Arithmetic with fractions

Arithmetic with fractions

Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zero.

Video: Fraction Arithmetic

Equivalent fractions

Video: How to find equivalent fractions in 30 sec with no calculator

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number , the fraction  . Therefore, multiplying by is equivalent to multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction . When the numerator and denominator are both multiplied by 2, the result is , which has the same value (0.5) as . To picture this visually, imagine cutting a cake into four pieces; two of the pieces together () make up half the cake ().

Dividing the numerator and denominator of a fraction by the same non-zero number will also yield an equivalent fraction. This is called reducing or simplifying the fraction. A simple fraction in which the numerator and denominator are coprime (that is, the only positive integer that goes into both the numerator and denominator evenly is 1) is said to be irreducible, in lowest terms, or in simplest terms. For example, is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1.

Using these rules, we can show that = =   = .

A common fraction can be reduced to lowest terms by dividing both the numerator and denominator by their greatest common divisor. For example, as the greatest common divisor of 63 and 462 is 21, the fraction   can be reduced to lowest terms by dividing the numerator and denominator by 21:

The Euclidean algorithm gives a method for finding the greatest common divisor of any two positive integers.

Comparing fractions

Video: Comparing fractions 

Comparing fractions with the same positive denominator yields the same result as comparing the numerators:

3/4 > 2/4 because 3 > 2, and the equal denominators are positive.

If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions:

because and .

If two positive fractions have the same numerator, then the fraction with the smaller denominator is the larger number. When a whole is divided into equal pieces, if fewer equal pieces are needed to make up the whole, then each piece must be larger. When two positive fractions have the same numerator, they represent the same number of parts, but in the fraction with the smaller denominator, the parts are larger.

One way to compare fractions with different numerators and denominators is to find a common denominator. To compare and , these are converted to and . Then bd is a common denominator and the numerators ad and bc can be compared. This modification of the two fractions is known as "cross multiplying"[, and it is not necessary to determine the value of the common denominator to compare fractions – one can just compare ad and bc, without evaluating bd, e.g., comparing  ? gives  .

For the more laborious question   ? multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding  ? . It is not necessary to calculate  – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is .

Because every negative number, including negative fractions, is less than zero, and every positive number, including positive fractions, is greater than zero, it follows that any negative fraction is less than any positive fraction. This allows, together with the above rules, to compare all possible fractions.

Addition

The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows:

.

If  of a cake is to be added to   of a cake, the pieces need to be converted into comparable quantities, such as ........

 

 

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