Problem 64 - Prealgebra

Question

There are two ways to work the problems below. You can combine the fractions inside the parentheses first and then multiply, or you can apply the distributive property first, then add. Find the sum of \(6, \frac{6}{11},\) and 11

Step-by-Step Solution

Answer

The sum is 17 \(\frac{6}{11}\).

Step 1: Identify the Problem Components

First, identify the numbers we are working with: 6, \(\frac{6}{11}\), and 11. We aim to find their sum by considering these components together.

Step 2: Convert Whole Numbers to Fractions

Express the whole numbers 6 and 11 as fractions with the same denominator as \(\frac{6}{11}\). This means rewriting 6 as \(\frac{66}{11}\) and 11 as \(\frac{121}{11}\).

Step 3: Add the Fractions

Now add the fractions: \(\frac{66}{11} + \frac{6}{11} + \frac{121}{11}\). Combine the numerators to get a new fraction: \(\frac{66 + 6 + 121}{11}\).

Step 4: Simplify the Equation

Calculate the sum of the numerators: \(66 + 6 + 121 = 193\). Therefore, the sum is \(\frac{193}{11}\).

Step 5: Convert the Fraction to a Mixed Number

Divide 193 by 11 to convert the improper fraction \(\frac{193}{11}\) to a mixed number. Dividing gives 17 with a remainder of 6, so the mixed number is 17 \(\frac{6}{11}\).

Key Concepts

Adding Fractions

When adding fractions, it is important to work with a consistent denominator - this is called the common denominator. Before you can add the fractions together, you need to ensure that the fractions involved share the same denominator.

• For example, when adding a whole number such as 6 to a fraction like \( \frac{6}{11} \), convert the whole number into a fraction with the same denominator. This changes 6 into \( \frac{66}{11} \).
• Performing this conversion allows you to easily add the fractions: \( \frac{66}{11} + \frac{6}{11} + \frac{121}{11} \).
• Once the denominators are the same, add the numerators together: \( 66 + 6 + 121 = 193 \).

Now, your sum is \( \frac{193}{11} \), which is an improper fraction. We'll turn our attention to two important fraction forms next that will help with this.

Improper Fractions

An improper fraction occurs when the numerator (the top number) is greater than or equal to the denominator (the bottom number). In our example, we have the improper fraction \( \frac{193}{11} \) because 193 is greater than 11.

• This type of fraction is called "improper" simply because it has a value greater than one whole unit.
• When you encounter an improper fraction, it's often helpful to convert it into a mixed number as it is easier to understand and work with.
• To convert an improper fraction to a mixed number, divide the numerator by the denominator.

In this exercise, 193 divided by 11 equals 17 with a remainder of 6, which leads us to the next concept: mixed numbers.

Mixed Numbers

Mixed numbers combine a whole number and a proper fraction, providing a clear way to express quantities larger than one. In our example, \( \frac{193}{11} \) is converted into a mixed number.

• To convert, divide the numerator (193) by the denominator (11). The quotient (17) is the whole number part of the mixed number.
• The remainder (6) over the original denominator (11) forms the fractional part: \( \frac{6}{11} \).
• Therefore, \( \frac{193}{11} \) becomes the mixed number 17 \( \frac{6}{11} \).

Mixed numbers are intuitive as they show how many whole parts exist alongside any remaining fraction part that fits within the whole unit. Understanding how to convert between improper fractions and mixed numbers is an essential skill in dealing with fractions efficiently.