Problem 50 - Prealgebra

Question

Add and subtract as indicated. $$\frac{7}{8}+\frac{1}{8}-\frac{1}{16}$$

Step-by-Step Solution

Answer

The result is \( \frac{15}{16} \).

Step 1: Identify a Common Denominator

In order to add and subtract fractions, they must have a common denominator. The fractions \( \frac{7}{8} \) and \( \frac{1}{8} \) already have a common denominator. However, \( \frac{1}{16} \) has a different denominator. The least common denominator (LCD) of 8 and 16 is 16.

Step 2: Convert Fractions to the Common Denominator

Convert \( \frac{7}{8} \) and \( \frac{1}{8} \) to equivalent fractions with a denominator of 16. Multiply both the numerator and the denominator of each fraction by 2: \( \frac{7}{8} = \frac{7 \times 2}{8 \times 2} = \frac{14}{16} \) and \( \frac{1}{8} = \frac{1 \times 2}{8 \times 2} = \frac{2}{16} \).

Step 3: Add the First Two Fractions

With the common denominator established, add the fractions: \( \frac{14}{16} + \frac{2}{16} = \frac{14+2}{16} = \frac{16}{16} \).

Step 4: Subtract the Third Fraction

Subtract \( \frac{1}{16} \) from the result obtained: \( \frac{16}{16} - \frac{1}{16} = \frac{16-1}{16} = \frac{15}{16} \).

Step 5: Simplify the Result

The fraction \( \frac{15}{16} \) is already in its simplest form because 15 and 16 have no common factors other than 1.

Key Concepts

Adding Fractions

When adding fractions, it is essential to have a common denominator. This means that the denominators of the fractions you're adding must be the same. Once the denominators are the same, you can simply add the numerators directly.

• Let's say we want to add \(\frac{7}{8}\) and \(\frac{1}{8}\).
• Here, the denominators are already the same, which is 8.
• We then just add the numerators: \(7 + 1 = 8\).
• This gives us \(\frac{8}{8}\), which simplifies to 1.

No need to change the denominators if they are already the same, making the addition quick and easy. Remember, after adding, check if the fraction can be simplified for the final answer.

Subtracting Fractions

Subtracting fractions is similar to adding them. You need a common denominator to easily subtract the numerators. Here's a quick example to clarify:

• Take \(\frac{16}{16}\) and subtract \(\frac{1}{16}\).
• Since the denominators are both 16, we can directly subtract the numerators.
• This means we subtract 1 from 16, which equals 15.
• Thus, \(\frac{16}{16} - \frac{1}{16} = \frac{15}{16}\).

The critical thing to remember is ensuring the denominators match before you perform subtraction. Then, simply subtract the top numbers, ensuring to keep the denominator the same.

Common Denominator

A common denominator is crucial for adding or subtracting fractions. It is the shared multiple among the denominators of a set of fractions. Finding a common denominator involves a few steps:

• Identify the denominators in your fractions, for example, 8 and 16.
• Find the least common multiple, which is the smallest number that both denominators can divide into evenly.
• For 8 and 16, the least common multiple is 16.
• Convert all fractions to equivalent fractions with this common denominator.

For instance, \(\frac{7}{8}\) becomes \(\frac{14}{16}\) by multiplying both its numerator and denominator by 2. This step sets the stage for straightforward addition or subtraction, as all fractions are proportionately adjusted to a shared base.

Simplifying Fractions

Once you've added or subtracted fractions, you might end up with a fraction that isn't in its simplest form. Simplifying a fraction means making it as small as possible while maintaining its value.

• To simplify \(\frac{15}{16}\), you check if 15 and 16 have common factors.
• If they do, divide both the numerator and the denominator by their greatest common factor.
• In this case, 15 and 16 don’t have common factors other than 1, so \(\frac{15}{16}\) is already simplified.

Simplifying helps in reducing fractions to their most concise form, making them easier to understand and work with. Checking if a fraction can be simplified is a good final step after subtracting or adding.