Problem 70 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{5}{16}+\frac{1}{16}$$

Step-by-Step Solution

Answer

The result of \( \frac{5}{16}+\frac{1}{16} \) is \( \frac{3}{8} \).

Step 1: Analyze Fractions

The fractions that are to be added are \( \frac{5}{16} \) and \( \frac{1}{16} \). Since these fractions have the same denominator, they can be straightforwardly combined.

Step 2: Add the Numerators

The numerators of the fractions are added by keeping the denominator constant to get \( \frac{5+1}{16} \).

Step 3: Simplify the Fraction

Simplify \( \frac{5+1}{16} \) to obtain \( \frac{6}{16} \). This fraction can be simplified further by dividing the numerator and denominator by their greatest common divisor; in this case, it is 2. Doing so leads to \( \frac{3}{8} \).

Key Concepts

Adding Fractions

When adding fractions, the first step is to determine if the fractions have the same denominator. This is often referred to as a common denominator. If the fractions share a common denominator, the process is simple: you keep this denominator and only add the numerators (the numbers above the fraction line). For instance, when adding \( \frac{5}{16} \) and \( \frac{1}{16} \) the denominator is 16 for both, so you would add 5 and 1 to get 6, which yields the combined fraction \( \frac{6}{16} \).

If the fractions don't have the same denominator, you first need to find equivalent fractions that do have a common denominator before you can add them. This might involve finding the least common multiple (LCM) of the denominators, which ensures that no fraction is 'over-converted,' making subsequent reduction easier.

Lowest Terms Reduction

Reducing a fraction to its lowest terms means to simplify the fraction so that the numerator and denominator are as small as possible while still keeping the same value. This can often make the fraction easier to understand or further work with. To accomplish this, you need to divide both the numerator and the denominator by their greatest common divisor (GCD). For example, once you've added \( \frac{5}{16} \) and \( \frac{1}{16} \) to get \( \frac{6}{16} \) as in our exercise, you must then identify the GCD of 6 and 16.

How do you find the GCD? It's the largest number that can divide into both the numerator and the denominator without leaving a remainder. In this case, it's 2. When you divide both the numerator 6 and the denominator 16 by 2, you're left with the fraction \( \frac{3}{8} \), which is in its simplest form. It is essential to reduce fractions to improve their interpretability and make subsequent calculations easier.

Greatest Common Divisor

The greatest common divisor (GCD), also known as the greatest common factor (GCF), is a key concept when dealing with fractions. It is the largest number that evenly divides two or more numbers. When working with fractions, the GCD is used to simplify fractions to their lowest terms. To find the GCD, list the factors of each number, looking for the largest factor that the numbers have in common. For the numbers 6 and 16, we can list their factors:

• Factors of 6: 1, 2, 3, 6
• Factors of 16: 1, 2, 4, 8, 16

The largest factor present in both lists is 2, thus the GCD of 6 and 16 is 2. Using the GCD to reduce fractions ensures that you obtain the simplest form of the fraction, making your mathematical expressions clear and equations easier to solve.

You can also use the Euclidean algorithm to find the GCD of two numbers, which is a more systematic and often faster process than listing out the factors, especially for larger numbers.