Question

Subtract the following fractions and mixed numbers. Reduce to lowest terms. \(12 \frac{1}{2}-\frac{3}{10}=\) ______

Short answer

\(\frac{61}{5}\) or \(12 \frac{1}{5}\) depending on the preferred form.

Step-by-step solution

Convert Mixed Number to Improper Fraction

First, let's convert the mixed number \(12 \frac{1}{2}\) to an improper fraction. To do this, multiply the whole number part (12) by the denominator of the fractional part (2), then add the numerator (1). This gives us:\[12 \times 2 + 1 = 24 + 1 = 25\]Thus, \(12 \frac{1}{2} = \frac{25}{2}\).

Find a Common Denominator

To subtract the fractions \(\frac{25}{2}\) and \(\frac{3}{10}\), we first need a common denominator. The least common multiple (LCM) of 2 and 10 is 10. We'll convert both fractions to have this common denominator.

Convert Fractions

Convert \(\frac{25}{2}\) to a fraction with a denominator of 10 by multiplying both the numerator and denominator by 5:\[\frac{25}{2} \times \frac{5}{5} = \frac{125}{10}\]The fraction \(\frac{3}{10}\) already has a denominator of 10, so it remains \(\frac{3}{10}\).

Subtract Fractions

Subtract the fractions with the common denominator:\[\frac{125}{10} - \frac{3}{10} = \frac{122}{10}\]

Simplify the Result

Simplify \(\frac{122}{10}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2:\[\frac{122 \div 2}{10 \div 2} = \frac{61}{5}\]

Convert Improper Fraction to Mixed Number (Optional)

Convert \(\frac{61}{5}\) back to a mixed number by dividing the numerator by the denominator. 61 divided by 5 is 12 with a remainder of 1:\[61 \div 5 = 12 \text{ remainder } 1\]This can be written as the mixed number \(12 \frac{1}{5}\).

Key concepts

Mixed Numbers

Mixed numbers are numbers that combine both a whole number and a fraction. For example, in the problem given, we have the mixed number:

• 12 as the whole number part
• \(\frac{1}{2}\) as the fractional part

Mixed numbers can be quite handy when dealing with everyday quantities, like measuring ingredients or distances. However, when performing operations like addition or subtraction, it is often necessary to convert them into improper fractions. This simplifies the calculation.To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fraction.

This gives you the numerator of the improper fraction, while the denominator remains the same as the original fraction.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. They might look a bit confusing at first, but they're very useful in mathematical calculations. For our exercise, converting the mixed number \(12 \frac{1}{2}\) yields the improper fraction \(\frac{25}{2}\). Here's how it works:

• The result from the whole number times the fraction's denominator is added to the fraction's numerator: \(12 \times 2 + 1 = 25\)
• This total becomes the numerator of the improper fraction, over the original denominator.

Improper fractions make operations like addition and subtraction straightforward, especially when finding a common denominator to complete the operation.

Common Denominator

To perform operations such as addition or subtraction on fractions, they must have the same denominator. This is called having a 'common denominator.' For our fractions \(\frac{25}{2}\) and \(\frac{3}{10}\), the least common multiple (LCM) of 2 and 10 is 10.To achieve a common denominator:

• Multiply the numerator and denominator of \(\frac{25}{2}\) by 5, resulting in \(\frac{125}{10}\).
• Keep \(\frac{3}{10}\) as is, since its denominator is already 10.

Having both fractions with a denominator of 10 allows for the straightforward subtraction: \(\frac{125}{10} - \frac{3}{10}\). This step is vital for ensuring that the fractions being worked on are compatible for subtraction.

Simplification

Once the fractions have been subtracted, the result often needs simplification. Simplification involves reducing the resultant fraction to its most basic form. For our final answer, the subtraction yields \(\frac{122}{10}\), which is not yet simplified.To simplify:

• Find the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD of 122 and 10 is 2.
• Divide both the numerator and the denominator by this GCD to get \(\frac{61}{5}\).

This simplification process is crucial to present the final answer in the simplest terms possible. Optionally, this improper fraction can be converted back to a mixed number for better understanding, giving \(12 \frac{1}{5}\). This step returns our fraction to a familiar format, often preferred in practical situations.