the denominator of the factors.  The foregoing describes the first method for multiplying fractions. After that, we would simplify the resulting fraction by canceling common factors in the products numerator and denominator. The rule here is multiply first and cancel second.  But this order can be changed.  Cancellation first   leads to smaller numbers and a quicker way (usually) to get the simplified form of the product. 

Example:  

Now instead of compute the products of the numerators and denominators (and then factoring the products to cancel common factors), we take advantage of the situation that the original numerators and denominators provide factors of the product numerators, and factor further to locate common factors that will cancel. Cancelled factors are crossed-out. 

Here we kept the original numerators and denominators and then factored them in a way that would help simplifcation (lowering terms) in the product fraction. So we cancelled the 25 and 11 after factorization. Then after no further factors could be cancelled, computed the decimal representation of the product numerator and denominator in reduced form.

 Here is the above product computation revisited with in place cancellation - the same calculation with a cosmetic change.

The first way we did the cancellation (that is,  multiplying the fractions together and then factoring to reduce) provides justification for the cancellation of common factors in the original fractions before multiplication is done. 

Algebraic Shorthand Description (rather complicated, can be ignored. None the less, the challenge is to understand what is says or suggests, good luck).

10. Division of Fractions

The following diagram indicates that the fraction ¾ goes into 3½ units, 4 full times with ½ left over. The ½ is two-thirds of ¾.

We see that 

So we put

We say 3½ divided by ¾ is 

We also say 3½  is ¾ is  of 

Algebraic Shorthand Description of Ideas

when  and only when 

Here the reciprocal 

works.  

Check: 

First Example Revisited:  How many times does   ¾ goes into 3½ = (7/2)?

Our conclusion is that division by a fraction is computed by multiplying by its reciprocal. 

Another Examples:

Check:  

The foregoing says (¹³/8) is exactly (²/3)rds of (³⁹/16).

One More Example:

Check: 

Remember: division by a fraction is computed by multiplying by its reciprocal. 

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