Problem 104
Question
Add. $$\frac{16}{8}+\frac{1}{8}$$
Step-by-Step Solution
Verified Answer
The sum is \( \frac{17}{8} \).
1Step 1: Understand the Common Denominator
Both fractions \( \frac{16}{8} \) and \( \frac{1}{8} \) have the same denominator, 8. This means you can directly add their numerators.
2Step 2: Add the Numerators
Since the denominators are the same, add the two numerators: \( 16 + 1 = 17 \).
3Step 3: Combine the Result
Keep the denominator as 8 and write the result as a single fraction: \( \frac{17}{8} \).
4Step 4: Simplify (if necessary)
Check if \( \frac{17}{8} \) can be simplified. In this case, it is already in its simplest form as 17 and 8 have no common factors other than 1.
Key Concepts
Understanding Common DenominatorsAdding the NumeratorsSimplifying Fractions
Understanding Common Denominators
When adding fractions, the importance of a common denominator cannot be overstated. The denominator is the bottom number in a fraction. It tells you the total number of equal parts the whole is divided into. To add fractions directly, they must have the same denominator. This is known as having a common denominator. Only when the denominators are the same, can you directly add or subtract the numerators, which are the top numbers of the fractions. For example, in the problem \( \frac{16}{8} + \frac{1}{8} \), both fractions share the common denominator of 8. This allows the numerators to be added directly without any further adjustments to the fractions' structure.
Adding the Numerators
The numerator of a fraction represents the number of equal parts being considered. When fractions to be added have the same denominator, the key operation involves simply adding the numerators together. This process preserves the fraction's common denominator, ensuring that you are still dealing with the same type of parts. In the provided problem, \( \frac{16}{8} + \frac{1}{8} \), the numerators are 16 and 1. By performing the basic addition of these numerators (16 + 1), you arrive at 17. Therefore, the numerators determine the count of these common parts, and by adding them, you effectively combine the number of parts you have from each fraction.
Simplifying Fractions
After arriving at the resulting fraction, it's a good habit to check if the fraction can be simplified. Simplifying means reducing the fraction to its simplest form where the numerator and denominator are as small as possible. This is done by finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by this number.
In the case of \( \frac{17}{8} \), the numerator is 17 and the denominator is 8. Since 17 is a prime number and has no common factors with 8 other than 1, \( \frac{17}{8} \) is already in its simplest form. Sometimes, simplifying may involve converting an improper fraction (where the numerator is larger than the denominator) to a mixed number, but that's not always necessary as in this exercise. Always check for possible simplification to ensure the fraction is as straightforward as possible.
In the case of \( \frac{17}{8} \), the numerator is 17 and the denominator is 8. Since 17 is a prime number and has no common factors with 8 other than 1, \( \frac{17}{8} \) is already in its simplest form. Sometimes, simplifying may involve converting an improper fraction (where the numerator is larger than the denominator) to a mixed number, but that's not always necessary as in this exercise. Always check for possible simplification to ensure the fraction is as straightforward as possible.