Problem 12 - Elementary Algebra Within Reach

Question

Write the ratio as a fraction in simplest form. \(1 \frac{2}{7}-\frac{1}{2}\)

Step-by-Step Solution

Answer

\(\frac{11}{14}\)

Step 1: Convert the Mixed Number to an Improper Fraction

To convert a mixed number to an improper fraction, multiply the whole number (1) by the denominator of the fractional part (7) and add the numerator of the fractional part (2). This becomes the numerator of the improper fraction, while the denominator remains the same. So, \(1 \frac{2}{7}\) converts to \(\frac{7*1 + 2}{7} = \frac{9}{7}\).

Step 2: Find a Common Denominator

To subtract fractions, they must have the same denominator. Since the denominators 7 and 2 aren't the same, find the smallest number that's a multiple of both, which is 14. Multiply both the numerator and denominator of \(\frac{9}{7}\) by 2, and \(\frac{1}{2}\) by 7, to convert them into fractions that have the same denominator: \(\frac{9*2}{7*2} - \frac{1*7}{2*7} = \frac{18}{14} - \frac{7}{14}\).

Step 3: Subtract the Fractions

Now that the denominators are the same, you can subtract the numerators: \(\frac{18 - 7}{14} = \frac{11}{14}\).

Step 4: Simplify the Result

Since the numerator 11 and the denominator 14 don't have any common factors other than 1, \(\frac{11}{14}\) is already in simplest form.

Key Concepts

Converting Mixed Numbers to Improper Fractions

When dealing with mixed numbers, it is often necessary to convert them to improper fractions for performing operations such as addition, subtraction, multiplication, or division. A mixed number has a whole number and a fraction, like
Step to Convert a Mixed Number to an Improper Fraction:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.
• Place the total over the original denominator to get your improper fraction.

For example, the mixed number

is converted in the following way: Multiply 1 (the whole number) by 7 (the denominator) to get 7, and then add 2 (the numerator), resulting in 9. Place this over the original denominator (7) to get the improper fraction . This is an essential step because it makes further calculations much simpler.

Finding Common Denominators

The process of adding or subtracting fractions requires them to have the same denominator. This is why finding a common denominator is a critical step. The common denominator should be the least common multiple (LCM) of the original denominators that allows for the fractions to be compared or combined.How to Find the Common Denominator:

• List multiples of each denominator.
• Identify the least multiple that is common to both denominators – that's your LCM.
• Convert each fraction to an equivalent fraction with the LCM as the new denominator.

In our example, the fractions and do not have a common denominator. Multiples of 7 are 7, 14, 21, ... and multiples of 2 are 2, 4, 6, 8, ..., with 14 being the smallest common multiple. Therefore, 14 is the common denominator needed to proceed with subtracting these fractions.

Subtracting Fractions

Subtracting fractions becomes straightforward once you have a common denominator. Keep in mind that you only subtract the numerators, the denominators will stay the same.Steps to Subtract Fractions:

• Ensure both fractions have the same denominator. If not, find a common denominator.
• Subtract the numerators of the two fractions.
• Place the result over the common denominator.
• Simplify the fraction if necessary.

For subtraction using our exercise, you would take the numerators 18 and 7 (from the equivalent fractions and with a common denominator of 14) and subtract them to get 11, giving us a final answer of which must then be checked to ensure it’s in its simplest form.

Simplifying Fractions

The last step in the process is often to simplify the fraction, which means reducing it to its simplest form where the numerator and the denominator are as small as possible, but still carry the same value as the original fraction.How to Simplify a Fraction:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.
• If the GCF is 1, the fraction is already in its simplest form.

In our exercise example, the fraction cannot be simplified further because the greatest common factor of 11 and 14 is 1, indicating that 11/14 is already in its simplest form. Understanding how to simplify a fraction is essential as it provides the most reduced version of the fraction, making it easier to understand and interpret in various mathematical contexts.