If we divide one whole unit into several equal portions, then one or more of these equal portions can be represented by a fraction.
A fraction is composed of the following three parts:
For example, [latex] \displaystyle{\frac{3}{8}} [/latex] is a fraction.
The numerator ‘3’ indicates that the fraction represents 3 equal parts of a whole unit and the denominator ‘8’ indicates that the whole unit is divided into 8 equal parts, as shown above. The fraction bar indicates that the numerator ‘3’ is divided by the denominator ‘8’.
[latex] \displaystyle{\frac{3}{8}} [/latex] is read as “three divided by eight”, “three-eighths”, or “three over eight”. All of these indicate that 3 is the numerator, 8 is the denominator, and the fraction represents 3 of 8 pieces of the whole. The numerator and denominator are referred to as the terms of the fraction.
Note: The denominator of a fraction cannot be zero, since a number cannot be divided into zero equal parts.
Any number that can be represented as a fraction is known as a rational number.
For example,
Fractions can be represented on a number line. They are plotted in-between whole numbers.
For example, [latex] \displaystyle{\frac{1}{2}} [/latex] is represented on a number line as:
When one unit is divided into two equal portions, each portion represents one-half of that unit. One of such equal portions is written as [latex] \displaystyle{\frac{1}{2}} [/latex].
Represent the following fractions on a number line:
[latex] \displaystyle{\frac{2}{3}} [/latex]
[latex] \displaystyle{\frac{3}{5}} [/latex]
When one unit is divided into three equal portions, each portion represents one-third of that unit. Two of such equal portions is two-thirds and is written as [latex] \displaystyle{\frac{2}{3}} [/latex].
When one unit is divided into five equal portions, each portion represents one-fifth of that unit. Three of such equal portions is three-fifths and is written as [latex] \displaystyle{\frac{3}{5}} [/latex].
A proper fraction is a fraction in which the numerator is less than the denominator.
For example,
An improper fraction is a fraction in which the numerator is greater than or equal to the denominator; i.e., the value of the entire fraction is more than 1.
For example,
A mixed number consists of both a whole number and a proper fraction, written side-by-side, which implies that the whole number and the proper fraction are added.
For example,
Follow these steps to convert a mixed number to an improper fraction: | Example: Convert [latex] 3{\frac{5}{8}} [/latex] to an improper fraction. |
Step 1: Multiply the whole number by the denominator of the fraction and add this value to the numerator of the fraction. | 3(8) = 24 24 + 5 = 29 |
Step 2: The resulting answer will be the numerator of the improper fraction. | The numerator will be 29. |
Step 3: The denominator of the improper fraction is the same as the denominator of the original fraction in the mixed number. | The denominator will be 8. Therefore, [latex] 3{\frac{5}{8}} [/latex] = [latex] \displaystyle{\frac{29}{8}} [/latex] |
Note: You can perform the above steps in a single line of arithmetic, as follows:
[latex] 3{\frac{5}{8}} [/latex] = [latex] \displaystyle{\frac{3(8) + 5}{8}} [/latex] = [latex] \displaystyle{\frac{24 + 5}{8}} [/latex] = [latex] \displaystyle{\frac{29}{8}} [/latex]
There is a total of 29 pieces, each piece being one-eighth in size.
Follow these steps to convert an improper fraction to a mixed number: | Example: Convert [latex] \displaystyle{\frac{29}{8}} [/latex] to an improper fraction. |
Step 1: Divide the numerator by the denominator. | |
Step 2: The quotient becomes the whole number and the remainder becomes the numerator of the fraction portion of the mixed number. | |
Step 3: The denominator of the fraction portion of the mixed number is the same as the denominator of the original improper fraction. | Therefore, [latex] \displaystyle{\frac{29}{8}} [/latex] = [latex] 3{\frac{5}{8}} [/latex] |
When both the numerator and denominator of a fraction are either multiplied by the same number or divided by the same number, the result is a new fraction known as an equivalent fraction. Equivalent fractions have the same value.
That is, the same part (or portion) of a whole unit can be represented by different fractions.
For example,
[latex] \displaystyle{\frac{1}{2}} [/latex], [latex] \displaystyle{\frac{2}{4}} [/latex], [latex] \displaystyle{\frac{3}{6}} [/latex], [latex] \displaystyle{\frac{4}{8}} [/latex], [latex] \displaystyle{\frac{5}{10}} [/latex], ... are equivalent fractions.
Find two equivalent fractions of [latex] \displaystyle{\frac{2}{5}} [/latex] by raising to higher terms.
(2 portions of 5 equal parts of the whole)
Multiplying both the numerator and denominator by 2,
(4 portions of 10 equal parts of the whole)
Multiplying both the numerator and denominator by 3,
(6 portions of 15 equal parts of the whole)
Therefore, [latex] \displaystyle{\frac{4}{10}} [/latex] and [latex] \displaystyle{\frac{6}{15}} [/latex] are equivalent fractions of [latex] \displaystyle{\frac{2}{5}} [/latex].
Find two equivalent fractions of [latex] \displaystyle{\frac{12}{30}} [/latex] by reducing to lower terms.
(12 portions of 30 equal parts of the whole)
Dividing both the numerator and denominator by 3,
(4 portions of 10 equal parts of the whole)
Further dividing both the numerator and denominator by 2,
(2 portions of 5 equal parts of the whole)
(2 portions of 5 equal parts of the whole)
Or, the lowest terms can be found by dividing both the numerator and denominator of the original fraction, [latex] \displaystyle{\frac{12}{30}} [/latex], by 6,
Therefore, [latex] \displaystyle{\frac{4}{10}} [/latex] and [latex] \displaystyle{\frac{2}{5}} [/latex] are equivalent fractions of [latex] \displaystyle{\frac{12}{30}} [/latex].
Note: The fraction [latex] \displaystyle{\frac{2}{5}} [/latex] cannot be further reduced, as 2 and 5 do not have any common factors.
Therefore, [latex] \displaystyle{\frac{2}{5}} [/latex] is a fraction in its lowest (or simplest) terms. We will learn more about this further in the section.
If the cross products of two fractions are equal, then the two fractions are equivalent fractions, and vice versa (i.e., if the fractions are equivalent, then their cross products are equal).
That is, if [latex] \displaystyle{\frac{a}{b}} [/latex] = [latex] \displaystyle{\frac{c}{d}} [/latex], then a × d = b × c a × d and b × c are known as cross products
For example, [latex] \displaystyle{\frac{3}{5}} [/latex] and [latex] \displaystyle{\frac{12}{20}} [/latex] are equivalent fractions because their cross products, 3 × 20 and 5 × 12, are equal:
3 × 20 = 60
5 × 12 = 60
Classify the pair of fractions as ‘equivalent’ or ‘not equivalent’ by using their cross products.
[latex] \displaystyle{\frac{2}{5}} [/latex] and [latex] \displaystyle{\frac{12}{20}} [/latex]
[latex] \displaystyle{\frac{5}{4}} [/latex] and [latex] \displaystyle{\frac{20}{12}} [/latex]
[latex] \displaystyle{\frac{3}{8}} [/latex] and [latex] \displaystyle{\frac{9}{24}} [/latex]
[latex] \displaystyle{\frac{2}{5}} [/latex] and [latex] \displaystyle{\frac{12}{20}} [/latex]
The cross products are 2 × 30 and 5 × 12.
2 × 30 = 60
5 × 12 = 60
Therefore, the two fractions are equivalent.
The cross products are equal.
[latex] \displaystyle{\frac{5}{4}} [/latex] and [latex] \displaystyle{\frac{20}{12}} [/latex]
The cross products are 5 × 12 and 4 × 20.
5 × 12 = 60
4 × 20 = 80
Therefore, the two fractions are not equivalent.
The cross products are not equal.
[latex] \displaystyle{\frac{3}{8}} [/latex] and [latex] \displaystyle{\frac{9}{24}} [/latex]
The cross products are 3 × 24 and 8 × 9.
3 × 24 = 72
8 × 9 = 72
Therefore, the two fractions are equivalent.
The cross products are equal.
Dividing both the numerator and denominator of a fraction by the same number, which results in an equivalent fraction, is known as reducing or simplifying the fraction.
For example, we saw in Example 2.1-c that [latex] \displaystyle{\frac{4}{10}} [/latex] and [latex] \displaystyle{\frac{2}{5}} [/latex] are reduced fractions of [latex] \displaystyle{\frac{12}{30}} [/latex].
A fraction in which the numerator and denominator have no factors in common (other than 1) is said to be a fraction in its lowest (or simplest) terms.
Any fraction can be fully reduced to its lowest terms by one of the following two methods:
Method 1: Dividing both the numerator and denominator by the greatest common factor (GCF).
Method 2: Writing the numerator and denominator as products of prime factors and reducing by the common prime factors.
Reduce the following fractions to their lowest terms.
[latex] \displaystyle{\frac{40}{45}} [/latex]
[latex] \displaystyle{\frac{54}{24}} [/latex]
[latex] \displaystyle{\frac{40}{45}} [/latex]
Method 1:
Factors of 40 are: 1, 2, 4, 5, 8, 10, 20, and 40
Factors of 45 are: 1, 3, 5, 9, 15, and 45
The GCF is 5.
Dividing the numerator and denominator by the GCF, 5,
[latex] \displaystyle{\frac{40}{45} = \frac{40 ÷ 5}{45 ÷ 5} = \frac{8}{9}} [/latex]
Method 2:
Prime factors of 40 are: 2 × 2 × 2 × 5
Prime factors of 45 are: 3 × 3 × 5
The common prime factor is one 5.
Therefore, [latex] \displaystyle{\frac{40}{45}} [/latex] is equal to [latex] \displaystyle{\frac{8}{9}} [/latex] reduced to its lowest terms.
[latex] \displaystyle{\frac{54}{24}} [/latex]
Method 1:
Factors of 54 are: 1, 2, 3, 6, 9, 18, 27, and 54
Factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24
The GCF is 6.
Dividing the numerator and denominator by the GCF, 6,
[latex] \displaystyle{\frac{54}{24} = \frac{54 ÷ 6}{24 ÷ 6} = \frac{9}{4}} [/latex]
Method 2:
Prime factors of 54 are: 2 × 3 × 3 × 3
Prime factors of 24 are: 2 × 2 × 2 × 3
The common prime factors are one 2 and one 3.
Therefore, [latex] \displaystyle{\frac{54}{24}} [/latex] is equal to [latex] \displaystyle{\frac{9}{4}} [/latex] reduced to its lowest terms.
Two numbers whose product is equal to 1 are known as reciprocals of each other. Every non-zero real number has a reciprocal.
For example,
When the numerator and denominator of a fraction are interchanged, the resulting fraction is the reciprocal of the original fraction.
For example,
The reciprocal of a positive number is always positive and the reciprocal of a negative number is always negative.
Note:
The reciprocal of a number is not the negative of that number.
(The reciprocal of [latex] 3 ≠ -3 [/latex]. The reciprocal of [latex] \displaystyle{3 = \frac{1}{3}} [/latex].)
The reciprocal of a fraction is not an equivalent fraction of that fraction.
(The reciprocal of [latex] \displaystyle{\frac{2}{5} ≠ \frac{4}{10}} [/latex]. The reciprocal of [latex] \displaystyle{\frac{2}{5} = \frac{5}{2}} [/latex].)
Number | 5 | −3 | [latex] \displaystyle{\frac{2}{3}} [/latex] | [latex] \displaystyle{-\frac{3}{8}} [/latex] |
---|---|---|---|---|
Negative of the Number |
−5 | 3 | [latex] \displaystyle{-\frac{2}{3}} [/latex] | [latex] \displaystyle{\frac{3}{8}} [/latex] |
Reciprocal of the Number |
[latex] \displaystyle{\frac{1}{5}} [/latex] | [latex] \displaystyle{-\frac{1}{3}} [/latex] | [latex] \displaystyle{\frac{3}{2}} [/latex] | [latex] \displaystyle{-\frac{8}{3}} [/latex] |
For Problems 1 to 6, classify the fractions as proper fractions, improper fractions, or mixed numbers.
a. [latex] 15\frac{12}{13} [/latex] b. [latex] \displaystyle{\frac{29}{30}} [/latex]
a. [latex] \displaystyle{\frac{16}{35}} [/latex] b. [latex] 3\frac{2}{9} [/latex]
a. [latex] \displaystyle{\frac{21}{22}} [/latex] b. [latex] \displaystyle{\frac{52}{25}} [/latex]
a. [latex] \displaystyle{\frac{19}{16}} [/latex] b. [latex] 9\frac{7}{8} [/latex]
a. [latex] 6\frac{1}{2} [/latex] b. [latex] \displaystyle{\frac{20}{75}} [/latex]
a. [latex] 4\frac{2}{5} [/latex] b. [latex] \displaystyle{\frac{7}{3}} [/latex]
For Problems 7 to 10, convert the mixed numbers to improper fractions.
a. [latex] 3\frac{2}{5} [/latex] b. [latex] 7\frac{5}{8} [/latex]
a. [latex] 2\frac{2}{7} [/latex] b. [latex] 3\frac{1}{8} [/latex]
a. [latex] 4\frac{3}{7} [/latex] b. [latex] 9\frac{5}{6} [/latex]
a. [latex] 5\frac{4}{5} [/latex] b. [latex] 6\frac{3}{4} [/latex]
For Problems 11 to 14, convert the improper fractions to mixed numbers.
a. [latex] \displaystyle{\frac{23}{7}} [/latex] b. [latex] \displaystyle{\frac{34}{3}} [/latex]
a. [latex] \displaystyle{\frac{19}{7}} [/latex] b. [latex] \displaystyle{\frac{45}{8}} [/latex]
a. [latex] \displaystyle{\frac{26}{4}} [/latex] b. [latex] \displaystyle{\frac{29}{5}} [/latex]
a. [latex] \displaystyle{\frac{23}{3}} [/latex] b. [latex] \displaystyle{\frac{31}{6}} [/latex]
For Problems 15 to 22, classify the pair of fractions as ‘equivalent’ or ‘not equivalent’ by first converting the mixed numbers to improper fractions.
[latex] \displaystyle{\frac{47}{8}} [/latex] and [latex] 5\frac{7}{8} [/latex]
[latex] \displaystyle{\frac{44}{5}} [/latex] and [latex] 4\frac{4}{5} [/latex]
[latex] \displaystyle{\frac{41}{4}} [/latex] and [latex] 10\frac{3}{4} [/latex]
[latex] 11\frac{5}{7} [/latex] and [latex] 7\frac{5}{7} [/latex]
[latex] \displaystyle{\frac{17}{8}} [/latex] and [latex] 2\frac{3}{8} [/latex]
[latex] \displaystyle{\frac{54}{7}} [/latex] and [latex] 7\frac{5}{7} [/latex]
[latex] \displaystyle{\frac{45}{11}} [/latex] and [latex] 4\frac{3}{11} [/latex]
[latex] \displaystyle{\frac{37}{9}} [/latex] and [latex] 4\frac{1}{9} [/latex]
For Problems 23 to 30, classify the pair of fractions as ‘equivalent’ or ‘not equivalent’ by first converting the improper fractions to mixed numbers.
[latex] \displaystyle{\frac{43}{6}} [/latex] and [latex] 7\frac{5}{6} [/latex]
[latex] \displaystyle{\frac{15}{4}} [/latex] and [latex] 3\frac{1}{4} [/latex]
[latex] \displaystyle{\frac{45}{7}} [/latex] and [latex] 6\frac{3}{7} [/latex]
[latex] \displaystyle{\frac{18}{5}} [/latex] and [latex] 3\frac{3}{5} [/latex]
[latex] \displaystyle{\frac{34}{8}} [/latex] and [latex] 4\frac{1}{8} [/latex]
[latex] 3\frac{8}{9} [/latex] and [latex] \displaystyle{\frac{35}{9}} [/latex]
[latex] 7\frac{3}{9} [/latex] and [latex] \displaystyle{\frac{67}{9}} [/latex]
[latex] \displaystyle{\frac{41}{12}} [/latex] and [latex] 3\frac{5}{12} [/latex]
For Problems 31 to 36, (i) reduce the fractions to their lowest terms and (ii) write their reciprocals.
a. [latex] \displaystyle{\frac{44}{12}} [/latex] b. [latex] \displaystyle{\frac{42}{70}} [/latex]
a. [latex] \displaystyle{\frac{30}{20}} [/latex] b. [latex] \displaystyle{\frac{48}{84}} [/latex]
a. [latex] \displaystyle{\frac{75}{105}} [/latex] b. [latex] \displaystyle{\frac{144}{48}} [/latex]
a. [latex] \displaystyle{\frac{56}{48}} [/latex] b. [latex] \displaystyle{\frac{84}{21}} [/latex]
a. [latex] \displaystyle{\frac{131}{84}} [/latex] b. [latex] \displaystyle{\frac{54}{126}} [/latex]
a. [latex] \displaystyle{\frac{36}{63}} [/latex] b. [latex] \displaystyle{\frac{60}{96}} [/latex]
For Problems 37 to 44, classify the pair of fractions as ‘equivalent’ or ‘not equivalent’.
[latex] \displaystyle{\frac{6}{10}} [/latex] and [latex] \displaystyle{\frac{9}{15}} [/latex]
[latex] \displaystyle{\frac{6}{12}} [/latex] and [latex] \displaystyle{\frac{15}{30}} [/latex]
[latex] \displaystyle{\frac{12}{18}} [/latex] and [latex] \displaystyle{\frac{18}{27}} [/latex]
[latex] \displaystyle{\frac{8}{10}} [/latex] and [latex] \displaystyle{\frac{15}{12}} [/latex]
[latex] \displaystyle{\frac{35}{15}} [/latex] and [latex] \displaystyle{\frac{28}{12}} [/latex]
[latex] \displaystyle{\frac{15}{12}} [/latex] and [latex] \displaystyle{\frac{36}{45}} [/latex]
[latex] \displaystyle{\frac{16}{24}} [/latex] and [latex] \displaystyle{\frac{25}{30}} [/latex]
[latex] \displaystyle{\frac{20}{25}} [/latex] and [latex] \displaystyle{\frac{24}{30}} [/latex]
For Problems 45 to 50, determine the missing values.
a.[latex] \displaystyle{\frac{42}{36} = \frac{14}{?}} [/latex] b. [latex] \displaystyle{\frac{42}{36} = \frac{?}{30}} [/latex]
a.[latex] \displaystyle{\frac{4}{9} = \frac{?}{27}} [/latex] b. [latex] \displaystyle{\frac{4}{9} = \frac{20}{?}} [/latex]
a.[latex] \displaystyle{\frac{45}{75} = \frac{?}{25}} [/latex] b. [latex] \displaystyle{\frac{45}{75} = \frac{18}{?}} [/latex]
a.[latex] \displaystyle{\frac{9}{12} = \frac{18}{?}} [/latex] b. [latex] \displaystyle{\frac{9}{12} = \frac{?}{4}} [/latex]
a.[latex] \displaystyle{\frac{25}{15} = \frac{?}{3}} [/latex] b. [latex] \displaystyle{\frac{25}{15} = \frac{35}{?}} [/latex]
a.[latex] \displaystyle{\frac{3}{2} = \frac{12}{?}} [/latex] b. [latex] \displaystyle{\frac{3}{2} = \frac{?}{12}} [/latex]
For Problems 51 to 58, express the answer as a fraction reduced to its lowest terms.
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