Problem 50 - Beginning and Intermediate Algebra

Question

Add or subtract as indicated. Write the answer in lowers ferms. See Example 7. $$ 5 \frac{2}{9}-3 \frac{1}{6} $$

Step-by-Step Solution

Answer

The result is \(\frac{37}{18}\).

Step 1: Convert Mixed Numbers to Improper Fractions

To begin with, convert each mixed number into an improper fraction. For the first mixed number, \(5 \frac{2}{9}\): - Multiply the whole number 5 by the denominator 9: \(5 \times 9 = 45\).- Add the result to the numerator 2: \(45 + 2 = 47\). Thus, \(5 \frac{2}{9} = \frac{47}{9}\).For the second mixed number, \(3 \frac{1}{6}\): - Multiply the whole number 3 by the denominator 6: \(3 \times 6 = 18\).- Add the result to the numerator 1: \(18 + 1 = 19\). Thus, \(3 \frac{1}{6} = \frac{19}{6}\).

Step 2: Find the Least Common Denominator (LCD)

Next, find the least common denominator of the fractions \(\frac{47}{9}\) and \(\frac{19}{6}\). The denominators are 9 and 6. The multiples of 9 are 9, 18, 27, and so on. The multiples of 6 are 6, 12, 18, and so on. The smallest common multiple is 18. Thus, the least common denominator (LCD) is 18.

Step 3: Convert Fractions to Equivalent Fractions with the LCD

Convert each fraction to an equivalent fraction with 18 as the denominator.For \(\frac{47}{9}\): - Multiply both the numerator and the denominator by 2: \(\frac{47 \times 2}{9 \times 2} = \frac{94}{18}\).For \(\frac{19}{6}\): - Multiply both the numerator and the denominator by 3: \(\frac{19 \times 3}{6 \times 3} = \frac{57}{18}\).

Step 4: Subtract the Fractions

Now, subtract \(\frac{57}{18}\) from \(\frac{94}{18}\):\[\frac{94}{18} - \frac{57}{18} = \frac{37}{18}\].

Step 5: Simplify the Fraction (if necessary)

Check if \(\frac{37}{18}\) can be simplified. In this case, the greatest common divisor of 37 and 18 is 1, so the fraction is already in its simplest form.

Key Concepts

Mixed Numbers

Mixed numbers are numbers that consist of a whole number combined with a fraction, like "5 \(\frac{2}{9}\)". They simplify the representation of amounts greater than one, making them easy to read and understand. To work with mixed numbers in mathematical calculations, convert them to improper fractions first. This involves:

• Multiplying the whole number by the fraction's denominator.
• Adding the numerator to this product.
• Writing the result over the original denominator.

This conversion simplifies operations, like addition or subtraction, on fractions.

Improper Fractions

Improper fractions are those where the numerator (top number) is greater than or equal to the denominator (bottom number), such as "\(\frac{47}{9}\)". They are useful for calculations as they remove the complexity of whole numbers, allowing easier mathematical manipulations. When dealing with mixed numbers, converting them to improper fractions is the first important step. This maintains the true value while expressing it in a uniform format, facilitating addition or subtraction.

Least Common Denominator

The least common denominator (LCD) is crucial when working with fractions, especially when they need to be added or subtracted. It’s the smallest number that is a multiple of the denominators in the given fractions. Finding the LCD helps in expressing fractions with different denominators in a common format. This allows straightforward calculations:

• List the multiples of each denominator.
• Identify the smallest common multiple.
• Use this common multiple as the new denominator.

The simplification of fractions into a common denominator is essential for efficient calculation and clarity.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator share no common factors other than 1. It’s a vital skill after performing operations like addition or subtraction. To simplify:

• Determine the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.

For example, if two numbers share no common divisors except 1, they're considered already simplified. This enhances the clarity and neatness of the result, essential for precise mathematical communication.