Arithmetic Of Fractions

Fractions are parts of a whole. For example, 1 orange is a whole orange. When we cut the orange into two equal parts, we have two fractions of the whole orange. In this case, each part is one-half (1/2) of the orange. In general, to have fractions, we must have a whole that is divided into parts. Each part or combination of parts less than the whole are fractions of the whole.

Fractions are formally expressed as a ratio of two numbers: a numerator and a denominator. The denominator is always equal to the number of parts into which the whole has been divided. For example, we divided the orange into two parts, so, the denominator is 2. The numerartor can be any whole number greater than 0 but less than the denominator. The value of the numerator depends on the number of parts of the whole under consideration. For example, if we divide our orange into 7 parts, then 1/7, 2/7, 3/7, 4/7, 5/7 and 6/7 are fractions of the whole orange.

Modern technology has made the evaluation of fractions and other numerical evaluations very easy. For example, to evaluate 4/7, one need only to divide 4 by 7 using a calculator and presto!, the result is displayed. Nonetheless, it is always beneficial to understand the concepts underlying the evaluations. Fundamentally, the arithmetic of fractions is based on their being parts of a whole.

Addition of two fractions (there are two addition scenarios)

(1) The denominators of the fractions to be added are the same
(2) The denominators of the fractions to be added are not the same.
Consider scenario (1). Suppose that the whole is divided into 4 parts as shown in figure 1a. The sum of two or more fractions all having the same denominator is equal to the sum of their nuemrators divided by their common denominator.
Now Consider scenario (2). In this case the denominators are not the same. Adding a fraction from figure 1a to a fraction from figure 1b is an example of this scenario. Since we know how to add fractions when the denominators are the same, our objective will be to find a way to make the denominators the same without changing the result of the addition. Suppose we want to add the fractions 1/4 and 1/8. We will carry out the addition as follows: 1/4 + 1/8 = (2/2)1/4 + (1/1)1/8 = 2/8 + 1/8 = 3/8. This fraction can not be simplified further because the numerator and denominator do not have a common divisor.
What we have done is multiply each fraction by two different ratios that evaluates to 1 so that we can have both denominators to be the same without changing the result of the original addition. Once the denominators are made the same, the result of the addition is the sum of the numerators divided by the common denominator.

Subtraction of a fraction from another fraction
Subtraction is similar to addition. First the denominators are made the same if they are different, then the numerator of the fraction to be subtracted is subtracted from the other numerator and the result is divded by the common denominator. For example (using figure 1) 1/4 -1/8 = 2/8 - 1/8 = 1/8.

Multiplication of two fractions
The multiplication of two fractions is straight forward. The result of the multiplication is the product of the numerators divided by the product of the denominators. For example, 5/7 x 9/11 = 45/77.

Division of a fraction by a fraction
The result of the division is determined by multiplying the fraction that is the numerator by the reciprocal or inverse of the fraction that is the denominator. For example, (1/7)/(2/7) = 1/7 X 7/2 = 1/2. The same technique used to simplify the addition of two fractions with different denominator is used to simply the division of a fraction by a fraction. The reciprocal of the fraction that is the denominator is divided by itself . This ratio is then used to multiply the original division. For example, (1/7)/(2/7) x (7/2)/(7/2) = 1/7 x 7/2 = 1/2. The two denominators cancel out leaving just the product of the numerators. The ratio of the reciprocals evaluates to 1 so the result of the original division is not changed.

Fractions are often expressed as decimal numbers and as percentages because in many practical situations, they are more meaningful and easier to handle in these formats. A direct way to convert a fraction to its decimal equivalent is to divide the numerator by the denominator and then place a point (".") in front of your result. To covert a fraction to a percentage multiply its decimal equivalent by 100 and place a percent symbol (%) after your result.

Fractions occupy the positions between 0 and 1 and between -1 and 0, on the number line. The rules that apply to negative numbers also apply to negative fractions.
Fractions play a fundamental role in mathematics. They are in essence constituencies of whole numbers.

Peter Oye Sagay

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