Basic Operations On Fractions

a) Equivalent Fractions

Equivalent fractions are fractions that represent the same value despite having different numerators and denominators.

A fraction remains equivalent if the numerator and the denominator are multiplied or divided by the same number.  

Example:

Consider the fractions \( \frac{2}{4} \) and \( \frac{3}{6} \). These fractions may appear different, but they represent the same portion of a whole.

Determining Equivalent Fractions

To determine if two fractions are equivalent, we check if they represent the same value. This can be achieved by cross-multiplying and comparing the products.

For \( \frac{2}{4} \) and \( \frac{3}{6} \):

\( 2 \times 6 = 12 \) and \( 3 \times 4 = 12 \)

Since the products are equal, the fractions \( \frac{2}{4} \) and \( \frac{3}{6} \) are equivalent.

b) Simplification of Fractions

A fraction is written in its simplified form if the numerator and the denominator have no common factor. In other words, it is impossible to find a number that is a divisor to both the numerator and the denominator in a fraction’s simplified form.  

Example:

Let's simplify the fraction \( \frac{12}{24} \).

Step 1: Determine the Greatest Common Divisor (GCD)

To simplify a fraction, we first identify the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides evenly into both numbers.

Factors of 12: \(1, 2, 3, 4, 6, 12\) 

Factors of 24: \(1, 2, 3, 4, 6, 8, 12, 24\)

The common factors of 12 and 24 are \(1, 2, 3, 4\), and \(6\). The greatest among these is \(12\).

Step 2: Divide Numerator and Denominator by the GCD

We divide both the numerator and the denominator of the fraction by the GCD (\(12\)) to simplify the fraction.

\[\frac{12}{24} = \frac{12 \div 12}{24 \div 12} = \frac{1}{2}\]

Therefore, the simplified form of \( \frac{12}{24} \) is \( \frac{1}{2} \).

Techniques for Simplifying Fractions:

1. Greatest Common Divisor (GCD): Identify the largest number that divides evenly into both the numerator and the denominator.

2. Prime Factorization: Express both the numerator and the denominator as products of their prime factors, then cancel out common factors.

3. Division Method: Continuously divide both the numerator and the denominator by their common factors until further simplification is unattainable.

Example

Simplify the following fractions:

\( \frac{18}{36} \)

Solution:

\( \frac{18}{36} \):

   Factors of \(18\): \(1, 2, 3, 6, 9, 18\) 

   Factors of \(36\): \(1, 2, 3, 4, 6, 9, 12, 18, 36\) 

   GCD = \(18\) 

   \[\frac{18}{36} = \frac{18 \div 18}{36 \div 18} = \frac{1}{2}\]

c) Fraction Addition

Addition of fractions is a fundamental operation in mathematics, allowing us to combine fractional quantities or parts of a whole.

Example:

Let's add the fractions \( \frac{3}{5} \) and \( \frac{2}{3} \).

Step 1: Finding a Common Denominator

To add fractions, we need to ensure they have the same denominator. If the fractions already share a common denominator, we proceed to the next step. Otherwise, we find the least common multiple (LCM) of the denominators to obtain a common denominator.

Denominators: 5 and 3

LCM of 5 and 3 = 15

Step 2: Adjusting the Fractions

We rewrite each fraction with the common denominator of 15.

\( \frac{3}{5} \) becomes \( \frac{9}{15} \) by multiplying the numerator and denominator by 3.

\( \frac{2}{3} \) becomes \( \frac{10}{15} \) by multiplying the numerator and denominator by 5.

Step 3: Adding the Adjusted Fractions

Now that both fractions have the same denominator, we add their numerators.

\( \frac{9}{15} + \frac{10}{15} = \frac{9 + 10}{15} = \frac{19}{15} \)

Since the numerator is greater than the denominator, we can express the result as a mixed number:

\( \frac{19}{15} = 1 \frac{4}{15} \)

Therefore, \( \frac{3}{5} + \frac{2}{3} = 1 \frac{4}{15} \).

Techniques for Adding Fractions:

1. Finding a Common Denominator: Ensure that all fractions have the same denominator before adding.

2. Adjusting Fractions: Modify fractions by multiplying both the numerator and denominator to obtain a common denominator.

3. Adding Numerators: Add the numerators of the adjusted fractions while keeping the common denominator unchanged.

Examples

Add the following pairs of fractions:

a) \( \frac{1}{4} + \frac{3}{8} \)

b) \( \frac{2}{3} + \frac{5}{6} \)

Solution:

a) \( \frac{1}{4} + \frac{3}{8} \)

   Common denominator: \(4 \times 8 = 32\)

   Adjusted fractions: \( \frac{8}{32} + \frac{12}{32} \)

   Sum: \( \frac{8 + 12}{32} = \frac{20}{32} \)

   Simplified: \( \frac{20}{32} = \frac{5}{8} \)

b) \( \frac{2}{3} + \frac{5}{6} \)

   Common denominator: \(3 \times 6 = 18\)

   Adjusted fractions: \( \frac{12}{18} + \frac{15}{18} \)

   Sum: \( \frac{12 + 15}{18} = \frac{27}{18} \)

   Simplified: \( \frac{27}{18} = 1 \frac{9}{18} = 1 \frac{1}{2} \)

d) Fraction Subtraction

Subtraction of fractions is a fundamental arithmetic operation that allows us to find the difference between fractional quantities or parts of a whole.

Example:

Let's subtract the fraction \( \frac{5}{6} \) from \( \frac{2}{3} \).

Step 1: Finding a Common Denominator

To subtract fractions, we need to ensure they have the same denominator. If the fractions already share a common denominator, we proceed to the next step. Otherwise, we find the least common multiple (LCM) of the denominators to obtain a common denominator.

Denominators: 3 and 6

LCM of 3 and 6 = 6

Step 2: Adjusting the Fractions

We rewrite each fraction with the common denominator of 6.

\( \frac{2}{3} \) becomes \( \frac{4}{6} \) by multiplying the numerator and denominator by 2.

\( \frac{5}{6} \) remains \( \frac{5}{6} \) as it already has the common denominator of 6.

Step 3: Subtracting the Adjusted Fractions

Now that both fractions have the same denominator, we subtract the numerators.

\( \frac{4}{6} - \frac{5}{6} = \frac{4 - 5}{6} = \frac{-1}{6} \)

Since the numerator is negative, we can express the result as a negative fraction.

Therefore, \( \frac{2}{3} - \frac{5}{6} = -\frac{1}{6} \).

Techniques for Subtracting Fractions:

1. Finding a Common Denominator: Ensure all fractions have the same denominator before subtraction.

2. Adjusting Fractions: Modify fractions by multiplying both the numerator and denominator to obtain a common denominator.

3. Subtracting Numerators: Subtract the numerators of the adjusted fractions while keeping the common denominator unchanged.

Examples

Subtract the following pairs of fractions:

a) \( \frac{3}{4} - \frac{1}{2} \)

b) \( \frac{5}{8} - \frac{3}{8} \)

Solution:

a) \( \frac{3}{4} - \frac{1}{2} \)

   Common denominator: \(4 \times 2 = 8\)

   Adjusted fractions: \( \frac{6}{8} - \frac{4}{8} \)

   Difference: \( \frac{6 - 4}{8} = \frac{2}{8} \)

   Simplified: \( \frac{2}{8} = \frac{1}{4} \)

b) \( \frac{5}{8} - \frac{3}{8} \)

   Common denominator: Already the same (\(8\))

   Difference: \( \frac{5 - 3}{8} = \frac{2}{8} \)

   Simplified: \( \frac{2}{8} = \frac{1}{4} \)

e) Fraction Multiplication

Multiplication of fractions is a fundamental arithmetic operation that allows us to find the product of fractional quantities or parts of a whole.

Example:

Let's multiply the fractions \( \frac{2}{3} \) and \( \frac{4}{5} \).

Step 1: Multiplying Numerators and Denominators

To multiply fractions, we multiply the numerators together and the denominators together.

\( \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} \)

Step 2: Simplifying the Result

We simplify the resulting fraction, if possible, by canceling out common factors between the numerator and the denominator.

Numerator: \(2 \times 4 = 8\)

Denominator: \(3 \times 5 = 15\)

Therefore, \( \frac{2}{3} \times \frac{4}{5} = \frac{8}{15} \).

Techniques for Multiplying Fractions:

1. Multiply Numerators: Multiply the numerators of the fractions together to obtain the numerator of the product.

2. Multiply Denominators: Multiply the denominators of the fractions together to obtain the denominator of the product.

3. Simplify the Result: Simplify the resulting fraction by canceling out common factors between the numerator and the denominator, if possible.

Practice Exercise:

Multiply the following pairs of fractions:

a) \( \frac{1}{4} \times \frac{3}{5} \)

b) \( \frac{2}{3} \times \frac{5}{8} \)

Solution:

a) \( \frac{1}{4} \times \frac{3}{5} \)

   \( \frac{1 \times 3}{4 \times 5} = \frac{3}{20} \)

b) \( \frac{2}{3} \times \frac{5}{8} \)

   \( \frac{2 \times 5}{3 \times 8} = \frac{10}{24} \)

   Simplified: \( \frac{10}{24} = \frac{5}{12} \)

f) Fraction Division

Division of fractions is a fundamental arithmetic operation that allows us to find the quotient of fractional quantities or parts of a whole.

Example:

Let's divide the fraction \( \frac{2}{3} \) by \( \frac{4}{5} \).

Step 1: Invert the Second Fraction

To divide fractions, we invert the second fraction (the divisor) and then multiply the first fraction (the dividend) by the inverted fraction.

\( \frac{2}{3} \div \frac{4}{5} = \frac{2}{3} \times \frac{5}{4} \)

Step 2: Multiply the Fractions

We multiply the numerators together to get the numerator of the quotient and the denominators together to get the denominator of the quotient.

\( \frac{2}{3} \times \frac{5}{4} = \frac{2 \times 5}{3 \times 4} \)

Step 3: Simplify the Result

We simplify the resulting fraction, if possible, by canceling out common factors between the numerator and the denominator.

Numerator: \(2 \times 5 = 10\)

Denominator: \(3 \times 4 = 12\)

Therefore, \( \frac{2}{3} \div \frac{4}{5} = \frac{10}{12} \).

Techniques for Dividing Fractions:

1. Invert the Second Fraction: To divide fractions, invert the second fraction (the divisor) and change the division sign to multiplication.

2. Multiply Numerators: Multiply the numerators of the fractions together to obtain the numerator of the quotient.

3. Multiply Denominators: Multiply the denominators of the fractions together to obtain the denominator of the quotient.

4. Simplify the Result: Simplify the resulting fraction by canceling out common factors between the numerator and the denominator, if possible.

Practice Exercise:

Divide the following pairs of fractions:

a) \( \frac{1}{4} \div \frac{3}{5} \)

b) \( \frac{2}{3} \div \frac{5}{8} \)

c) \( \frac{3}{7} \div \frac{4}{9} \)

Solution:

a) \( \frac{1}{4} \div \frac{3}{5} \)

   \( \frac{1}{4} \div \frac{3}{5} = \frac{1}{4} \times \frac{5}{3} = \frac{5}{12} \)

b) \( \frac{2}{3} \div \frac{5}{8} \)

   \( \frac{2}{3} \div \frac{5}{8} = \frac{2}{3} \times \frac{8}{5} = \frac{16}{15} \)

c) \( \frac{3}{7} \div \frac{4}{9} \)

   \( \frac{3}{7} \div \frac{4}{9} = \frac{3}{7} \times \frac{9}{4} = \frac{27}{28} \)

If you're looking to ace your ATI TEAS test and get accepted into the nursing program of your dreams, try ExamGates today. Tutors who have taken the exam before wrote and prepared the practice questions on ExamGates. Therefore, you have 100% relevant content, vivid images and illustrations, and in-depth explanations for right and wrong answers.