Problem 77 - Prealgebra

Question

Find each of the following sums and differences. (Add or subtract.) $$4 \frac{3}{10}+2 \frac{1}{100}$$

Step-by-Step Solution

Answer

The sum is \(6 \frac{31}{100}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers to improper fractions. A mixed number like \(4 \frac{3}{10}\) can be expressed as \(\frac{43}{10}\) by multiplying the whole number by the denominator and adding the numerator. Similarly, \(2 \frac{1}{100}\) becomes \(\frac{201}{100}\).

Step 2: Find a Common Denominator

Identify a common denominator for \(\frac{43}{10}\) and \(\frac{201}{100}\). The least common multiple of 10 and 100 is 100. Express both fractions with this common denominator: \(\frac{43}{10} = \frac{430}{100}\) and \(\frac{201}{100}\) remains as \(\frac{201}{100}\).

Step 3: Add the Fractions

Now that both fractions share a common denominator, add them: \(\frac{430}{100} + \frac{201}{100} = \frac{631}{100}\).

Step 4: Convert the Improper Fraction to a Mixed Number

Convert \(\frac{631}{100}\) back to a mixed number. Divide the numerator by the denominator: \(631 \div 100 = 6\) with a remainder of 31. Therefore, the fraction is \(6 \frac{31}{100}\).

Key Concepts

Understanding Improper Fractions

Improper fractions are a type of fraction where the numerator (the top number) is larger than the denominator (the bottom number). For example, when we convert the mixed number \(4 \frac{3}{10}\) into an improper fraction, we get \(\frac{43}{10}\). This means there is more than one whole in the fraction itself. To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and then add the numerator. Thus, for \(4 \frac{3}{10}\), you calculate \(4 \times 10 + 3 = 43\), resulting in \(\frac{43}{10}\). Improper fractions are quite useful in mathematical operations like addition and subtraction because they simplify the process of using a common denominator. Converting all fractions to improper ones before any operation is a great strategy to ensure accuracy and ease.

Finding a Common Denominator

When dealing with fractions, finding a common denominator is essential for addition or subtraction. A common denominator is a shared multiple of the denominators of the fractions you are working with. In our example, we have the fractions \(\frac{43}{10}\) and \(\frac{201}{100}\). To proceed with the addition, we need both fractions to share the same denominator. Here, the least common multiple (LCM) of 10 and 100 is 100.

By converting \(\frac{43}{10}\) to \(\frac{430}{100}\), we align it with \(\frac{201}{100}\). Having the same denominator allows us to add or subtract the numerators directly. Consistently using a common denominator simplifies the process and helps avoid errors during calculations.

Adding Fractions with a Common Denominator

Adding fractions becomes straightforward once the denominators align. Let's illustrate with our example: we have fractions \(\frac{430}{100}\) and \(\frac{201}{100}\). Now that both fractions share the same denominator, we simply add the numerators together:

• \(430 + 201 = 631\)

So, \(\frac{631}{100}\) is your result. Additionally, it’s useful to convert this sum back into a mixed number as fractions are often easier to interpret this way.

To convert \(\frac{631}{100}\) back into a mixed number, divide 631 by 100. This gives us 6 with a remainder of 31, so the mixed number is \(6 \frac{31}{100}\). Thus, expressing fractions in terms of a shared denominator not only aids in adding them but also allows for easy interpretation of the results in different forms.