Question

Change the following fractions to decimals. Carry division three decimal places as necessary. \(6 \frac{1}{2}\) ______

Short answer

6.5

Step-by-step solution

Convert Mixed Number to Improper Fraction

The given number is a mixed number, which means it has a whole number and a fraction part. First, convert the mixed number \(6 \frac{1}{2}\) to an improper fraction. Multiply the whole number 6 by the denominator 2 and add the numerator 1: \( 6 \times 2 + 1 = 13\). Thus, the improper fraction is \(\frac{13}{2}\).

Set Up Long Division

To convert the fraction \(\frac{13}{2}\) to a decimal, we perform long division. The numerator, 13, acts as the dividend, and the denominator, 2, is the divisor. Set up the division: 13 divided by 2.

Perform Long Division

Divide 13 by 2: 2 goes into 13, 6 times (as 2 \(\times\) 6 = 12). Write 6 above the division line. Subtract 12 from 13 to get a remainder of 1. Bring down a zero to make it 10. Divide 10 by 2, which is 5 (as 5 \(\times\) 2 = 10). Subtract 10 from 10 to get 0 remainder. Thus, \( 13 \div 2 = 6.5 \).

Present the Decimal

The result of the division shows that the decimal representation of \(6 \frac{1}{2}\) is 6.5. Since we carried the division to completion, there are no additional decimal places needed.

Key concepts

Understanding Improper Fractions

Improper fractions are a key concept in mathematics, particularly when dealing with mixed numbers and converting them into decimals. An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, in the conversion process of the mixed number \(6 \frac{1}{2}\), we transformed it into the improper fraction \(\frac{13}{2}\). This is done by multiplying the whole number (6) by the denominator (2), then adding the numerator (1). The result, 13, becomes the new numerator. Hence, \(\frac{13}{2}\) is an improper fraction because 13 (numerator) is greater than 2 (denominator). Improper fractions can be helpful because they allow for easier computation in operations like addition, subtraction, and especially division, which is necessary for converting fractions to decimals.

Mastering Long Division

Long division is a crucial method used to convert fractions into decimals. When dealing with an improper fraction like \(\frac{13}{2}\), the goal is to perform division using long division techniques. Long division involves dividing larger numbers step by step, proving crucial when fractions are involved. To start with long division, use the numerator (13) as the dividend and the denominator (2) as the divisor. Here's a simple breakdown of the steps involved:

• First, determine how many times the divisor (2) fits into the dividend (13). In this case, 2 fits into 13, 6 times because 2 \(\times\) 6 equals 12.
• Subtract 12 from 13 to get a remainder of 1.
• To continue, "bring down" a zero, making it 10, and divide by the divisor (2) again. Now, 2 fits into 10, 5 times as 2 \(\times\) 5 equals 10.
• Subtracting leaves a remainder of 0, signaling the end of the division.

Thus, \(13 \div 2 = 6.5\), resulting in the decimal 6.5. This step-by-step approach makes long division intuitive and manageable, especially when you continue the process until the desired precision is achieved.

Exploring Mixed Numbers

Mixed numbers form an integral part of mathematical concepts, blending whole numbers with fractions seamlessly. A mixed number incorporates a whole number alongside a fractional component. In our exercise, the mixed number was \(6 \frac{1}{2}\) comprising a whole number (6) and a fraction (\(\frac{1}{2}\)).Understanding mixed numbers is essential because they often occur in real-life situations. For instance, measurements such as baking or length often appear as mixed numbers. Transforming them into improper fractions, as shown earlier, aids in clinical mathematical operations like conversions between formats such as fractions to decimals.When dealing with mixed numbers, remember:

• They help communicate quantities in a more relatable format.
• Conversion into improper fractions simplifies arithmetic operations.
• Knowing how to revert the process might help in applications where results need to be expressed back in mixed numbers.

In essence, mixed numbers deliver a straightforward and meaningful representation of mathematical quantities, enhancing accessibility and comprehension.