Problem 26 - Prealgebra

Question

Find each sum or difference. Write in simplest form. $$\frac{7}{8}-\left(-\frac{3}{16}\right)$$

Step-by-Step Solution

Answer

The sum is \( \frac{17}{16} \) or 1 \( \frac{1}{16} \).

Step 1: Identify and Simplify the Expression

The given expression is \( \frac{7}{8} - \left(-\frac{3}{16}\right) \). Since subtracting a negative is equivalent to adding, we can rewrite it as \( \frac{7}{8} + \frac{3}{16} \).

Step 2: Find a Common Denominator

The denominators in the fractions are 8 and 16. The least common denominator of these is 16. To have the same denominator, convert \( \frac{7}{8} \) to \( \frac{14}{16} \).

Step 3: Add the Fractions

Now, add the fractions \( \frac{14}{16} + \frac{3}{16} \). Since they have the same denominator, you can add the numerators: \( 14 + 3 = 17 \). So, \( \frac{14}{16} + \frac{3}{16} = \frac{17}{16} \).

Step 4: Simplify the Result

The fraction \( \frac{17}{16} \) is already in simplest form as an improper fraction. It can also be expressed as a mixed number: 1 \( \frac{1}{16} \).

Key Concepts

Addition of Fractions

Adding fractions can be tricky at first, but it's easy once you get the hang of it. Essentially, when you add fractions, you are summing up their values. It's important to make sure the fractions have a common denominator before you add them. This way, you can simply add the numerators and keep the denominator the same.

In our example, we started with \( \frac{7}{8} - \left(-\frac{3}{16}\right) \). Subtracting a negative is equivalent to adding a positive. So, we rewrote the expression as addition: \( \frac{7}{8} + \frac{3}{16} \). By aligning these fractions to have the same base (denominator), we could proceed to add the numerators directly once the denominators matched.

Simplifying Fractions

Simplifying fractions makes them easier to work with and understand. A fraction is simplified when the numerator and the denominator have no common factors other than 1. This means you can't divide them both by the same number to make them smaller.

In the exercise, our result was \( \frac{17}{16} \), which is already in its simplest form as the numbers 17 and 16 don't share any common divisors other than 1. This fraction can also be immediately recognized as an improper fraction, where the numerator is larger than the denominator. In such cases, it can be helpful to convert it to a mixed number for clarity, resulting in the mixed number 1 \( \frac{1}{16} \).

Common Denominator

To add fractions, they must have a common denominator. This means both fractions are expressed over the same base in the fraction part. Finding the common denominator involves finding a number that is a multiple of both denominators involved.

In our example, the denominators were 8 and 16. We identified 16 as the least common denominator. This is because 16 is the smallest number that both 8 and 16 can divide into. To align the fractions with this new denominator, we converted \( \frac{7}{8} \) to \( \frac{14}{16} \). Now, both fractions were compatible for addition. Using a common denominator transforms the fractions into the same 'language,' which makes them ready for simple addition or subtraction.