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Magazine Chapter 6: Problem 50
Add and subtract as indicated. $$\frac{7}{8}+\frac{1}{8}-\frac{1}{16}$$
Short Answer
Expert verified
The result is \( \frac{15}{16} \).
Step by step solution
01
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.
02
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 \( \frac{1}{8} = \frac{1 \times 2}{8 \times 2} = \frac{2}{16} \).
03
Add the First Two Fractions
With the common denominator established, add the fractions: \( \frac{14}{16} + \frac{2}{16} = \frac{14+2}{16} = \frac{16}{16} \).
04
Subtract the Third Fraction
Subtract \( \frac{1}{16} \) from the result obtained: \( \frac{16}{16} - \frac{1}{16} = \frac{16-1}{16} = \frac{15}{16} \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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, \(\frac{16}{16} - \frac{1}{16} = \frac{15}{16}\).
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.
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鈥檛 have common factors other than 1, so \(\frac{15}{16}\) is already simplified.
