Problem 35 - Prealgebra

Question

Find the following sums. (Add.) \(7 \frac{1}{10}+8 \frac{3}{10}+2 \frac{7}{10}\)

Step-by-Step Solution

Answer

The sum is \(18 \frac{1}{10}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Convert each mixed number to an improper fraction. For \(7 \frac{1}{10}\), convert by multiplying the whole number 7 by the denominator 10, and then add the numerator 1: \(7 \times 10 + 1 = 71\). Thus, it becomes \(\frac{71}{10}\).Repeat for the second term \(8 \frac{3}{10}\): \(8 \times 10 + 3 = 83\), which is \(\frac{83}{10}\).Finally, for \(2 \frac{7}{10}\): \(2 \times 10 + 7 = 27\), which is \(\frac{27}{10}\).

Step 2: Add the Improper Fractions

Since all fractions have the same denominator, add the numerators:\[71 + 83 + 27 = 181\]Thus, the sum of the fractions is given by:\[\frac{181}{10}\]

Step 3: Convert the Improper Fraction to a Mixed Number

Divide the numerator by the denominator to convert \(\frac{181}{10}\) back to a mixed number:\[181 \div 10 = 18 \text{ remainder } 1\]So, \(\frac{181}{10}\) converts to the mixed number \(18 \frac{1}{10}\).

Key Concepts

Converting Mixed Numbers to Improper Fractions

When adding mixed numbers, the first step is to convert them into improper fractions. This makes it easier to handle and combine them. A mixed number consists of a whole number and a fraction. To convert it to an improper fraction, follow these steps:

• Multiply the whole number by the fraction's denominator.
• Add the product to the numerator of the fraction.
• The result becomes the new numerator, while the denominator remains the same.

For example, to convert the mixed number \(7 \frac{1}{10}\):

• Multiply \(7 \times 10 = 70\).
• Add the numerator: \(70 + 1 = 71\).
• This gives us the improper fraction \(\frac{71}{10}\).

Repeat this process for other mixed numbers in the problem to get them ready for addition.

Adding Fractions with Like Denominators

After converting mixed numbers to improper fractions, you'll often find that they share the same denominator. This is ideal for addition, as it simplifies the process significantly.

To add fractions with like denominators, simply add the numerators together while keeping the denominator the same. In the given exercise:

• Fractions \(\frac{71}{10}\), \(\frac{83}{10}\), and \(\frac{27}{10}\) all share the denominator 10.
• Add the numerators: \(71 + 83 + 27 = 181\).
• This results in the fraction \(\frac{181}{10}\).

Adding fractions like this keeps the arithmetic straightforward and manageable.

Converting Improper Fractions to Mixed Numbers

Once you have a total as an improper fraction, converting it back to a mixed number gives a clearer, more intuitive result. This conversion follows these steps:

• Divide the numerator by the denominator.
• The quotient gives the whole number part of the mixed number.
• The remainder becomes the new numerator, still using the original denominator.

In our problem, we have the improper fraction \(\frac{181}{10}\):

• Divide: \(181 \div 10 = 18\) with a remainder of 1.
• This results in the mixed number \(18 \frac{1}{10}\).

This process transforms the fraction into a form that is often easier to understand at a glance.