Fraction (mathematics)

The concepts surrounding fractions date to antiquity. Babylonians handled fundamental arithmetic operations in a manner not unlike that which is employed today. Division was carried out through an easy multiplication of the dividend by the reciprocal of the divisor. Babylonians worked with sexagesimal fractions (the root of our method of timekeeping). According to early Greeks mathematicians, "number" had been the domain of the natural numbers and the unit fractions; yet among the Babylonians it had been the field of all rational fractions. In Greece the word number was used only for the integers. A fraction was not looked upon as a single entity, but as a ratio or relationship between two whole numbers.

Fractions on the Number Line

Fractions are just numbers. Once this is understood, it becomes easier to perform sums with them. A number line has markings made along the length of it, ranging from zero to +ve ∞, in whole number subdivisions (ignoring the left-hand side for now, which reaches from zero to – ve ∞). It can be used to develop and test an understanding of fractions as numbers.

Here, the number line is in black, and the fractions are coloured and have been positioned on the number line. So a fraction represents part of the gap between the stated whole number (for example, 3) and the next one on the number line (4). The denominator (here, 2) defines how the gap between the two whole numbers is subdivided, and the numerator defines how many subdivisions from the stated whole number one needs to go to position the fraction on the number line.

Decomposing Fractions

To decompose a fraction means to take that fraction apart.

For example, 5/9 can be represented as:

Decomposing a fraction is another way of simplifying that fraction making it more straightforward to perform summation or subtraction on that fraction. Instead of having what might seem to be an intractable fraction, decomposition shows a way forward. For example: 5/9 – 2/3 is not obvious but 2/3 decomposes into 4/9 (among other fractions) yielding 5/9 – 4/9, or 1/9.

Understanding Fractions as Division

It is important to understand the relative nature of fractions: If one child gets half of a big cake and another child gets half of a small one, they do not receive the same amount. Also one can share equal amounts of something, even if pieces are cut in different ways. Referring to Section 1 above, a fraction is a number in its own right with a position on a number line. Here the "whole" is one unit on the number line. Even though one half may look like a fixed amount, it is still in fact one half of a whole and the actual amount does not depend on the size of the proportion but on the size of the whole. Finally, the inverse relation between the denominator and the quantity needs to be understood: The more people there are sharing something, the less each one will get. Moreover, fractions can refer to objects, quantities, or shapes, thus extending their complexity.

Simplifying Fractions

Simplifying (also known as reducing) a fraction means making it as simple as possible. For example, 1/3 rather than 3/9. There are two ways of simplifying a fraction, as below.

Method 1. Try to evenly divide both the numerator and denominator of the fraction by 2, 3, 5, 7,... until it is not possible to go any further while resulting numerator and denominator are whole numbers.

Example: Simplify 24/96

Divide by 2......12/48

Divide by 2.......6/24

Divide by 6.......1/4

So 24/96 simplifies to 1/4

Example: Simplify 10/35

2, 3, 4 do not work, but 5 does, so 10/35 simplifies to 2/7.

Method 2. Divide numerator and denominator by the greatest common factor. For example, for the fraction 8/12, the greatest common factor is 4. So 8/12 simplifies to 2/3.

Multiplying Fractions by Fractions

Conceptually, multiplying two fractions involves finding the fraction of another fraction: 1/3 × 1/8 is the same as 1/3 of 1/8. In practice, multiplying fractions is simple. Multiply the numerators to get the new numerator, and multiply the denominators to get the new denominator. So 1/3 × 3/4 = 3/12. This can then be reduced (in this case, to 1/4). When two fractions less than one are multiplied, the product is always less than either fraction. Finally, multiplication (like division) is commutative, so problems can be simplified by changing the order of the fractions.

Converting Mixed Numbers and Improper Fractions

A mixed number is made up of a whole number and a fraction, for example, 4 3/4. An improper fraction has the numerator greater than or equal to the denominator, for example, 15/4). Convert from an improper fraction to a mixed number requires simple division.

First, calculate how many times the denominator can go into the numerator. For example, in the case of the improper fraction 7/5, 5 can go into 7 only once. For larger improper fractions, such as 47/4, divide the numerator by the denominator (47 divided by 4). The quotient (11) serves as the integer of the converted mixed number.

Next, calculate the remainder, the amount left over after dividing. In the first example, after 5 goes into 7 once, there is a remainder of 2. In the second example, after 4 goes into 47 a total of 11 times, there is a remainder of 3.

The new mixed fraction is composed of both a whole number and a fraction. The quotient is the whole number and the remainder is the numerator. The denominator remains the same as it was with the improper fraction. For the first example, the improper fraction 7/5 converts to the mixed number 1 2/5. In the second example, the improper fraction 47/4 converts to the mixed number 11 3/4.

To convert a mixed number (for example, 5 3/4) to an improper fraction, multiply the whole number with the denominator (5 × 4 = 20), then add the remainder (20 + 3 = 23), and make the new number the numerator (23/4).

Complex Fractions

A complex fraction is a fraction in which either the numerator or the denominator or both contain a fraction. For example: 3/(2/7) or (4/9)/(8/18) or (3/4/9/8) / (8/7/4/5). When we find this kind of complexity in a fraction, we have to simplify it to a proper or improper fraction before we do anything else.

Take the first example above: 3/(2/7). This can be rewritten as 3 ÷ 2/7 hence 3/1 ÷ 2/7 hence 3/1 × 7/2 hence 21/2 hence 10 1/2. This can easily be checked using a calculator.

In the second example,

In the third example,

In this case, the original complex fraction simplifies to fraction that is still complex (2/3/10/7). However, it can be further reduced. Write the fraction as:

Bibliography

Brousseau, Guy, Virginia Warfield. Teaching Fractions Through Situations: A Fundamental Experiment. New York: Springer, 2013.

Petit, Marjorie M. A Focus on Fractions: Bringing Research to the Classroom. New York: Routledge, 2010.

Small, Marian.Uncomplicating Fractions to Meet Common Core Standards in Math. New York: Teachers College, 2013.