Problem 65 - 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. Give the difference of \(\frac{7}{8}\) and \(\frac{1}{4}\)

Step-by-Step Solution

Answer

The difference of \( \frac{7}{8} \) and \( \frac{1}{4} \) is \( \frac{5}{8} \).

Step 1: Understand the Problem

We need to find the difference between two fractions: \( \frac{7}{8} \) and \( \frac{1}{4} \). This means we need to subtract the second fraction from the first.

Step 2: Find a Common Denominator

The fractions have different denominators (8 and 4). The least common denominator for 8 and 4 is 8. We need to convert \( \frac{1}{4} \) to have a denominator of 8.

Step 3: Convert \( \frac{1}{4} \) to \( \frac{2}{8} \)

Multiply both the numerator and the denominator of \( \frac{1}{4} \) by 2 to get \( \frac{2}{8} \). Now both fractions \( \frac{7}{8} \) and \( \frac{2}{8} \) have the same denominator.

Step 4: Subtract the Fractions

Now that both fractions have the same denominator, subtract the numerators: \( 7 - 2 = 5 \). The difference is \( \frac{5}{8} \).

Step 5: Verify the Result

Check the calculation by ensuring the subtraction was done correctly: \( \frac{7}{8} - \frac{2}{8} = \frac{5}{8} \). Both the problem and the operation are correct.

Key Concepts

Common Denominator

When subtracting fractions, it is crucial to ensure that both fractions have the same denominator. This makes the operation straightforward, as you can directly subtract the numerators. The denominator is the bottom part of a fraction, and a common denominator means both fractions have this same part. To make the denominators the same, you may need to adjust one or both fractions. In this problem, we have two fractions: \( \frac{7}{8} \) and \( \frac{1}{4} \). The denominators here are 8 and 4. To subtract them, we need to use the same denominator for both fractions, which requires finding a common one. By doing this, we place both fractions on a common "ground," making our subtraction easy and straightforward.

Least Common Denominator

The least common denominator (LCD), also known as the smallest common multiple of the denominators, is the smallest number that both denominators divide evenly into. It significantly simplifies the subtraction process of fractions. To find the LCD, look at each denominator: 8 and 4. In this exercise, the LCD is 8 since 8 divided by 8 is 1, and 8 divided by 4 is 2. This commonality is why 8 is the smallest number both denominators can divide into without leaving a remainder. Once we identify the LCD, we convert fractions to have this same denominator. Specifically, we converted \( \frac{1}{4} \) to \( \frac{2}{8} \) by multiplying the numerator and the denominator by 2. This step ensures that both fractions share the least common denominator, allowing for straightforward subtraction.

Distributive Property

The distributive property is a fundamental principle in math, describing how multiplication interacts with addition and subtraction. It states that multiplying a number by a sum is the same as doing each multiplication separately and then adding the results: \( a(b + c) = ab + ac \). While not directly used in the primary solution here, understanding this property offers an alternative approach. You could choose to distribute before dealing with separate terms, particularly when working with expressions involving fractions and variables. In the context of this fractional exercise, applying the distributive property means first setting terms of a common factor and then facilitating operations, be they addition or subtraction. For straightforward fraction work, though, simplifying using the common and least common denominators usually saves time.