Problem 62 - Prealgebra

Question

The following problems all involve the concept of borrowing. Subtract in case. \(21 \frac{2}{5}-20 \frac{5}{6}\)

Step-by-Step Solution

Answer

The result is \(\frac{17}{30}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, convert each mixed number into an improper fraction. For the number \(21 \frac{2}{5}\), multiply 21 by 5 and add the numerator 2 to get \( \frac{107}{5}\). For the number \(20 \frac{5}{6}\), multiply 20 by 6 and add the numerator 5 to get \( \frac{125}{6}\).

Step 2: Find a Common Denominator

The denominators are 5 and 6. The least common denominator of 5 and 6 is 30. Convert \(\frac{107}{5}\) to \(\frac{642}{30}\) by multiplying both the numerator and the denominator by 6. Convert \(\frac{125}{6}\) to \(\frac{625}{30}\) by multiplying both the numerator and the denominator by 5.

Step 3: Subtract the Improper Fractions

Now subtract the two improper fractions: \(\frac{642}{30} - \frac{625}{30} = \frac{17}{30}\). The result does not need further simplification as 17 and 30 have no common factors other than 1.

Key Concepts

Improper Fractions

Improper fractions play a crucial role when performing operations with mixed numbers. They are fractions where the numerator is greater than or equal to the denominator. This means that the value of an improper fraction is greater than or equal to 1. For example, when dealing with mixed numbers such as \(21 \frac{2}{5}\), it is often easier to work with improper fractions because it simplifies the subtraction, addition, or other arithmetic operations.

Here's how you convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator.
• Add the numerator to the result from the first step.
• Place this new value over the original denominator.

Using this method, \(21 \frac{2}{5}\) becomes \(\frac{107}{5}\) and \(20 \frac{5}{6}\) becomes \(\frac{125}{6}\). Converting mixed numbers to improper fractions is an effective way to handle them in equations.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. It allows for two fractions to be compared or combined since it puts them under a common base. The least common denominator (LCD) is the smallest multiple that the denominators share. In our example, the denominators are 5 and 6. Their least common multiple is 30.

Here's how to convert to a common denominator:

• Calculate the least common multiple of the denominators.
• Adjust each fraction so that both have this common denominator.
• To adjust, multiply the numerator and denominator by the necessary amount to reach the LCD.

This results in \(\frac{107}{5}\) being converted to \(\frac{642}{30}\) and \(\frac{125}{6}\) to \(\frac{625}{30}\). This step makes subtraction straightforward as it aligns the fractions up neatly for subtraction.

Mixed Numbers Conversion

Converting back from improper fractions to mixed numbers can be necessary to understand the answer in a more intuitive form. While the calculation is often simpler with improper fractions, mixed numbers can make the result more accessible at a glance.

Here's how to perform the conversion:

• Divide the numerator by the denominator.
• The quotient will be your whole number.
• The remainder becomes the new numerator.
• Place the remainder over the original denominator to form the fractional part.

However, in subtraction problems, it's often more efficient to leave a simple fraction result as it is, especially if it doesn't need further simplification like \(\frac{17}{30}\). In this case, no conversion is necessary, and you get a clear, concise answer.