Problem 54 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$8 \frac{9}{10}-1 \frac{1}{6}$$

Step-by-Step Solution

Answer

The simplified difference is \( \frac{116}{15} \).

Step 1: Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers into improper fractions. The mixed number \( 8 \frac{9}{10} \) can be written as: \( 8 + \frac{9}{10} = \frac{80}{10} + \frac{9}{10} = \frac{89}{10} \).For \( 1 \frac{1}{6} \): \( 1 + \frac{1}{6} = \frac{6}{6} + \frac{1}{6} = \frac{7}{6} \).

Step 2: Find a Common Denominator

Identify a common denominator for the fractions \( \frac{89}{10} \) and \( \frac{7}{6} \). The least common multiple of 10 and 6 is 30.

Step 3: Convert to Equivalent Fractions

Convert each fraction to an equivalent fraction with the common denominator of 30.\( \frac{89}{10} = \frac{89 \times 3}{10 \times 3} = \frac{267}{30} \).\( \frac{7}{6} = \frac{7 \times 5}{6 \times 5} = \frac{35}{30} \).

Step 4: Subtract the Fractions

Subtract \( \frac{35}{30} \) from \( \frac{267}{30} \).\( \frac{267}{30} - \frac{35}{30} = \frac{267 - 35}{30} = \frac{232}{30} \).

Step 5: Simplify the Result

Simplify \( \frac{232}{30} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 2.\( \frac{232}{30} = \frac{232 \div 2}{30 \div 2} = \frac{116}{15} \).Since \( \frac{116}{15} \) cannot be simplified further, this is the simplest form.

Key Concepts

Improper Fractions

An improper fraction has a numerator larger than or equal to its denominator. This means the fraction is greater than or equal to one. For example, in the fraction \( \frac{89}{10} \), 89 is greater than 10, making it an improper fraction.

Why Do We Use Improper Fractions?

• They're easier to work with in equations. You can perform operations like addition or subtraction without converting back and forth from mixed numbers.
• They simplify calculations because you are working directly with the numerators and denominators.

When you have mixed numbers, like \( 8 \frac{9}{10} \), converting them to improper fractions is the first step. You multiply the whole number by the denominator and add the numerator. Thus, for \( 8 \frac{9}{10} \): \( 8 \times 10 + 9 = 89 \), which gives \( \frac{89}{10} \). Mixing converting allows you to easily perform arithmetic operations.

Least Common Multiple

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. When dealing with fractions that have different denominators, finding the LCM helps us create equivalent fractions with a common denominator.

How to Find the LCM:

• List the multiples of each number. For example, for 10 and 6: 10, 20, 30, 40, ... and 6, 12, 18, 24, 30, ...
• Identify the smallest multiple that appears in both lists. In this example, it is 30.

Using the LCM in fractions is crucial for adding or subtracting them. Converting the given fractions \( \frac{89}{10} \) and \( \frac{7}{6} \) to equivalent fractions with a denominator of 30 involves multiplying each by a number that will achieve this common denominator, ensuring that no fractional value is lost.

Mixed Numbers

Mixed numbers consist of a whole number and a proper fraction. They are very practical when expressing quantities greater than one in an understandable way.

How to Convert to Improper Fractions

• Multiply the whole number by the denominator of the fraction.
• Add the numerator to the result of the multiplication.
• Place the sum over the original denominator.

This process aids in calculations. For \( 1 \frac{1}{6} \), multiplying 1 (whole number) by 6 gives 6, adding the fraction part, 1, results in 7. Thus, it becomes \( \frac{7}{6} \), allowing us to focus purely on fractions for operations.

Simplest Form

Simplifying a fraction means reducing it to its smallest possible equivalent by dividing both the numerator and the denominator by their greatest common divisor (GCD). This ensures that the fraction is expressed in its most reduced state.

Steps to Simplify a Fraction:

• Determine the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.

For \( \frac{232}{30} \), the GCD is 2. By dividing both terms by 2, you achieve \( \frac{116}{15} \), which cannot be reduced further. It's important in final answers because it presents the most straightforward representation of the value.