Problem 12 - Prealgebra

Question

Evaluate each expression if \(u=7 \frac{1}{7}\) and \(v=6 \frac{3}{7}\). \(v-u\)

Step-by-Step Solution

Answer

\(v - u = \frac{-5}{7}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, let's convert the mixed numbers to improper fractions. For \(u = 7\frac{1}{7}\):1. Multiply the whole number \(7\) by the denominator \(7\): \(7 \times 7 = 49\)2. Add the numerator \(1\) to this result: \(49 + 1 = 50\)3. This means \(u = \frac{50}{7}\).For \(v = 6\frac{3}{7}\):1. Multiply the whole number \(6\) by the denominator \(7\): \(6 \times 7 = 42\)2. Add the numerator \(3\) to this result: \(42 + 3 = 45\)3. This means \(v = \frac{45}{7}\).

Step 2: Subtract Improper Fractions

Now, we'll subtract \(u\) from \(v\):Given that \(v = \frac{45}{7}\) and \(u = \frac{50}{7}\), simply subtract:\[v - u = \frac{45}{7} - \frac{50}{7} = \frac{45 - 50}{7} = \frac{-5}{7}\]

Step 3: Interpret the Result

The result of \(v - u = \frac{-5}{7}\) means that \(v\) is \(\frac{5}{7}\) less than \(u\).

Key Concepts

Mixed Numbers

Mixed numbers are a combination of whole numbers and fractions. They represent values greater than a single fraction. Think of them as whole numbers with extra parts. For example, in our exercise, we have two mixed numbers: \(7\frac{1}{7}\) and \(6\frac{3}{7}\).

• The whole number is the number to the left of the fraction.
• The fraction is made up of a numerator (top number) and a denominator (bottom number).

Mixed numbers provide an easy way to express numbers that are not whole but have a simple fraction remaining. Converting them to improper fractions, which we'll see later, makes it easier to perform operations like addition and subtraction. It's important because improper fractions make numerical operations seamless.

Improper Fractions

Improper fractions have numerators larger than their denominators. This means they represent values greater than one whole. They can appear intimidating at first, but they're quite helpful for mathematical operations.

• To convert a mixed number to an improper fraction, multiply the whole number by the denominator.
• Add the numerator to this product.
• Use this sum as the new numerator, keeping the same denominator.

For example, converting \(7\frac{1}{7}\) involves multiplying \(7 \times 7\) and adding 1, resulting in \(\frac{50}{7}\). Doing the same for \(6\frac{3}{7}\) gives \(\frac{45}{7}\). Using improper fractions streamlines operations, especially subtraction and addition of mixed numbers.

Subtraction of Fractions

Subtracting fractions is straightforward when they share a common denominator. This commonality simplifies the subtraction process, as demonstrated in our problem where both fractions share a denominator of 7.

• Align the numerators above their respective denominators.
• Subtract the second numerator from the first while keeping the denominator the same.

For \(\frac{45}{7} - \frac{50}{7}\), it's just \(45 - 50\), resulting in \(\frac{-5}{7}\). If the result is negative, it simply means the second fraction (or the one being subtracted) is larger. This operation of subtracting fractions becomes intuitive once you grasp the importance of equal denominators and practice handling negative results.